Resonant X-ray Raman scattering

RESONANT X-RAY RAMAN SCATTERING

Faris GEL’MUKHANOV, Hans As GREN Institute of Physics and Measurement Technology, Linko( ping University, S-58183, Linko( ping, Sweden

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 312 (1999) 87—330

Resonant X-ray Raman scattering Faris Gel’mukhanov, Hans As gren Institute of Physics and Measurement Technology, Linko( ping University, S-58183, Linko( ping, Sweden Received September 1998; editor: J. Eichler Contents 1. Introduction 1.1. Plan of presentation 1.2. Scope of content 2. Synopsis of experimental techniques 2.1. Sources of monochromatic and polarized X-ray radiation 2.2. Optics and detection 3. Principles of resonant X-ray Raman scattering 3.1. Radiative and non-radiative X-ray Raman scattering channels 3.2. Role of spectral distribution of the incident X-ray beam 3.3. Fingerprints of resonant X-ray Raman scattering 3.4. Line-shape distortion. Stokes doubling effect 3.5. Moments of the RXS spectral function. 3-level system 4. Duration of resonant X-ray Raman scattering 4.1. Complex duration time 4.2. Decay and dephasing times 4.3. RXS duration as a mean time of scattering 4.4. RXS amplitude and wave packet versus the RXS duration time 4.5. Characteristic times of the wave-packet evolution versus RXS duration time 4.6. Space distribution of wave packets versus RXS duration time 4.7. Core hole-induced relaxation versus RXS duration 4.8. Nuclear dynamics versus RXS duration 5. Polarization features 5.1. Qualitative picture of RXS by randomly oriented samples

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5.2. Quantitative theory of RIXS polarization of free molecules 5.3. Point group symmetry and polarization anisotropy 5.4. Measurements of polarization anisotropy and angular distributions 6. Interference effects in resonant X-ray Raman scattering 6.1. Direct and interference contributions. Coherence between core-excited states 6.2. One-versus two-step models for the polarization anisotropy and the total cross section 6.3. Role of spin—orbit splitting for channel interference and polarization 6.4. Lifetime-vibrational interference 6.5. Integral interference 7. Elastic radiative X-ray scattering 7.1. Scattering amplitude. Thomson scattering 7.2. Subnatural linewidth resolution 7.3. Diatomic molecules 7.4. Profile of REXS spectral bands versus profile of excitation function 7.5. Polarization of elastically scattered Xray radiation 8. Profiles of RXS spectral bands 8.1. Moments of RXS profiles. Many-level systems 8.2. Center of gravity of vibronically broadened RXS resonances 8.3. Vibronically broadened RXS resonances. Collapse effect

0370-1573/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 0 3 - 4

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F. Gel+mukhanov, H. As gren / Physics Reports 312 (1999) 87—330 9. Role of symmetry in radiative X-ray Raman scattering 9.1. General symmetry analysis 9.2. Selection rules and core hole localization 9.3. Parity selection rules for fixed and randomly oriented molecules 9.4. RXS of diatomic molecules in the soft X-ray region 9.5. Symmetry selection involving electronic continuum resonances 9.6. Symmetry analysis and polarization anisotropy 9.7. Experimental observation of parity selection rules for RIXS 10. Breaking and restoration of electronic selection rules 10.1. Breaking of electronic selection rules due to orientational dephasing 10.2. Breaking of selection rules due to vibronic coupling 10.3. Restoration of selection rules through frequency detuning 11. X-ray resonant scattering involving dissociative states 11.1. Decay channels involving continuum and bound states 11.2. Space correlation between absorption and emission 11.3. Franck—Condon amplitudes 11.4. RXS cross sections for dissociative potentials 12. Time-dependent theory of resonant X-ray Raman scattering 12.1. Time-dependent representation of the RXS cross section 12.2. One-step nuclear dynamics 13. Direct versus resonant X-ray Raman scattering 13.1. Resonant photoemission 13.2. Fano problem for electro-vibrational transitions 13.3. Role of interference between direct and resonant photoemission 14. Doppler effects 14.1. Shortly about Doppler effects in X-ray spectra

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14.2. Phase analysis of the scattering amplitude and the Doppler effect 14.3. Anomalous anisotropy of Auger electron and ion yields 14.4. Averaging of the cross section over molecular orientations 14.5. Homonuclear diatomics. Role of channel interference 14.6. Super-narrowing of the atomiclike resonances 15. Screening, relaxation and chemical shifts 15.1. Screening in free molecules 15.2. Screening in extended systems 15.3. Localization, relaxation and correlation in a two-step picture 15.4. Role of chemical shifts in RXS 16. X-ray Raman scattering by crystalline solids and polymers 16.1. Independent-electron model of RXS for solids 16.2. RXS by polymers 16.3. Resonant and excitonic RIXS bands 16.4. Role of electron—phonon interaction 16.5. Zero-phonon line in RIXS 16.6. RIXS in the dipole approximation over molecular size 16.7. RXS for detuned incident radiation 17. X-ray Raman scattering by surface adsorbates 17.1. Theory for polarized RXS from adsorbed molecules 17.2. Dephasing of X-ray Raman scattering by surface adsorbed molecules 17.3. Sample applications: Ethylene and benzene on copper surfaces 18. X-ray absorption spectra measured in the Raman mode 18.1. High-resolution NEXAFS measured in the Raman mode 18.2. Polarization features of NEXAFS measured in the radiative Raman mode 18.3. EXAFS measured in the Raman mode Acknowledgements References Note added in proof

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Abstract An overview is presented of the theory of X-ray Raman scattering. Second-order perturbation theory for the interaction between matter and light is used as a common starting point, and the consequences of this theory are analytically and numerically analyzed for a variety of experimental situations. The review focuses on results from radiative and nonradiative scattering experiments conducted with 2nd and 3rd generation synchrotron radiation sources during the last couple of years, dealing with atomic, molecular, solid state and surface adsorbate targets. After giving a brief synopsis of relevant experimental techniques, some basic theoretical concepts and principles of X-ray Raman scattering are described, followed by a presentation of the various particular aspects associated with the resonant X-ray scattering process. That is: polarization — interference — role of symmetry — symmetry breaking and energy dependence — dissociation and time dependent interpretations — duration time and frequency detuning — formation of band profiles — Doppler effects — screening and chemical shifts — elastic scattering — solid state theory — application to surface adsorbates — absorption in the Raman mode — direct processes versus resonant X-ray scattering — many channel theory. Each aspect is described by a qualitative picture, a mathematical analysis, and by illustrative examples from experiment combined in some cases with results from simulations. Simple systems are chosen to demonstrate the consequences of various aspects of the theory.  1999 Elsevier Science B.V. All rights reserved. PACS: 31.70.Hq; 32.30.Rj; 32.30.Rm; 78.70.Ck; 78.70.En; 82.80.Ej

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1. Introduction X-ray physics constitutes one of a few scientific branches which has remained vital through the 20th century. Not only did it survive the many revolutions in physics during the century, but it has even experienced a renaissance at the very end. The great majority of X-ray experiments have concerned elastic scattering using traditional sources of X-ray radiation, and elastic, “Bragg”, scattering has served as one of the oldest and most useful tools for investigations of the microscopic properties of solids [1,2]. The notion of “X-ray physics” has become synonymous with X-ray diffraction, which, somewhat ironically, is known to the public mostly for its applications in inorganic chemistry and medicine. In contrast to elastic X-ray scattering, which is an intense coherent process if confined to certain angles, radiative inelastic X-ray scattering is very weak beyond the photoabsorption threshold. This is the main reason that X-ray inelastic scattering experiments were practically absent in the era when conventional X-ray tubes with fixed or rotating anodes were used. To penetrate into the resonant region, extremely bright, well-collimated sources are necessary. The prospects to fulfill these conditions increased considerably with the development of accelerator technology, since very-high-energy electrons confined in a storage ring radiate large amounts of energy in a beam which can be made sufficiently narrow and well collimated to serve as a source for resonant inelastic scattering experiments. This polarized synchrotron radiation has a broad energy spectrum extending into the hard X-ray region, and is easily tunable. In the last decade, the so-called 2nd and 3rd generation synchrotron sources have created nothing short of a revolution within X-ray spectroscopy, analogous to the breakthrough with laser sources in the optical region in the 1960s. Being much brighter than conventional rotational anode sources, synchrotron radiation has thus qualitatively changed the traditional X-ray emission and Auger spectroscopies, and at present time both radiative and nonradiative resonant X-ray scattering techniques of super-high resolution have become standard practice in the X-ray community, see Fig. 1. Several important steps in the development of X-ray scattering experiments can be identified, the first one obviously being the discovery in 1895 of X-rays by Wilhelm Conrad Ro¨ntgen [3], and then

Fig. 1. Radiative (a) and non-radiative (b) X-ray Raman scattering.

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followed by diffractional scattering [4] and spectroscopy [5] in the 1910s, first in emission and then in absorption. It was soon realized that X-ray energies were not only characteristic of the element, but also of the chemical environment (X-ray chemical shifts [6]) and could even manifest many-electron effects (X-ray satellites [7]). Successively, the field of non-resonant fluorescence spectroscopy in the hard X-ray region matured and became thoroughly documented by classification systems and databases [8,9]. Much of the scientific content of hard X-ray spectroscopy was focussed on the problem of the chemical shift of core-to-core X-ray emission. Such shifts became well characterized and the underlying mechanism basically understood [10—13]. Relevant for the present review are also two other effects named after well-known characters in physics, namely the Auger effect and the Raman effect. The former refers to the non-radiative decay channel of an inner shell vacancy, which, although discovered quite early in the century, remained largely unexplored in spectroscopy for a long time due to lack of adequate electron analyzers. The Raman effect, on the other hand, was rather quickly established after its discovery [14—17] as a common spectroscopic tool in the infrared and optical regions. The modern theory of X-ray Raman scattering is based on the classical investigations by Kramers and Heisenberg [18] and Weisskopf and Wigner [19,20], but unlike low-energy Raman spectroscopy there was a wide time lag between theory and applications due to the vastly increased technical requirements in the X-ray case: Sparks [21] used monochromatic Cu K -radiation on a different target materials to observe the resonant enhancement of radiative inelastic RXS near an absorption edge; Eisenberger et al. [22] used synchrotron radiation to investigate this phenomenon in the vicinity of the Cu K-edge resonance. Also for the non-radiative (Auger) channel, similar effects were observed [23,24]; Brown et al. [24] are ascribed the credit of having discovered the X-ray Raman effect. The theory of synchrotron radiation was already developed in the 1940s, but its usefulness for spectroscopic research was not realized until the studies by Tomboulian and Hartman in 1956, and by now numerous general as well as detailed descriptions of the theory of synchrotron radiation are available [25—27]. The brightness of synchrotron radiation is of the order 10 larger than that of bremsstrahlung radiation and other low-intensity X-ray radiation sources, like X-ray tubes, and this fact accounts for the step-wise replacement of the old type of equipment in the 1970s. At present, synchrotron radiation seems to constitute the only alternative for anyone interested in resonant X-ray Raman scattering experiments. The high intensity implies that the monochromatization can be driven far, not only so as to exclusively excite single electronic states, but also to produce sub-natural-width resolution, i.e. using widths of the exciting photon beam that are narrower than the lifetime width of the resonant level. The latter fact has become important for the realization of several so-called “Raman” effects. Another important parameter is the polarization of synchrotron radiation. This, being elliptic in general, depends on the orientation of the X-ray beam relative to the storage ring, and can, just as the frequency, be continuously changed. An additional advantage of synchrotron radiation often advocated is its very smooth spectrum as depicted in Fig. 2 [28]. The creation of high power X-ray sources has greatly challenged a concomitant development of dispersive and detection elements, which was a necessary development to achieve balanced spectroscopic setups. The early inclination towards the use of hard X-rays rested not only on the sources but also on other aspects, such as larger penetrability, easier vacuum requirement, higher fluorescence yields, and, perhaps most importantly, that hard X-ray wavelengths matched the

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Fig. 2. Calculated resolved flux of ALS after the exit slit for different diffraction gratings. The resolved flux is computed [28] as the width of the entrance slit is varied to fix the slit-width-limited resolving power at 10 000. The calculations are based on the predicted flux from the undulator, neglecting field errors and using the first, third, or fifth harmonics for the first, second, and third gratings, respectively. Calculations include mirror absorption, aberration losses at the entrance slit, and a diffraction efficiency for square-wave gratings, in first order, with shadowing. In practice, grating aberrations and slope errors prevent this value from ever being achieved.

atomic plane distances of normal crystals, and which therefore could be used as dispersive elements. Radiative X-ray scattering measurements were accordingly for a long time restricted to non-resonant excitation in the hard X-ray region. Although many attempts with soap film crystals with large interplanar spacings were made, by large the resolving power and efficiency of crystal spectrometers turned out to be too poor for studying individual levels in molecular spectra or bands in solids. Such spectra are best obtained in the soft X-ray region, since the penultimate main shell is sufficiently narrow for the necessary resolution. The development took a turn in the 1970s when sufficient resolution was achieved by using techniques with gratings as dispersive elements mounted at grazing incidence [29—31], i.e. with basically the same technique as later employed for monochromators for synchrotron radiation. Being able to use soft X-rays, one obtained spectra which are strongly influenced by the chemical environment in terms of electronic and geometrical structures given as molecular or solid states properties [32—35], rather than by chemical shifts as with hard X-ray spectroscopy. With an inherent resolution of 5000 in the soft X-ray wavelength region it was then possible to monitor such influences both on a molecular orbital and a vibronic level of energy resolution and to study energy bands of samples in the solid state. These techniques later carried over to resonant X-ray emission making studies possible of molecular orbital and vibronic contributions to the X-ray Raman process [36,37]. A concomitant development in the resonant case has also taken place for applications on solids [38] and surface adsorbates [39]. The modern history of X-ray Raman scattering investigations has spanned a comparatively short time period, and has thus tightly followed the developments of synchrotron radiation sources.

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In spite of this short time span of investigations, many new physical phenomena in the field have been unraveled and studied for both the gas and condensed phases. Especially the 1990s has seen a revelation of new phenomena and an accumulation of data. It therefore seems timely to try to overview some fundamental results reached so far and, as much as possible, attempt to put them into a unified presentation. This review can be regarded as an attempt in that direction. 1.1. Plan of presentation After giving a brief synopsis of the experimental techniques (Section 2), we go through some basic principles (Section 3) of X-ray Raman scattering, and present successively the sections describing various aspects of the process. We start by elucidating the important concept of a duration time for the RXS process, thereby hinting some crucial aspects of the time-dependent formulations to follow (Section 4). Polarization, polarization anisotropies and angular distributions are salient features of the spectroscopy, especially in the radiative mode, and are discussed in Section 5. Channel interference is a concept that is very central to almost all features and measurable quantities connected with the RXS process; some special interference features are discussed in Section 6. The properties of elastic scattering, as opposed to inelastic scattering, are treated in Section 7. Moments of spectral bands and dispersion relations are derived in Section 8 and there used to describe the formation of RXS band profiles. Symmetry plays a central role for defining selection rules and for spectral classifications, see Section 9. The symmetry concept is, however, just as important as its break-down: As discussed in Section 10 the breaking of the symmetry selection rules are given by two major effects; orientational dephasing and non-adiabatic, vibronic, coupling. The restoration of the symmetry selection rules by frequency detuning is a most conspicuous feature outlined in that connection. The many intriguing aspects of RXS involving dissociative states and the dual appearance of molecular and fragment contributions to the RXS cross section are described in Section 11, followed by a more complete time-dependent formulation for the RXS cross sections (Section 12). The possibility to manipulate processes at different microscopic time scales by detuning the frequency and thereby shortening the duration time is novel aspect of the spectroscopy in this respect, leading to effects like “symmetry restoration”, “vibrational collapse”, and “control of dissociation”. In future high-resolution experiments we anticipate to observe various phenomena associated with Doppler shifts, as proposed in Section 14. The many common aspects for radiative and non-radiative RXS have been discussed from the outset of the one-step model with the golden rule for the scattering cross section and the Kramers—Heisenberg (KH) expression for the scattering amplitude. A special feature distinguishing the two processes is discussed in Section 13, namely the contribution of direct processes and their interference with the resonant X-ray scattering. The special roles of screening, relaxation and chemical shifts are discussed in Section 15. In Sections 16 and 17 we turn the attention to other aggregates; to solids, polymers and surface adsorbates and discuss the many features of RXS of such species, for example, momentum selection, band mapping, dispersion relations, excitons, electron—hole- and electron—phonon coupling, coherence length of the radiation interaction, and — for partially and fixed order surface adsorbates — polarization anisotropies and orientational dephasings. The last section (Section 18) is devoted to X-ray absorption — near-edge or extended X-ray absorption — in the Raman mode, which shows some special high-resolution features and which potentially can give new information of both bond distances and orientation of samples.

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In each section we have tried to follow a three-fold division; a simple qualitative explanation, a deeper mathematical presentation, and illustrative examples from experiment combined with results from simulations. Simple systems are chosen to illustrate the various aspects of the theory. Since this is basically a theoretical review the emphasis is put on the mathematical formulations. 1.2. Scope of content The review is evidently limited in many respects, concerning wavelength region, samples, processes, theoretical approaches, and in the reference to various experiments. The review focuses on modern X-ray Raman scattering and is most relevant for those experiments carried out with the second and third generations of synchrotron sources during the time period 1993—1998, although obvious and necessary links to earlier work are made. In this context it is relevant to mention older reviews in the field of X-ray experiment and theory but with different emphasis and content, and which are recommended for consultation for a broader understanding of various branches of X-ray spectroscopies. As examples we mention reviews by As berg [40—43], Almbladh and Hedin [44], Cowan [45,46], Crasemann [47], Eberhardt [48], Ha¨ma¨la¨inen et al. [49], Kane [50,51], Kotani [52], Ma rtensson [53], Nordgren et al. [36,37,54], Pratt et al. [55], Svensson and Ausmees [56], and Walker and Specht [2]. Concerning processes, resonant X-ray scattering with both photons and electrons as scattered particles are treated. Apart from the fact that unified descriptions of the two types in terms of single scattering events are available [41], also more simplified phenomenological descriptions can encompass them both simultaneously; the Kramers—Heisenberg amplitude constitutes the main crossing point between them two. Thus, much of the text covers both radiative and non-radiative scattering, with some chapters focusing on one of them; this should be evident by the presentation in each case. Out of several possible acronyms we prefer to use RXS — resonant X-ray scattering or resonant X-ray Raman scattering, and so radiative and non-radiative RXS. RIXS and REXS define inelastic and elastic (radiative) RXS, respectively. For the special case of non-radiative RXS with spectator decay we have occasionally also used RPE — resonant photoemission. Concerning theoretical approaches, we note that modern atomic theory formulates the Auger and X-ray emission effects in terms of multichannel resonance scattering processes [57,40] along the ideas originally outlined by Fano [58], Feshbach [59] and Mies [60]. The extension of this view to molecular [61,62] and solid state systems is feasible although the introduction of additional degrees of freedom due to the nuclear motion certainly makes the theory more complicated, since it is then necessary to handle many electronically bound vibrational or phononic states that simultaneously interact with many open channels. The present review adopts non-relativistic second-order perturbation theory for the interaction between matter and light as the main starting point and reviews the consequences of this ansatz for a variety of different experimental situations and findings. It seems that, so far, this has been a profitable path to proceed for analyzing results that have come across with 2nd and 3rd generation synchrotron sources. Multi-channel scattering theory is briefly reviewed in Section 13 describing interference between direct and resonant photoemission. Electronic structure theory is out of the scope of the review; it enters indirectly only in some of the illustrating examples. Likewise, comments on the merits and limitations of various simulation techniques are avoided due to space limitations.

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Concerning samples, the review focuses on atoms, molecules, surface adsorbates and solids, with some emphasis on the molecular scatterers; for other aggregates the theory is, in our opinion, not sufficiently developed yet to be reviewed. Concerning wavelength region, the review deals mostly with soft X-ray scattering involving transitions between penultimate and outermost mains shells, roughly between 100 and 1000 eV, for which molecular orbital levels and energy bands, and sometimes even fine structures, can be resolved. Some aspects of scattering of harder X-rays beyond 1 keV, involving deeper core levels, are considered, but still “soft” enough for the atomic dipole approximation to be valid. The core excited intermediate states include discrete levels below the edge, like valence-type resonances and Rydberg progressions, as well as resonances above the edge — shape resonances and extended absorption fine structures. Natural circular dichroism in RXS is considered, while static electric and magnetic field-induced effects are not. Although the review is written with the best of intentions it will be biased by our own experience and can therefore not be exhaustive. We apologize in advance for any omissions.

2. Synopsis of experimental techniques The primary requirement to conduct an RXS experiment is to have a narrowband source of tunable X-ray radiation available, to have tunable monochromators and an equipment for measuring RXS cross sections for different scattering angles and for different polarization (or spin) of incident X-ray photons and final particles. It is necessary that the light source has high intensity, and that the spectrometers are equipped for high resolution. 2.1. Sources of monochromatic and polarized X-ray radiation In principle, a synchrotron facility consists of a ring where electrons orbit at speeds close to the speed of light. The electron trajectories are bent in the ring by magnets and the electrons are accelerated by a cavity in one of the straight sections of the ring to compensate for the energy losses due to the emitted synchrotron light. The critical energy (E ) for which the intensity of the  synchrotron radiation drops and the total radiated power (P) from a synchrotron ring are given by E+GeV, , E +eV,"2.218  R+m,

P+kW,"8.85;10\

E+GeV, I+mA, , R+m,

(1)

where E is the kinetic energy of the electrons, R the curve radius — which depends on the strength of the magnetic field — and I is the current in the ring. A high energy, a small radius and a large current is thus required to maintain high power. Fig. 3 shows two different devices used to produce synchrotron radiation. The simplest device is a bending magnet, which bends the electron orbit by a strong magnetic field (directed down in the figure) (MaxI in Lund, Sweden). An important device is the wiggler, which is placed in a straight section of the synchrotron ring and which consists of several strong magnets placed one after another. The magnets have alternating directions to make the electrons wiggle back and forth. By using strong superconducting magnets, the radius of the electron movements can be made small. A multipole wiggler consists of several wigglers placed after

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Fig. 3. Angular divergence of bending magnet and undulator/wiggler radiation.

each other in order to increase intensity (Stanford Synchrotron Radiation Laboratory (SSRL)). The third important device is the undulator (Advanced Light Source (ALS) in Berkeley and MaxII in Lund), which is a multipole wiggler tuned to achieve interference. It is also placed in a straight section of the synchrotron ring. The electrons move between two rows of small, but strong, magnets which have a variable gap. A typical undulator can have 100 periods, each 5 cm broad. The synchrotron light due to all the oscillations of the electrons in the undulator will come out with an interference pattern, having intensity only at the odd nodes. By varying the gap the nodes can be set to different energies. The intensity from an undulator increases in principle with the square of the number of wigglings, and is therefore much higher than from an ordinary bending magnet or a wiggler. Broadband synchrotron radiation (see Fig. 2) is emitted in a narrow but divergent beam and has to be collected, monochromatized and focused to be of any real use for RXS spectroscopy. This is achieved by a set of optical elements — mirrors, slits, gratings etc. — in the beamline. Fig. 4 gives an illustration of a monochromator connected to a storage ring and shows that the mirrors and grating are operated at grazing incidence. In the X-ray range the reflectivity is very low unless the incident angle is small, below the total reflection angle. The refractive index is below 1 at these energies enabling total external reflection conditions. Fig. 4 shows the X-ray source in an insertion device installed in straight sections of the storage ring. The insertion device is an undulator. All components in the storage ring and beamline section are operated in ultra high vacuum to obtain working conditions that do not disturb the operation. After passing the monochromator the X-ray beam is focused to a small spot where the experimental section is placed. At present many synchrotron radiation machines operate in the world, for example the Photon Factory in Tsukuba, facilities at Daresbury and Novosibirsk, the Advanced Light Source (ALS) in Berkeley, APS in Chicago, Max laboratory in Lund, Super-Aco in Orsay, Stanford Synchrotron Radiation Laboratory (SSRL), ESRF in Grenoble, HASYLAB in Hamburg, CHESS in Ithaca, and the recently commissioned Sincrotrone Trieste (ELETRA). Descriptions of these facilities can be found in [63—65] and in many other articles.

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Fig. 4. Schematic illustration of an experimental set-up using monochromatized synchrotron radiation [66,67,36].

2.2. Optics and detection There are several kinds of spectrometers currently in use for X-ray Raman scattering, for radiative as well as for non-radiative detection. They obey some general boundary conditions set by the fluorescence yield (which below 1 keV is typically below 1%), detection efficiencies, low grating reflectivity and small solid angle of acceptance. Grating instruments have to be used to obtain sufficiently high resolution at low energies. These are operated at grazing incidence geometry to achieve reasonable sensitivity. The geometry causes a low solid angle of acceptance which calls for a careful alignment procedure in order to enhance intensity. To be able to perform RXS experiments it is necessary to optimize not only the detection but also the whole set-up consisting of the source, interaction region and detector to increase the total efficiency. That is, to use a high intensity source with a spot that matches the spectrometer acceptance angle and that has a region where as much radiation as possible interacts with the sample in a region observable with the spectrometer. A modern spectrometer used for soft X-ray measurements is shown in Fig. 5 (constructed at Uppsala University and described in [66,67,36]). The instrument is based on three gratings mounted at fixed angles of incidence and a large two-dimensional detector, which can be accurately positioned and oriented tangential to the Rowland circle of the respective grating by means of motorized coordinate tables. The gratings are mounted so as to have different suitable angles matched to the respective wavelength region covered, 50—1000 eV [66,67,36]. Due to the grazing incident geometry it is impossible to reach the Brewster angle and to select a certain linear polarization of the emission. Thus contrary to the hard X-ray region with crystals as analyzers, this type of spectrometer does not allow one to perform polarization measurements of emitted photons. Gas-phase experiments [68,69] are most difficult experiments, and rely on a careful optimization also of the small target. Typical entrance slits have lengths of the order of 1 cm, and are aligned parallel to the synchrotron radiation. To absorb the radiation in an observable region an adequate pressure of the sample gas has to be maintained. Pressures in the order of 0.1—10 Torr are necessary for many gases in order to absorb most of the radiation within that length. In gas-phase

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Fig. 5. An overview of an X-ray emission spectrometer [66,67,36].

experiments [68,69] the sample is included in a restricted volume with transmitting windows. The elastic scattering emission can in many cases be heavily absorbed, making line profile- and intensity determination difficult. The gas-cell is mounted on a special manipulator that enables an accurate alignment with respect to the chamber, which is crucial due to the small size of the entrance window, and with respect to the beam direction with two degrees of rotation. The whole experimental chamber housing the gas-cell arrangement and the spectrometer is rotatable 90° around the axis defined by the incoming synchrotron radiation. This enables angular dependence measurements with respect to the polarization of the radiation. This type of spectrometer has been used as end stations at, for instance, the X1B beamline at the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory [70] and at beamline 7.0 at ALS in Berkeley [28]. A second class of equipment has been developed with the intention to conduct RXS investigations in the hard X-ray region (u'1 keV) [71,46]. A schematic picture of the instrumentation is shown in Fig. 6. X-rays from a bending magnet in the NSLS storage ring are filtered in energy, and their linear polarization is enhanced by a tunable two-crystal monochromator. The scattering from a gas target, confined in a cell, is then observed by a curved single-crystal spectrometer. The secondary-spectrometer crystal disperses the emitted X-rays onto a position-sensitive proportional counter (PSPC). With the sample placed well inside the Rowland circle of the instrument, the PSPC simultaneously collects all photons within a given spectral region. In some measurements the secondary spectrometer can also perform polarization analysis and/or determine the observationangle, b, dependence of the spectra with respect to incident polarization. The polarization

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Fig. 6. Schematic diagram of instrumentation used for inelastic radiative RXS observations in the hard X-ray region [71]. Anisotropy measurements involve b-rotations of the secondary spectrometer.

selectivity of the secondary spectrometer stems from the fact that the X-ray equivalent of an optical Brewster angle is very close to 45°. The use of this angle results in an almost complete suppression of the component of linear polarization having its electric vector parallel to the plane of incidence. Even when the incident angle is several degrees off 45°, sufficient polarization selectivity is obtained to observe the polarization effects that occur at sub-threshold resonances (see below). Equipment of this kind has been used for RXS measurements at the NIST-ANL X-ray beamline, denoted as X-24A [71,46], at NSLS in Brookhaven. Since the final particles of the non-radiative RXS process are electrons, the measurement of this process is obviously different from radiative RXS. In fact, such measurements are carried out with photoelectron spectrometers. A description of an electron analyzer used for the non-radiative experiment can be found, e.g., in [72—74]. The photoelectrons pass through an electrostatic lens in a hemispherical electrostatical analyzer. A four-element electrostatic lens retards/accelerates the electrons before they enter the hemispherical analyzer, which only allows electrons with a certain energy to pass through the detector. Electrons with too high a kinetic energy will hit the outer part of the sphere, whereas electrons with too low a kinetic energy will hit the inside. By scanning the retardation/acceleration potential of the lens and measuring the intensity, the kinetic energy of the electrons can be studied. The electron analyzer collects all electrons; photoelectrons from direct photoionization as well as Auger electrons. The analyzing chamber has rotary seals which makes it possible to rotate it around the incoming light, and to select any desired emission angle with respect to the polarization vector of the incoming radiation. The transmission of the spectrometer decreases rather slowly with increasing kinetic energy, making it possible to record electrons with high initial kinetic energy at reasonable intensity, which is of importance for studies of weak resonant processes. Due to the high degree of linear polarization of the light, direct measurements of the relative differences in cross sections can be obtained from the spectra recorded at the magic angle. A high-resolution setting for studies of non-radiative Raman spectra of first row species is

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typically obtained with a constant analyzer pass energy of 75 eV and with 100 meV spectrometer resolution [72—74,56,75]. A total experimental resolution in “a high-resolution setup” can typically be 150 meV, arising from a convolution of the monochromator, spectrometer and Doppler broadening of +40 meV. Thus, the high-resolution spectra are recorded with vibrational resolution whereas a more moderate resolution (with larger yield) is often sufficient to distinguish between transitions to different electronic levels.

3. Principles of resonant X-ray Raman scattering 3.1. Radiative and non-radiative X-ray Raman scattering channels Resonant X-ray scattering (RXS) occurs due to an interaction of X-ray photons with a target consisting of atoms, molecules or a solid — for convenience we refer here to the target as a “molecule”. The molecule is excited from the ground state "o2 to a core excited state "i2 by absorption of the incoming X-ray photon (c) with frequency u, wave vector p and polarization vector e. The core excited state is metastable due to vacuum zero vibrations or interelectron Coulomb interaction and can therefore decay to final states " f 2 in two different ways. In the first type of decay channel a final X-ray photon (c) with frequency u, wave vector p and a polarization vector e is emitted M#cPM PM #c . (2) G D The energy of the core excited state can also be released by ionization of one of the electrons e\ of the molecule: M#cPM PM #e\ . (3) G D These two decay channels constitute the radiative and the non-radiative X-ray scattering processes, respectively. When the frequency of the incident X-ray photons is tuned below or closely above the core ionization threshold resonant, core excitation takes place. It is natural to refer to this case as to resonant X-ray scattering (RXS) or X-ray Raman scattering. The RXS process thus consists of two steps. In the first step the molecule absorbs an X-ray photon and in the second step it emits a final particle, an X-ray photon or an Auger electron. The processes and the energy relations are sketched in Fig. 1 and described in detail in this review. 3.1.1. Radiative X-ray Raman scattering In non-resonant X-ray emission spectroscopy, the energy of the bombarding particles exceeds the core ionization threshold. In this situation, the X-ray emission spectral profile is practically independent of the excitation energy and the excitation probe (photon, electron or heavy particle). Because of this fact the spectral shape of X-ray emission was for a long time only described as transitions between discrete core ionized and final states using a two-step model with the emission decoupled from the excitation. In contrast, the shapes of resonant X-ray emission spectra are dependent on the way of preparation of the core excited state of the target, and in later experiments using energy tunable photon sources, the X-ray emission spectra were observed to be strongly frequency dependent, when the excitation frequency approached the X-ray photoabsorption

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threshold. It was also realized that the accompanying theoretical descriptions must switch from a two-step to a one-step model with the excitation and the deexcitation treated as one, nonseparable, scattering event. The essential features of soft X-ray radiation scattering can be described by non-relativistic quantum mechanics. The interaction Hamiltonian between the electrons and the plane-wave monochromatic fields of incident e cos(ut!p ) r) and final spontaneously emitted e cos(ut!p ) r) X-ray photons (4)

!(a/2)(P ) A#A ) P)#(a/2)A ) A

can be obtained from the standard prescription PPP!aA with P, A and a"  as the total  electronic momentum, vector potential and the fine-structure constant, respectively (atomic units are used unless otherwise stated). According to standard perturbation theory [76] the differential RXS cross section for scattering into a solid angle do can be written as dp(u,u) u "r "F "D(u#u !u,C ) . (5) Mu D DM D dudo D Here r "aK2.82;10\ cm is the Thomson radius (the classical radius of the electron), M u "E !E is the resonant frequency of transition jPi with E as the total molecular energy, GH G H G C is the final state lifetime broadening, and D D(X,C)"C/n(X#C) (6) is a Lorentzian function with the half-width at half-maximum (HWHM) C. The scattering amplitude F is given by the Kramers—Heisenberg (KH) formula [18] D DR D DR D DG GM F "F2# F , F "u u eιq  RL ! DG GM , D D DGL DGL GD GM u!u #iC u#u GM G GM L G ιqr F2"(e ) e)1 f " e "o2 , (7) D where q"p!p is the change of the X-ray photon momentum due to the scattering, D"(e ) D), D"(e ) D) with D being the total electronic dipole moment of the molecule, and R is L the coordinate of the core excited atom. The sum at the right-hand side of the expression for F2 implies summation over coordinates of all Z electrons. D The KH scattering amplitude consists of three qualitatively different contributions. The time sequences resulting in these three terms can be represented schematically as shown in Fig. 7. The first term in Eq. (7) is the amplitude of the Thomson scattering which in the ultra-soft X-ray region is allowed only for the elastic channel





F2Kd Z(e ) e) . (8) D DM The Thomson scattering (diagram I in Fig. 7) arises from the A ) A term in the interaction Hamiltonian (4). The existence of this term in the KH dispersion formula was first demonstrated by Waller [77]. The other two terms originate in the (p ) A#A ) p) interaction, the first of which is resonant and complies with the ordinary time sequence (absorption before emission, see Fig. 7,II).

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Fig. 7. Diagrams corresponding to each of the three terms in the Kramers—Heisenberg formula (7) of radiative RXS. The time arrow is directed upwards.

The intermediate state of the compound system including the molecule and the photons is essentially virtual and hence the time for incident photon absorption is not known exactly nor is the time when the final photon is emitted. This means that the off-resonant process permuted in the time events of absorption and emission also is possible (last term in Eq. (7), see Fig. 7,III). The energy conservation law plays a fundamental role in the description of the RXS process u"u!u (9) DM as reflected by the argument of the D-function in Eq. (5). This so-called Raman—Stokes shift of the scattered radiation, given by the energy difference u between final and ground states, DM constitutes the Raman effect. The energy conservation law (9) is, however, not implemented strictly because the time—energy uncertainty relation implies that the final-state energy E is known only D with the accuracy up to the inverse lifetime C of the final state. The energy conservation law is thus D strictly fulfilled only for elastic scattering. As a rule, the expression for the scattering cross section is then given by Eq. (9) and Eq. (5) without the D-function [78—81]. Such a representation for the scattering cross section was first obtained by Kramers and Heisenberg [18] using the correspondence principle. Out of several subsequent researchers it seems that only Weisskopf and Wigner [19,20] realized the importance of the form of the cross section given by (5) including the D-function (however, without the Thomson—Waller term). As can be understood from the following description the D-function in Eq. (5) plays a fundamental role in a formation of the RXS profiles, and the realization of this fact has lead to a number of new features in the description of the RXS processes. Eq. (9) explains the fact that the scattered radiation is mainly red shifted (Stokes lines). One can understand that blue shifts (anti-Stokes lines) also are possible in principle, for example, due to thermal population of ground state vibrational levels in extended systems or of low-lying electronic states in solids, although such lines probably are much less important than for the vibrational Raman effect in the infrared (IR) region.

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Two important special cases for elastic scattering can be distinguished dp(u,u) "rd(u!u) u ) const if u;u , M GM dudo dp(u,u) "rd(u!u) "F2" if u
(10)

As first observed by Lord Rayleigh the scattering cross section at long wavelengths varies as the fourth power of the frequency (because of this law the sky is blue and the sunset is red), while in the high-energy limit Thomson scattering by free electrons dominates. In many cases both nonresonant contributions (Fig. 7,I,III) to the scattering amplitude can be neglected in the vicinity of a strong photoabsorption resonance. For example, the ratio of the Thomson—Waller term to the resonant amplitude is of the order of Zr C /(jC);1 (the wavelength of the soft X-rays is M G j&10\!10\ cm). In the general case, see below, one must be careful with this estimation because of the dependency of this ratio on Z and on the branching ratio C /C (C  is here the rate G G G of radiative decay of core excited state i while C is the total, radiative # non-radiative, rate). One G can thus neglect the Thomson—Waller term as well as the last non-resonant term in the cross section for inelastic soft X-ray scattering, Eq. (7). 3.1.2. Non-radiative X-ray Raman scattering Besides spontaneous X-ray photons, Auger electrons can also be produced (3), see Fig. 1. The photoelectron spectrometer can measure the energy e, the direction of propagation and the spin orientation of the outgoing electron. If the incident photon frequency is tuned close to the core ionization threshold, this process is commonly referred to as non-radiative RXS or as the Auger Raman effect. Direct photoionization or photoionization accompanied by excitation of a second electron (Fig. 8) dominates over non-radiative RXS (resonant photoemission) if the incident X-ray photon frequency u is tuned far from the resonant frequency u of the transition from the core GM level to the unoccupied level. The amplitude of this direct channel is given by the dipole matrix element D "1U "e ) D"W 2, E"e#E (11) D# D# M M for a radiative transition from the ground molecular state W to the final continuum state U with M D# the ingoing-wave boundary condition. The resonant scattering channel starts to compete with, or even dominate over, the non-resonant one when u tends to u . There are two qualitatively GM different resonant channels. The first is the so-called spectator channel, when the core excited electron remains in the “same” unoccupied MO (Fig. 8). The final state of the spectator Auger channel can also be reached by a direct “shake-up” photoionization. The core excited electron can be ejected from the molecule due to Auger decay (Fig. 8) — it then participates in the Auger transition — and the corresponding resonant scattering channel is therefore called the participator channel. The final state of this process is the single-hole state that can be populated also by direct one-electron photoionization. When the excitation frequency is close to the excitation energy of the intermediate discrete core excited state i this state is also populated. Due to the strong interaction between this discrete state and the continuum states close by in energy, autoionization channels are opened. The final state

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Fig. 8. Non-resonant and resonant contributions to non-radiative RXS (resonant photoemission or resonant Auger effect).

continuum wave function W now differs from U and will have a resonant feature caused by the D# D# autoionization of the discrete state. This results in the following resonant structure of the non-radiative RXS amplitude:

t GD# . (12) D "D # D# D# u!u #iC GM G G The strength of the resonant contribution t is difficult to calculate and depends on the dipole GD# moment of the oPi transition and the Coulomb interaction between discrete and continuum states. A strict description is given by the theory of Fano [58,82] which is recapitulated briefly in Section 13. According to the Fermi golden rule the cross section of non-radiative RXS is given by dp(e,u) "4nau "D "D(e#E !E !u,C ) , (13) D# D M D de do D where E denotes the energy of the ionized final state of the molecule. The structure of the cross D sections of radiative and non-radiative RXS is essentially the same, see Brown et al. [24] and As berg et al. [12]. Eq. (13) shows directly that the kinetic energy of the electron also follows a Raman—Stokes law (9), e"u!u . (14) DM The width of the spectral profile plays a fundamental role for interpreting radiative and nonradiative RXS. As a rule the lifetime broadening of the final state is considerably smaller than the lifetime broadening of the core excited state C ;C . D G

(15)

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This leads to the possibility of performing X-ray spectroscopy below the natural width of the core excited state when monochromatic excitation is used [83,84,24,85,41,86,42]. The general case of arbitrary spectral distributions of the incident photons will be considered in the following. As mentioned, the radiative and non-radiative RXS cross sections are described by similar equations, and they have many common spectral features. However, a strong qualitative distinction exists between the processes because of the different interactions leading to emission of X-ray photons, and Auger electrons, respectively. In the first case, the interaction is of electromagnetic origin, while in the second case it is a non-radiative Coulomb interelectron interaction. The latter type of interaction is stronger for low-Z elements, leading to a fluorescence yield in the order of K10\ for first row molecules. One of the main, indirect, consequences of this fact is that the overall quality and resolution becomes better for resonant Auger spectra in comparison with radiative RXS spectra. However, the second distinction between the two processes favours the radiative case, namely the selection rules. Contrary to radiative RXS with dipole selection rules for the emission step, symmetry selection in Auger spectroscopy is ineffective because of the Coulomb interaction, and the Auger spectral profile often consists of many overlapping lines which can make the interpretation of Auger spectra difficult. In the radiative case, symmetry selection will act both by strict rules in terms of irreducible representations of the involved states and the operator, and as local effective selection rules. Thus X-ray emission provides a local probe of the valence states, in which the atomic dipole selection rule governing the process provides a means to extract information about the partial atomic density of states, e.g. p density of states for first row species. Interpretations of the two spectroscopies require also different levels of electronic structure theory, e.g. molecular orbital and many-body theory, with non-radiative RXS being the more complex with respect to electron correlation effects and breakdown of the MO picture. 3.2. Role of spectral distribution of the incident X-ray beam In the previous section, the simplest possible interaction with a monochromatic plane X-ray wave was considered. In a real experimental situation, the resolution of the RXS spectra is considerably more complex than that given by Eq. (5). One can distinguish two qualitatively different mechanisms for the spectral broadening, the first, trivial, mechanism refers to the resolution of the spectrometer. At the present time this resolution has reached such magnitude that the spectrometer broadening c often is smaller than the lifetime broadening C [87]. For example, E G in the non-radiative RXS experiment with krypton presented in [87] the spectrometer HWHM was c "23 meV, while the lifetime broadening of the Kr 3d\5pP4p\5p transition is C "41.5 E  G meV. The second type of spectral broadening connected with the experimental equipment is caused by the finite width c of the spectral function U(X,c) of the incident photon beam. The analysis of the role of the spectral function U(X,c) in resonant Raman spectroscopy is very important, since, as reviewed below, the shape of the incident radiation can be used to actively govern the spectral shape of RXS process as a whole. To describe a realistic experimental situation, one must consider the convolution



dp(u,u ) u  U(u !u,c)"r "F (u)"U(u#u !u,c) , p(u,u)" du  dudo  Mu D DM D

(16)

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of the RXS cross section (5) with the normalized-to-unity spectral function U(u !u,c) centered at  frequency u. The RXS amplitude F (u) is given by Eq. (7) with u"u#u (9). A consideration D DM of case (15): C ;C ,c, which is common in the X-ray region, simplifies the analysis. The correD G sponding convolution for non-radiative RXS gives an expression similar to Eq. (16). Due to the finite resolution of the spectrometer, the cross section (16) must also be convoluted with the spectrometer function G(X,c ) E



p(u,u)" du p(u ,u)G(u !u,c ) .    E

(17)

At the present time theoretical evaluations of the spectral functions U(X,c) seem to be lacking, but some intuitive speculations and experimental measurements [88] (see below) indicate that this function is sharper than a Lorentzian. Sometimes the Voigt profile is used for the spectral function but most often in numerical simulations a simple Gaussian form is used 1 U(X,c)" ((ln 2/n) exp(!(X/c) ln 2) c

(18)

with c as the HWHM. 3.2.1. Self-absorption Photoabsorption of scattered radiation distorts the spectral shape given by the Kramers— Heisenberg (KH) equation. In the gas-phase experiments one can identify two reasons behind this distortion. When the photoabsorption profile of the window has structure, the RXS spectral shape can differ strongly from the KH profile. In the following we neglect this effect and assume that the photoabsorption by the window is flat and structureless at the photon energies spanned by the RXS band. A second reason for distortion is the photoabsorption of incident and scattered radiations by the gas, something that is commonly referred to as self-absorption. Contrary to resonant inelastic X-ray scattering (RIXS) the spectral shape of the resonant elastic X-ray scattering (REXS) profile is strongly influenced by self-absorption since both the incoming and the final X-ray photon frequencies reside in the region of strong photoabsorption. In some cases, for example in metals, it is therefore difficult to exactly distinguish REXS from RIXS. The self-absorption depends on the photon energies (u and u), the gas pressure, the geometry of the cell, the geometry of optical measurements, and on the scattering angle 0 between the propagation directions of incident and final X-ray photons. For example, in experiments conducted by the Uppsala group (see Section 2.2) the incident radiation passes through a small entrance window, 250 lm in square [68], and only X-ray radiation scattered at the angle 0"90° is detected. The gas cell has a width 2h(¸, where ¸"1 cm is the length of the narrow exit window. In such measurements the absorption of the incident X-ray photons and of emission radiation decreases the REXS cross section (5) by the factor 1 . q(u,u)"l(u)l(u)(1!exp(!¸/l(u)))(1!exp(!h/l(u))), l(u)" np(u)

(19)

The first multiplier in this equation is caused by photoabsorption of incident radiation during propagation along the cell axis, while the photoabsorption of X-rays emitted in the cross direction

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is described by the second term. It is evident that the photoabsorption length l(u) depends on the gas concentration n and the photoabsorption cross section p(u). A qualitatively different frequency dependence of the self-absorption factor q(u,u) takes place for the forward scattering in a narrow cell h/¸;1 h e\*JSY!e\*JS , 0; . q(u,u)" ¸ l\(u)!l\(u)

(20)

Contrary to Eq. (19), the self-absorption function can now be nonmonotonic. Simulations [89] show that for an optically thick (¸/l(u)'1) gas target a spectral hole appears in q(u,u). The self-absorption can strongly influence the ratio of RIXS and REXS bands as well as the spectral shape of the REXS profile. 3.2.2. Measurements of the spectral function of the incident X-ray beam The analysis outlined above is well exemplified by the experiment of Ref. [88] (see also [87]) devoted to measurements of the spectral function U(X,c). Such measurements were carried out at the “Finnish beam line” at the MAX synchrotron laboratory, Lund, Sweden, which is equipped with a modified Zeiss SX-700 plane grating monochromator having a plane elliptical focusing mirror. The resonance Auger spectra were collected with a high-resolution SES-144 spherical mirror electron spectrometer equipped with a position sensitive detector system. In order to keep the lineshape modifications caused by the electron spectrometer as small as possible, while maintaining a reasonable count rate, a pass energy of 10 eV was used. This corresponds to an electron spectrometer resolution (HWHM) of about 20 meV. The 4s photoelectron line of Kr, Fig. 9, provided a good measure for the energy distribution of incident X-ray photons, because the lifetime broadening of the 4s line is negligible and also the spectrometer broadening is relatively small for the large incident photon bandwidths. Fig. 9 shows the measured 4s photoelectron lines, with the mean photon energy u"91.20 eV, for exit slit widths of 20, 200, 400 and 600 lm. The photoelectron line becomes more and more square-like with increasing slit width (the SX-700 monochromator has only one slit, the exit slit). For wider slit widths the square-like slit contribution dominates in the convolution of partial broadenings from slit width, source size, and slope errors [90]. The latter two contributions are expected to be close to a Gaussian shape. In the observed photoelectron spectrum (Fig. 9), the photon band is further convoluted by the spectrometer function of the electron energy analyzer. The measured spectral function departs significantly from a Lorentzian profile, but can be approximated by a Gaussian function for a sufficiently small slit width. 3.3. Fingerprints of resonant X-ray Raman scattering There are thus two main fingerprints of radiative and non-radiative RXS: (1) The linear dispersion law (9), (14), and: (2) Narrowing of the RXS profile below the lifetime width of the core excited state, if the width of the incident photon spectral distribution is smaller than this width.

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Fig. 9. Spectral function. Kr 4s photoelectron spectra, recorded using different monochromator exit slit widths (marked next to the spectra) [88]. The tail at low photon energy for a 400 lm slit width is shadowed. For a further details see text, [88].

3.3.1. Experimental observation of Raman—Stokes shifts It is easy to understand that the statement above concerning the linear dispersion law and resonance narrowing is valid only if the final state of the RXS process is discrete with a sufficiently small lifetime broadening (C (C ). This assumption breaks down when the final state is in the D G continuum, as is the case when the excitation exceeds the core ionization threshold I (electronic continuum), if it goes above the Fermi level in solids, or if the final state is dissociative [91], see Section 11. In these cases the sum over final states in Eqs. (5) and (13) must be replaced by an integral over the final state energies. As a result of this integration the D-function disappears in the equations for the RXS cross sections (5) and (13). The width of the RXS spectral profile is then given by a sum of lifetime broadenings, C #C , of the core excited and final states and the peak positions G D then no longer depend on the excitation energy [92,41,42]. Such a step-like dispersion law was confirmed in a recent radiative RXS experiment by MacDonald et al. [93] near the ¸ edge of  Xenon at beam line X-24A of NSLS at Brookhaven National Laboratory. Fig. 10 shows the results of this experiment for the peak position of the ¸a and ¸b emission spectra as a function of   excitation energy. It is necessary to mention that the linear dispersion law (9) and (14) is valid if the excitation energy is tuned close to the single photoabsorption resonance. The dispersion is strongly nonlinear, and can even become “anti-Raman” (negative dispersion) when the region of excitation energy includes several intermediate states. One reason for this non-linear dependency is the so-called specific screening effect [94,95,69], namely the electron excited to the vacant MO t screens differently the subsequent decay of electrons from various occupied levels to the inner J shell. This specific screening effect leads to a non-linear dependence on l and hence on the excitation energy of the RXS peak positions and corresponding intensities. Other reasons for

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Fig. 10. Upper part: Variation of the position of the peak of the Xe ¸a radiative RXS as the incident X-ray energy is  scanned across the ¸ edge (circles, right-hand scale) and superimposed on the XE ¸ absorption spectrum (line,   left-hand scale) for comparison. Lower part: Same as upper frame except that the position of the Xe ¸b peak is plotted  (circles, right-hand scale).

a non-linear behavior of the dispersion can be found in the Stokes doubling effect (see Section 3.4) and the presence of vibrational structure (Section 6.5.1). The Raman—Stokes shift of emission lines has by now been verified in many experiments [24,85,93,87]. 3.3.2. Subnatural narrowing for core excitation near the vicinity of a step-like edge The second fingerprint of RXS, the resonant narrowing effect, was observed for the first time by Eizenberger, Platzman and Winick (EPW), who measured the radiative RXS in the vicinity of the metal Cu K-edge [22], Fig. 11. Their experiment can be described by considering the model given in Fig. 12, which mimics the main features of the EPW theory and experiment. We introduce here the detuning, X"u!I , of the incident photon frequency u from the Cu K-edge, I , and Q Q X"u!u as the the detuning of the final photon frequency from the resonant frequency, NQ u , of the 2p P1s X-ray emission. An approximation of the photoabsorption probability in NQ  Eq. (16) by the step function H(X) and Eq. (18) results in p(u,u)JD(X,C)H(X!X,c) ,








X!X 1 H(X!X,c)" 1#erf c 2

 AP  H(X!X) .

(21)

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Fig. 11. Narrowing effect: Cu Ka line. Comparison of the experimental results (points) with theoretical calculations  (line) [22]. Fig. 12. Narrowing effect and level scheme. FWHM (solid line) and the peak position (dashed line) versus detuning X (22).

Here c"c/(ln 2 and the lifetime broadening of the final state, C , is neglected. This equation leads D to the following simple expression for the width of the RXS band when the spectral function width is negligibly small, cP0: FWHM"2C, if X5C , FWHM"C#X, if C5X50 ,

(22)

FWHM"X#(2X#C, if X40 . One can here immediately obtain the narrowing below the lifetime broadening, 2C: FWHM "C/(2(2C when X"!C/(2, see Figs. 11 and 12. However, this narrowing is not

 complete, FWHM O0. It is relevant to compare the frequency dependence of the RXS width

 (22) with the peak position which is equal to X or 0 if X40 or X50, respectively, see Figs. 11 and 12. 3.3.3. Subnatural narrowing for core excitation near a single resonance The concept “resonant narrowing” is mostly understood as an effect connected with excitations to an isolated resonance. This narrowing is more complete than the “edge narrowing” reviewed in

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the previous subsection — up to the final state lifetime broadening or up to zero if C is negligible. D This effect was observed for the first time in the non-radiative case in the Kr M N N and Xe    N O O resonant Auger spectra following Kr 3d P5p and Xe 4d P6p resonant excita     tions [86] (see also Ref. [96]). These measurements were carried out with different spectral widths according to monochromator exit slit sizes of 25, 50, 200, 400, and 800 lm and a 10-eV pass energy. The total instrumental resolution was considerably smaller than the lifetime widths of the Kr 3d and Xe 4d (FWHM"111 meV) levels. As a reference line for the linewidth considerations Kivima¨ki et al. used the peak originating from the transitions to the Kr 4p(D)5p(D ,P )   and Xe 5p(P)6p(P ) final states. The widths of these Auger peaks and the Kr 4s and Xe 5s  photoelectron lines are plotted versus the exit slit width of the monochromator in Fig. 13. The fact that the photoelectron lines and the sharpest Auger features have very nearly the same widths for 25-, 50-, and 200-lm slits shows directly that the inherent broadening of the resonant Auger lines follows the width of the photon band and does not depend on the lifetime width of the core level (FWHM"111 meV). The measured FWHM for 50 and 100 lm slits are about the same (60—70 meV) and are narrower than the lifetime broadening (111 meV) which implies that it is the electron analyzer which gives the most significant contribution to the total linewidth in this experimental setup. 3.4. Line-shape distortion. Stokes doubling effect A simple 3-level model of RXS with one intermediate and one final state (Fig. 14) is sufficient to illustrate Stokes doubling. The RXS cross section then consists of only one direct term p(u,u)"p N(X,X), M

N(X,X)"D(X,C)U(X!X,c) .

(23)

Fig. 13. Narrowing effect. The width of the Kr and Xe resonant Auger lines, Kr 4s and Xe 5s photolines as function of slit size. [88].

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Fig. 14. A 3-level model for RXS.

Here X"u!u is the detuning of the incident photon frequency u from the resonant frequency GM u of the absorption transition oPi, while X"u!u is the detuning of the final photon GM GD frequency u from the emission resonant frequency u . All non-essential quantities are collected GD into the constant p . The non-radiative RXS is also described by Eq. (23) if the uPe replacement is M made. The RXS spectral shape (23) is thus simply given as the product of the photon frequency distribution U(X!X,c) and the Lorentzian D(X,C) function. These two multipliers introduce two resonant features for the RXS cross section. The first one follows the condition for an ordinary emission resonance X"0 ,

(24)

while the second, the singularity in (23), follows the energy conservation (Raman) law X"X

(25)

and is a fundamental property of the Raman effect. This condition is equivalent to the Raman—Stokes shift (9) of photon frequency for X-ray Raman scattering. The second spectral feature (25) does not influence the spectral shape of RXS in the case of broadband excitation c
(26)

The spectral function can be approximated by U(X,c)"const, and the spectral shape (23) is given only by a Lorentzian with a width equal to the lifetime broadening. In the opposite limiting case of narrowband excitation, c;C

(27)

the Lorentzian function in Eq. (23) can be considered as constant. Now, the RXS spectral shape coincides with the spectral function U(X!X,c) and the peak maxima of this so-called Stokes resonance follow a linear dispersion law (9) and (25). The spectral width of the RXS resonance is equal to c and obtained very narrow as discussed above.

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Fig. 15. Lineshape distortion caused by the Stokes doubling effect. C"0.15 eV, c"0.75 eV. The profile with Stokes doubling is shown as a solid line [97]. Fig. 16. The region of existence of Stokes doubling effect [97].

The spectral shape of RXS is more complex if the width c of the spectral function is comparable with the lifetime broadening C (Fig. 15). The dependence of the RXS cross section on the final particle energy u is governed by two dimensionless parameters: c/C and X/c. Independent of these parameters, the RXS profile has the shape of a single resonance or a single resonance with a shoulder (left or right). Under certain conditions the ordinary emission resonance (24) and the Stokes resonance (25), can be observed simultaneously. Indeed, the Stokes resonance (25) has red or blue shifts relative to the ordinary emission line (24) if the incoming X-ray photon frequency u is lower (X(0), respectively, higher (X'0) than the resonant frequency u . This results in a doublGM ing of the ordinary emission lines. This so-called “Stokes doubling” effect in RXS was predicted in Ref. [97]. In general, the conditions for this doubling are quite restrictive and depend critically on the shape of the spectral function [97]. For instance, the conditions for the appearance of Stokes doubling differ quite significantly for the Lorentzian and Gaussian spectral functions [97], and are much more restrictive in the latter case. For a realistic Gaussian spectral function (18), such doubling (Fig. 15) takes place in the energy region shown in Fig. 16. The spectral profile has the shape of a single resonance with a shoulder (see dashed profile in Fig. 15) if the excitation energy is tuned from the Stokes doubling region (Fig. 16). 3.4.1. Experimental observation of the Stokes doubling effect The first experimental evidence of the Stokes doubling effect was given by the resonant Auger spectrum of Kr [88]. The measurements were carried out at the Finnish beam line in the MAX laboratory. The details of this experiment are the same as those described in Section 3.2.2. The 3d\5pP4p\5p resonant Auger spectrum has some well-isolated lines in the kinetic energy  range from 55.5 to about 59 eV, especially a double peak structure between 58 and 59 eV

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Fig. 17. Computer simulated resonance Auger spectrum with its individual components (solid lines), together with an experimental spectrum (dotted line) [88].

corresponding to the 3d\P4p\(D ) normal Auger transition, and which splits due to the   coupling with the 5p spectator electron to the 3d\5pP4p\(D)5p(D ,P ) and    ,P ) transitions. 3d\5pP4p\(D)5p(F    The experimental spectrum in the kinetic energy range 58.1—59.1 eV, taken at the mean photon energy u"91.45 eV (detuning equal to 0.25 eV) and with the monochromator exit slit width of 400 lm is given by the dotted line in Fig. 17. The solid line in Fig. 17 displays the results of a computer simulation of the experimental spectral function U(X,c). The Auger line energies and the intensity ratios of the three peaks were taken from a high resolution Auger resonance Raman spectrum recorded at the photon energy u . According to the simulation, the double-peak structure appears GM only if the detuning is in the range 0.25—0.40 eV (u"91.45—91.6 eV). In other cases only heavily asymmetric profiles are seen. These results are confirmed by the experiment. It can be concluded that Stokes doubling of spectral lines can be anticipated in any highresolution RXS measurement. It complicates the RXS profiles and must be carefully taken into account when interpreting the RXS spectra. 3.5. Moments of the RXS spectral function. 3-level system In the previous section it was shown that the Stokes doubling effect distorts the spectral shape of the RXS cross section when the spectral function width c is comparable with the lifetime broadening C. The underlying reason for this distortion is the presence of the two resonant features of the RXS profile. The position of the first one (24) coincides with the peak maximum of the ordinary emission resonance and does not depend on the excitation energy. The second resonant feature follows a linear dispersion according to the Raman—Stokes law (25). The coexistence of these two qualitatively different phenomena leads to violations of the linear dispersion law. Due to Stokes doubling one can also expect the effective broadening of spectral profiles to depend on the

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excitation energy. This problem was solved numerically by Armen and Wang [90] for the peak maxima and FWHM. Unfortunately, the notions of peak maxima and FWHM become ambiguous in the region of Stokes doubling [90] (Fig. 15), and it is more natural to study the moments of the X-ray profiles; the center of gravity (CG)



 p (u) p (u)" e(u)"  , duuLp(u,u) L p (u) \ M and the spectral width








p(u,u)  . du(u!e(u)) p (u) \ M which also give simpler analytical properties than peak maxima and FWHMs. C(u)"

(28)

(29)

3.5.1. First moment of the RXS profile To understand the general properties of the frequency dependency of the center of gravity (first spectral moment) a system with 3 levels, o, i and f, is first reviewed — more general cases will be considered later on in Section 8.1. It is convenient to present the center of gravity (CG) in the following form: e(u)"u #u(X) . (30) GD As indicated below, the resonant frequency u of an emission transition iPf is essentially given by GD the center of gravity for broadband excitation (c/CPR). The function u(X) describes the deviation of the exact CG e(u) from u for broadband excitation and is defined by the following GD equation: mU(m,c)  dm (m#X)#C Re(zw(z)) c \ u(X)"X# "X#c , c" . Re(w(z)) U(m,c) (ln 2  dm \ (m#X)#C

(31)

Here z"(ιC!X)/c, w(z) is the error function for a complex argument [98]. A similar function u(X)!X appears in the theory of light-induced drift (LID) [99,100], and the properties of this antisymmetric function (u(!X)"!u(X)) can be found in Ref. [101], see also Fig. 18. As the most important limiting case, the spectral function of incoming X-ray photons, Eq. (31), is approximated by a Gaussian (18): U(X,c)Jexp(!(X/c)), where the connection between c and HWHM c is given in Eq. (31). The properties of the u-function govern the general spectral features of the center of gravity e(u) and C(u) [101]. Contrary to intuition the CG depends on the frequency u in the limiting case of broadband excitation (26). Indeed, in accordance with Eqs. (30) and (31), this dependence is given by 2C Ku if "X";c , e(w)"u #X GD GD (nc c e(w)"u #X! Ku #X if "X"
(32)

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Fig. 18. The dispersion of center of gravity (30),(31) for the 3-level model. The relative CG and relative detunings are (e(u)!u )/c and X/c. The broadband case was calculated according to Eq. (31) for C/c"0.1, while narrowband GD excitation was calculated for C/c"2. The dispersion of CG for a frequency-independent spectral function (c"R) is depicted by a dotted line. The slope of CG under broadband excitation was calculated according to Eq. (32).

Eq. (32) shows that the slope of e(u) increases strongly from 2C/((nc) to 1 if "X" passes through "X"&c (see solid lines in Fig. 18). When "X"(c, the center of gravity is very close to the resonant emission frequency u , and will follow the Raman—Stokes law (9) GD e(u)"u #X"u!u . (33) GD DM If "X" considerably exceeds c. The center of gravity is independent on the frequency u only if the spectral function U(X,c) is constant c/CPR (dotted line in Fig. 18). The frequency dependence of the CG is qualitatively different when the X-ray fluorescence is induced by a narrowband X-ray beam (27). In this case the CG has the asymptotic behavior:





 





c  c "u #X 1! e(u)"u #X 1! GD GD C X#C

if "X";C ,

c c "u #X! if "X"
(34)

The slope of CG e(u) increases slowly when the magnitude of detuning "X" increases. In the limit considered here, the CG follows closely the Raman law (33) (dashed line, Fig. 18). So both broadband and narrowband excitations lead to the same result, namely to the Raman law (33), when the incoming photon frequency is tuned sufficiently far from the X-ray absorption resonance "X"
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3.5.2. Second moment of the RXS profile The spectral width C(u) of the emission band is the second important characteristic quantity of the RXS profile. This quantity can be introduced in two different ways; firstly, by defining C(u) as the FWHM of the spectral band [90], and; secondly, as the second moment of the RXS profile (29). The main advantage of using the second moment is given by the simpler analytical properties of this quantity. Using a Gaussian spectral function (18) the spectral width (29) reads in the 3-level model [101] (Fig. 14), cC  . (35) !u(X)!C C(u)" (n Re w(z) The dependences of the second moment (35) on detuning and on the width of the spectral function are depicted in Fig. 19. To understand the properties of C(u) it is again useful to consider the limiting cases of narrowband and broadband excitations. As for the center of gravity one can use the properties of the u-function and the error function for a complex argument. The following asymptotes are then obtained: cC  if (X#C;c , C(u)" (p c C(u)" if (X#C








Fig. 19. The second moment (spectral width, C(u)) of RXS bands for the 3-level model. (a) The dispersion of C(u). (b) The dependence of C(u) (29) on spectral width c of incident radiation. A thin line shows the asymptote of C(u) (36) for large c (c<(X#C).

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For broadband excitation, (X#C;c, the second moment of the RXS band diverges as (c when c tends to infinity (36). This divergence is caused by the slowly decaying Lorentzian tail of the scattering amplitude (7). Contrary to the second moment this divergence is absent for the FWHM. As is well known the FWHM tends to the lifetime broadening 2C for broadband excitation. Going to the narrowband excitation case, (X#C
3.5.3. Experimental observation of non-linear dispersion relations According to theoretical predictions the dispersion relations can be strongly nonlinear if the spectral function width and lifetime broadening are comparable, see Fig. 19. The influence of the photon bandwidth on the dispersion relation was first observed in the non-radiative RXS experiment in [87]. The Kr 3d\5pP4p\5p resonant Auger spectrum has been measured at the  Finnish beamline (BL 51) at the MAX-I storage ring in Lund. The details of this experiment are very similar to those described in Section 3.4.1. The result of the dispersion measurements are shown in Fig. 20 (the simulated curve was here obtained with c"50 meV and C "41.5 meV). G

Fig. 20. Peak position of the Kr 3d\5pP4p\(D)5p(D ) Auger electron line plotted versus photon energy   detuning. Experimental values (filled circles) are shown relative to the nominal energy e , together with a simulated curve. M The straight line corresponds to linear dispersion. e is the photoelectron energy when the excitation frequency u is equal M to the photoabsorption resonant frequency u [87]. GM

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4. Duration of resonant X-ray Raman scattering One of the important characteristics of the dynamics of a resonant X-ray Raman scattering process is the “duration time” [102—107]. It presents a pure quantum notion based on the interference, or dephasing, suppression of large time contributions to a scattering amplitude. The concept has provided a deeper insight into the formation of the RXS spectral profile [103,105—110]. It has been proven that with the variation of the duration time through detuning the energy one can control — or manipulate — different microscopic dynamical processes responsible for the spectral shape of RXS. A principal aspect of this notion to be understood is the apparent contradiction between the time of the evolution of the wave packet at the core excited state and the duration time q of the RXS process (Fig. 21). A Contrary to the relaxation time of the wave packet the effective duration of RXS strongly depends on the detuning X of the frequency of incident radiation from the photoabsorption band. Moreover, q tends to zero for large "X". The relaxation of the wave packet is characterized by the A time of flight and the lifetime of the core excited state, C\, both of which can considerably exceed the RXS duration. This leads to the following paradox: To reach the fast limit for the RXS amplitude — short duration of RXS — the time of the wave packet evolution must exceed the lifetime C\ by several times. A goal of this section is to review the solution of such a contradiction, and to explain the notion of RXS duration and the characteristic times of the wave packet evolution on the core excited state in some detail. This section provides also a background for the timedependent formulation of the RXS scattering cross section, the several aspects of which are reviewed in Section 12 to follow. 4.1. Complex duration time Close to the resonant region the radiative and non-radiative RXS amplitudes have the same structures [42], and can be described using the same notion of a duration time. Physically, it is convenient to consider the Kramers—Heisenberg scattering amplitude as a projection of the stationary wave packet W (R) on the final state 2 Q"c21c"D"o2 . (37) F"!i1 f "W (R)2, W (R)"i 2 2 u!u #iC AM A

Fig. 21. Dependence of the RXS duration q on the excitation energy (40). A

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Here u "E !E , E is the energy of cth state. In order to describe the formal manipulations AD A D A more transparently, we drop the index f for the scattering amplitude: F PF. The operator D D describes the interaction of the target with the incident X-ray photon. In the case of nonradiative RXS. Q is the Coulomb operator and Q"DH when the emitted particle is the final X-ray photon [42]. The half-Fourier transform of the denominator at the right-hand side of Eq. (37) yields the time-dependent representation for the scattering amplitude F"F(R),

F(q)"!i1 f "W (q)2 . 2

(38)

Here



O (39) W (q)" dt e\R2Qt(t), t(t)"e\iHRD "o2, H"H!EM , 2  where H is the molecular Hamiltonian, and E is the average energy of a core excited state (see below). 4.2. Decay and dephasing times One can directly see from Eq. (39) that the complex time 1 ¹"¹C#i¹X" "q eiP , A C!ιX 1 X q ,"¹"" , tan u" A C (X#C

(40)

characterizes the time scale of the RXS process. Following [103—107] we refer to this time as the duration of RXS. X"u!u is defined here as the detuning of u relative to the characteristic frequency u"EM !E of the X-ray absorption band. A more precise definition of this characteristic M frequency depends on the problem of interest; u can be the position of the strongest peak or edge in the X-ray absorption spectrum. When the electron-vibrational band is analyzed it is convenient to choose u as the position of the center of gravity of the electronic peak (see below). Going back to the main subject we consider the fact that the duration time (40) is complex and that it consists of two qualitatively different contributions ¹C"C/(X#C),

¹X"X/(X#C) .

(41)

The real part ¹C of ¹ coincides with the lifetime of the core excited state C\ if X"0. It is interesting to note that ¹C is equal to the delay time [111,112] ¹ "dd/du for scattering in the B vicinity of a Breit—Wigner resonance: FJ1/(X#ιC). Here d"!arctan(C/X) is the phase shift under scattering. The imaginary contribution ¹X vanishes when X"0 and ¹X"1/X if the lifetime broadening is small. Thus ¹X originates mainly from detuning. This time can be named as the “dephasing time” since the contributions to F (38) from different times t and t interfere   destructively owing to the phase difference X(t !t ). This destructive interference suppresses the   long-time contribution to the scattering amplitude F if "X" is large and if a damping C (even infinitesimal) exists. We will see below that only decay transitions in the time domain 0(t("¹"

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Fig. 22. The dependence of the real and imaginary parts of the duration time (40), (41) on the detuning and lifetime broadening.

contribute significantly to F. The real, ¹C, and the imaginary, ¹X, parts of the RXS duration time depend differently on the detuning and the lifetime broadening, see Fig. 22. The complex nature of the duration time (40) deserves to be emphasized. This is not “accidental” but has a deeper physical reason. We can refer to the real (¹C) and imaginary (¹X) parts of ¹ as to the irreversible and reversible contributions, respectively. Indeed, the dephasing is a reversible process contrary to the decay which is irreversible. 4.3. RXS duration as a mean time of scattering The duration of RXS can be introduced also as the mean time ¹M over all RXS events with decay of the core excited state at moment t:



1  dt t F(t), F(t)"1 f "F(t,R)2, "F(t,R)2"!iQ"t(t)2e\R2 . (42) ¹M " F  Here F(t) is the amplitude of the X-ray scattering with decay at time t, and F"dtF(t). The use  of F(t) in the averaging procedure (63) instead of as the real distribution "F(t)" can be motivated by that in the latter case the coherent properties of the RXS amplitude — which play a crucial role in the notion of the RXS duration — are lost. Making use of the resolution of the identity 1" "c21c", one receives the stationary representaA tion for ¹M ¹M "!i

ι 1 f "Q"c21c"D"o2 * ln F" . F (u!u #iC) *u AM A

(43)

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This representation shows immediately that ¹M coincides with the duration time ¹ (40) for large detuning or lifetime broadening (¹M P¹). The absolute value "¹M " and the phase u"arcsin(Im ¹M /"¹M ") of this complex duration time are depicted in Fig. 23 assuming the harmonic approximation for the nuclear degrees of freedom. The duration of RXS increases up to the lifetime of the core excited state for an exact photoabsorption resonance. One notes a non-monotonous behavior of the duration time in the region of the strong photoabsorption as seen in this figure. One can also use an alternative definition of the RXS duration based on the averaging procedure ¹I "1t(R)"t(R)2,



"t(R)2"





dt t

"F(t,R)2 , 1F(R)"F(R)2

(44)

where "F(R)2"dt"F(t,R)2. Contrary to Eqs. (40) and (42) this mean duration time (44) is real  and gives a correct asymptote ¹I K(X#C)\ for large X or C. Such a definition for the RXS duration is convenient for the analysis of the decay transitions to continuum final states (Fig. 24) due to independence of ¹I on the final state. Figs. 23 and 24 show a strong asymmetry of ¹M and ¹I as function of the detuning. One can see that the RXS duration decreases faster for core excitation below the frequency of the vertical transition º (R )!º (R )!u /2 than above this crossing point. One reason for this behavior is A M M M M given by the different evolutions of the wave packet for core excitation below and above the crossing point. For example, in the case of a dissociative state and excitation above the crossing point, the wave packet can propagate back to the potential wall where it is reflected.

Fig. 23. The dependence of the absolute value "¹M " and phase u"arcsin(Im ¹M /"¹M ") of the mean duration time (43) on the detuning. The RXS duration decreases when the excitation energy is tuned far from the photoabsorption band. u "0.3 eV, C"0.1 eV, b"1.5. M Fig. 24. The dependence of the mean duration time ¹I (44) on the detuning. The resonant scattering through the Cl L (2p\pH) dissociative core excited state in the HCl molecule [113]. C"0.065 eV. The resonant frequency of the vertical transition, u "º (R )!º (R )!u /2, is equal to 202.5 eV. The mean duration time ¹I depends asymmetrically on  A M M M M the detuning u!u . 

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4.4. RXS amplitude and wave packet versus the RXS duration time 4.4.1. Dynamical representation We now consider an entirely different time-dependent representation for the RXS amplitude, one which is based on the expression (38) and an integration of the right-hand side of Eq. (39) F(q)"1 f "Q

1 (1!eiX>iC\HO)D"o2 . X#iC!H

(45)

This equation yields the following general dynamical representation for W (q), which is valid for all 2 values of the complex time ¹; W (q)"lim Q¹(1!¹*/*t)\(t(t)!e\O2t(q#t)) , 2 R W (q)"Q¹(t(0)#¹t(0)#2!e\O2+t(q)#¹t(q)#2,) . (46) 2 Such a time-dependent representation differs conceptually from the original representation (39) and allows to predict directly what time domain gives the main contribution to the stationary RXS amplitude F"F(R) (38) with the stationary wave packet W (R)"lim Q¹(1!¹*/*t)\t(t) . (47) 2 R This is most easily understood by considering the important special case of short RXS duration (see below). It should be pointed out that expression (47) is equivalent to the following differential equation: * 1

(t)" (t)!Qt(t), W (R)" (0) 2 *t ¹

(48)

with solution (39). The right-hand side of this equation consists of a decay-dephasing part, (C!ιX) (t), and a source, !Qt(t). 4.4.2. Fast RXS If the excitation energy is tuned to the wing of the photoabsorption band the duration of the scattering (40) can be shorter than the inverse width q "1/*u of the photoabsorption band close M to u "¹";q . (49) M The time q characterizes the quantum beats of the wave packets t(t) and W (q) and gives the main M 2 time scale for problems concerning the RXS duration. *u:1 eV in the problems connected with the nuclear dynamics, while this spectral width has the order 1—10 eV or more when the dynamics of the relaxation of the electronic shells due to the creation of a core hole is studied (see below). Keeping the first two terms in the expansion over ¹ in Eq. (46), the following remarkable result is immediately obtained: FK!ι¹1 f "Qt(¹)2,

W (R)KQ¹t(¹), 2

"¹";q . M

(50)

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One sees that the spectral shape of fast RXS is given by a simple projection of the wave packet Qt(¹) at the complex time ¹ (40) onto the final state " f 2. Eqs. (50) generalize the corresponding results from Ref. [102]. It is worthwhile to compare W (R) (50) and W (q) (46); 2 2 W (q)KQ¹(t(¹)!e\O2t(q#¹)), "¹";q . (51) 2 M One finds, contrary to intuition, that the wave packets W (R) (47), (50) and W (q) (46), (51) do not 2 2 coincide even when q essentially exceeds the RXS duration time "¹". A coincidence takes place only if the time of the wave packet evolution q is larger than the lifetime C\. One can say that the fast limit (50) for the RXS amplitude is obtained only when the wave packet (50) must go through the long time evolution (longer than the lifetime, q
*F(q)P0 if q<1/C .

(53)

According to Eqs. (52) and (53) (cf. also Eqs. (50) and (51)) all times from t"0 up to C\ are important for the scattering amplitude F(q). So it is apparent that the RXS duration time and the relaxation time of the wave packet do not coincide, and one can ask what times characterize the relaxation of the wave packet to the stationary value? 4.5.1. Bound core excited state. Time of relaxation A system with a bound core excited state and only one final state with the lowest vibrational level, f"o, is suitable for considering the relaxation time. Identical, harmonic, vibrational frequencies u and potentials for the ground, core excited and final states, are then assumed. Both F(q) and M W (q) (51) perform non-damped oscillations and never reach F and W (R) if C"0, see Fig. 25 2 2 [107]. Apparently, F(q)PF only when q
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Fig. 25. The dependence of the RXS amplitude F(t) (52) on the real time t for the bound core excited state shows damped revival oscillations. The potentials of the ground and final states have the same shape. The RXS scattering from the lowest ground state vibrational level to the lowest final state vibrational level (0—0 scattering). u "0.3 eV, C"0.03 eV, M X"3 eV, b"1.5. ¹Kι0.2 fs. b"(R !RA)/(a (2), a "1/(ku . M M M M M

when the wave packet propagates on the bound potential of the core excited state — or, more precisely, for core excitation below the dissociation threshold of a bound potential. This behavior stands in contrast to the qualitatively different picture obtained in the case of core excitation above the dissociation threshold. 4.5.2. Revival time Simulations (Fig. 25) for bound core excited and final states show that the scattering amplitude F(q) nearly recovers its fast X-oscillations with the period ¹(X)"2p/X. Such a “revival” occurs because of the discreteness of the spectrum and the constant value of the excitation frequency u. The revival time depends on both X and u and is equal to M n max+¹(X),¹(u ), , (55) M where the integer n also depends on X and u . In the special case of Fig. 25 the revival time is equal M to the vibrational period ¹(u )"2p/u . Due to the finite lifetime of the core excited state these M M “revival” oscillations become damped (Jexp(!Cq)) and are totally suppressed when q
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Fig. 26. The space distribution of "W (R)" versus the RXS duration (detuning) for the bound core excited state P of N . 2 S  The contribution of the region outside of the ground state vibrational wave function is suppressed when X is large. The parameters for N are taken from [161]. The details of the corresponding time-dependent calculations are described in  Ref. [113]. u "0.29 eV, RA!R "0.12 a.u. The resonant energy of the photoabsorption transition, M M M º (R )!º (R )!u /2, is equal to 400.95 eV. Here and in Figs. 28—30 a different definition of the detuning is used: A M M M M X"u!(º (R )!º (R )!u /2). C"0.065 eV. The RXS duration times are: "¹"K5.6 fs, 0.73 fs, and 0.35 fs for the A M M M M excitation frequencies u"401 eV, 402 eV, and 403 eV, respectively.

time domain (see discussion following Eq. (41)). This interference leads also to the suppression of the large distance contribution to W (R) if the RXS duration is short (large detuning), see Fig. 26. 2 The wave packet W (R) copies the space distribution of the ground state wave function "o2 in this 2 limit of fast RXS (59), Fig. 26c. At this point it should be noted that the role of the destructive interference in F(q) for q(1/C is strongly reduced due to the extra term *F(q) (52). The interference quenches the contribution to F(q) of the long time evolution ("¹"(t(q) only if q'C\. The physical picture for a dissociative core excited state is, however, more complex (see below). We can now select one of the main results of the discussed problem: The time-dependent amplitude F(q) (38) consists of two terms. The first one F"F(R) is characterized by the frequency depending duration time ¹ (40). However, the second term, the deviation *F(q) of the timedependent RXS amplitude F(q) has a different time scale: F(q) tends to the stationary value F exponentially with a characteristic time equal to the lifetime C\. One can say that to receive the sudden limit, the time of the “measurement” must be equal to infinity, qPR, when C is small. The destructive interference suppresses only in this case the contribution to F(R) for times larger than the RXS duration, q ""¹". Such an experiment with A q"R corresponds to the ordinary stationary RXS measurements (38).

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Another property of F(t) should be emphasized (Fig. 25): Simulations [113] show that the maximum value of F(t) can exceed by several times the stationary RXS amplitude F"F(R). This leads to the important conclusion that in time-resolved measurements one can obtain signals "F(t)" that are stronger than stationary value "F(R)". The destructive interference in the entire time domain is the reason behind this observation. 4.6. Space distribution of wave packets versus RXS duration time To emphasize the principal distinction between wave packet evolution for core excitation below and above the dissociation threshold it is suitable to consider a core excited state with a repulsive potential. To be specific, let us consider RXS by the HCl molecule close to the Cl L-edge. The excitation energy is tuned in the vicinity of the 2p\pH dissociative core excited state (details of the time-dependent calculations for this example can be found in Ref. [113]). The repulsive force F(R)"!º (R)'0 moves the atoms in opposite directions. Instead of the back and forth A periodic motion in a bound potential, the initial Gaussian wave packet t(0)""o2 now spreads in the forward direction only, see Fig. 27. It is necessary to mention that contrary to the absolute value "t(t)" (Fig. 27) the wave packet t(t) has strong space oscillations in the dissociative region. The amplitude of t(t) decreases due to this spread and the finite lifetime. The reason of the spread can be found in the different phase velocities of the Fourier components of the wave packet. 4.6.1. Time evolution behind and close to the wave front A propagation of t(t) only in the forward direction leads to a qualitatively different scenario for the formation of the wave packet W (q) (39). Recall that W (q) (39) is the sum of t(t)exp((iX!C)t) 2 2

Fig. 27. Time evolution of "t(t)" (39). CL L (2p\pH) dissociative core excited state in the HCl molecule. Maximal value of "t(0)" is equal to 0.039.

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over all times t. Due to the forward propagation the wave packets t(t) will never reach the point R of departure. This results in that the part of the wave packet W (q) which is behind the wave front, 2 R!R (vq, does not change for the later times q'q. Clearly, W (q)"0 for R!R 'vq since the M 2 M wave packets t(t) with t(q have no time to reach this region ahead of the wave front of W (q) (here 2 v is the some average speed of the wave packet propagation). One can summarize the results given above as follows: W (q)K q"o2, if q(q , 2 M W (q)K W (R), if t (R)(q , 2 2 D W (q)K 0 if t (R)'q , 2 D where



(56)






0 dR 2  R!R M, v(R)& (º(R )!º(R)) (57) & M v(R) k v 0M is the time of flight from the crossing point R to R and and q "(2ka /F(R ) is the time of flight M M M M through the width a of the initial wave packet "o2. This equation describes the three-step formation M of the wave packet W (q) (30): (1) the beginning of the formation of the “molecular” part; (2) the 2 termination of the formation of the “molecular” part and formation of the long distance or dissociative contribution; and (3) the formation in the region near the wave front. The results of numerical simulations (Figs. 28 and 29) confirm Eqs. (56) and (57) except for some transitions in the region close to the oscillatory front (R!R "vq) of the wave packet W (q). In addition to an M 2 oscillatory front, one sees a “molecular” peak close to R "2.4 a.u., and a flat dissociative M contribution which decreases as e\CRD0. The finite lifetime blurs the oscillations and reduces the amplitude of the front (Figs. 28 and 29). The molecular contribution increases when the RXS duration ¹ (40) decreases [102,104,106,113,114], and one finds indeed a quenching of the amplitude of the dissociative contribution when C or detuning "X" increases, see Figs. 28 and 29. The length of the dissociative contribution to W (q) increases as v(R)q up to v(R)C\. Figs. 28 and 29 2 show clearly the role of the RXS duration and relaxation times of the wave packet W (q) for the 2 formation of this wave packet. t (R)" D

4.6.2. Characteristic times for wave packet deformation Figs. 28 and 29 demonstrate another principal distinction between wave packet evolutions in bound and unbound potentials. The forward propagation in the dissociative potential changes drastically the role of the RXS duration on the formation of W (q). One can see in these figures 2 a strong suppression of the flat dissociative contribution when the RXS duration is small. It can be emphasized that this suppression takes place even for CP0 (an effect which is absent for discrete vibrational states). Moreover, contrary to the dissociative case, the space distribution W (q) 2 oscillates without damping in the case of a bound core excited state and infinite lifetime C\ (cf. Fig. 25 and Figs. 28, 29). It is necessary to mention that the suppression of the long distance contribution to W (q) for bound core excited states takes place only if q9C\, see Fig. 26. 2 One obtains here an important conclusion: The space shape of the wave packet W (q) (56) behind 2 its front (Figs. 28 and 29) does not depend on time (except for very small q): The amplitude of the flat dissociative contribution to W (q) is suppressed when the RXS duration decreases. This means 2

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Fig. 28. Time evolution of the absolute value of the wave packet W (q) (39) for different excitation energies. The shape of 2 the wave packet behind the wave front does not change. Cl L (2p\pH) dissociative core excited state in the HCl molecule. C"0.045 eV. (a) X"0 eV, "¹"K14.6 fs. X"4 eV, "¹"K0.16 fs. Fig. 29. Same as Fig. 28 but C"0.0045 eV. (a) X"0 eV, "¹"K146 fs. (b) X"4 eV, "¹"K0.16 fs.

that when the wave front has left the molecular region, the role of q is only to change the intensity of the atomic like peak. This peak is formed due to a flat long-distance contribution to W (q) (Figs. 28 2 and 29) which damps as exp(!(R!R )C/v(R)) (as well as the molecular contribution does). The M front of this contribution *R"v(R)q propagates with the velocity v(R). So we have the following times characterizing the relaxation of the wave packet W (q): The 2 molecular contribution to W (q) is formed during the time of propagation, q "(R !R )/v, into 2 K B M the dissociative region R . After this time the dissociative contribution begins to shape during the B lifetime C\. An important subsequent question refers to the role of the RXS duration ¹ in the formation of W (R) and in the time evolution of W (q). The effect of ¹ is seen directly from the 2 2 dependence of the space distribution of W (q) on the excitation energy. Both wave packets W (q) 2 2 (Figs. 28 and 29) and W (R) (Fig. 30) show the suppression of the amplitude of the long distance 2 contribution (for the dissociative core excited state) when the RXS duration decreases. 4.7. Core hole-induced relaxation versus RXS duration The electronic subsystem evidently does not relax instantaneously upon the creation of a core hole, and there is a “relaxation time” q associated to the process. A quantification of this relaxation  time for core excited states was introduced by Hayes [115] and by Manne [116]. Both authors connect q with the inverse second moment *E, or the effective width of the corresponding core 

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Fig. 30. The absolute value of the stationary wave packet W (R) (39) for different excitation energies. The dissociative 2 part of W (R) decreases faster than the molecular contribution when the RXS duration decreases. Cl L (2p\pH) 2 dissociative core excited state in the HCl molecule. C"0.045 eV. The RXS durations are: "¹"K14.6 fs, 0.3 fs for detuning X"0, 2 and 4 eV, respectively. The wave packet for X"0 eV is increased by 6 times.

level photoelectron band q K1/*E , (58)  which is a somewhat different definition than given previously. Since *E is a large quantity (*EK54.3 eV for the neon 1s photoelectron band) the relaxation time is short (q K10\s  [115,116]). So if the incident photon frequency is tuned below the photoabsorption threshold X(!*E the duration time is shorter than the relaxation time, q (q . In this case the electronic   subsystem has no time to relax due to the extremely fast RXS and hence one can expect that RXS is not influenced by the relaxation effect. Eq. (39) confirms this expectation since for large values of detuning the main contribution to the integral over time (39) originates from t"0: 1 f "W (0)2 1 f "PRP"o2 2 " . (59) F " D X#iC X#iC G G This equation corresponds to a sudden quadrupole transition (tK0) from the ground state to the final state. 4.8. Nuclear dynamics versus RXS duration Several characteristic time scales are responsible for the dynamics of the nuclear (or vibrational) wave packet on the potential surface for the core excited state. The first one is the period q of 
vibrations in the bound core excited state. The second one is the time of deformation of the initial nuclear wave packet q . This time period is defined by the difference of the nuclear potential  surfaces of the ground and core excited states [104]. It is easy to understand that both these time periods have the same order of magnitude, namely D\, where D is the vibrational broadening of 

the photoabsorption line. The time of propagation q of the nuclei between the absorption point 

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R and the emission point R is yet another dynamic characteristic which is important in the case of core excitation above the dissociation threshold or to a dissociative state. These characteristic times can be collected as



0Y dR , v(R) 0 where v(R) is the relative velocity of the nuclei at the point R. When the RXS duration time is smaller than these characteristic times q &q &D\, q " 
 


(60)

q ;q ,q ,q , (61)  
  the RXS process can be considered as a sudden process from the point of view of the nuclear degrees of freedom. Under condition (61) the nuclei have no time to change their conformation during the scattering process. Eq. (59) applied to this vibrational problem leads, in the Born— Oppenheimer (BO) approximation, to the following result [117,104]: 1n"W(0)2 1n"o2 F "a "a , a"u u DR D . (62) D GD GM DG GM X#iC X#iC G G This equation coincides with the results obtained by Cesar et al. [61] in the short lifetime limit C ;D . Here X"u!u is the detuning from the adiabatic frequency of the considered G 
GM electronic transition and "o2 and "n2 are vibrational states of the ground and final states, respectively. Now t(t) (39) is the nuclear wave packet. Expression (62) shows that the scattering amplitude of fast RXS is proportional to the Franck—Condon (FC) factor 1n "o2 between vibraD tional wave functions of the ground and the core excited states. This agrees with the FC principle applied to the sudden RXS process. It is important to note that the fast limiting cases (59) and (62) are direct consequences of the channel interference. This means also that the concept of a duration time is very closely connected to the interference effect, an effect which is manifested in many other ways in RXS spectra, as reviewed in some detail in the section following the next.

5. Polarization features It is well known that the spectral shape of ordinary non-resonant X-ray emission of ordered samples depends on the orientation of the emission polarization vector e relative to the sample through the scalar product of this vector and the transition dipole moment vector, e ) d . This was DG first observed in connection with measurements on graphite [118]. The polarization of ordinary nonresonant X-ray emission can be understood as being due to the alignment of the molecular moieties in the crystalline samples. Utilizing the primitive two-step model [12], the spectral shape of X-ray emission from randomly oriented samples, like gas phase atoms and molecules, is isotropic and does not depend on the absorption polarization vector e. This assumption connects to traditional theory of non-resonant X-ray emission in which the preparation of the core excited state is neglected. As shown early [119] and also more recently [120,46,121,95] the two-step assumption, holding for non-resonant X-ray emission, often breaks down for RXS because only molecules with

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certain space orientations are excited due to the spatial selective excitation. The alignment or orientation of core excited molecules [119] by the polarized incident X-ray beam thus leads to specific spectral and anisotropy features of the emitted photons [119,120,46,121,95,122] photoelectrons, and photoions [123—129,48,130—132]. 5.1. Qualitative picture of RXS by randomly oriented samples A qualitative picture of the polarization properties of radiative RXS can be obtained from a simple atomic system [133,134,100], containing three levels with non-degenerate ground and final S states and a triply degenerate intermediate P state. 5.1.1. Linearly polarized incident X-ray photons When the incident light beam is linearly polarized only absorption transitions to the core excited state with m"0 are allowed according to dipole selection rules (Fig. 31), leading to a nonequilibrium distribution over magnetic sublevels. The z-axis is here parallel to the incident polarization vector e. One can consider the process as a state selective excitation in angular momentum space, since only core excited states with m"0 are populated. The gas of core excited atoms (and the gas as a whole) obtains quadrupole electric momentum or alignment [134,100] directed along e. Because the p -atomic orbital (AO) of the core excited state is directed along e, the X 0—0 emission is linearly polarized with e primarlily along e . 5.1.2. Circularly polarized incident X-ray photons RXS of circularly polarized light differs qualitatively from the previous case. Only states with m"!1 (m"#1) are populated following photoabsorption of left (right) polarized radiation (Fig. 31) (the z-axis is here parallel to wave vector k). A state selective excitation in angular momentum space thus also takes place in this case, with a nonequilibrium distribution over magnetic sublevels due to the transfer of angular momentum from the left (right) photon to the atom. The gas of core excited atoms (and the gas as a whole) has now permanent magnetic

Fig. 31. Polarization of X-ray scattered radiation by atoms.

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momentum or orientation. Because only m"!1 (m"#1) core excited states are populated, the emitted X-ray photons have left (right) circular polarization (Fig. 31). 5.1.3. Polarization and anisotropy of RXS by molecules The molecular field splits atomic levels, eliminating the spherical symmetry of the atomic wave function, and causes qualitatively different anisotropic and polarization features in the RXS process [119,135,46,95,122]. To understand the origin of the anisotropy of polarization of molecular RXS it instructive to consider K-fluorescence by a simple diatomic molecule like CO. With tunable frequency the incident X-ray radiation can in principle select unoccupied MO levels of certain symmetries. For example, employing dipole excitation with a transition matrix element e ) D , molecules mainly aligned parallel to e are excited when the frequency is tuned in resonance MG with an unoccupied p orbital, see Fig. 32. In other words, the incoming radiation transforms an initially unordered molecular gas to a spatially aligned gas of core excited molecules. This space ordered gas emits X-ray photons with polarization in certain directions depending on the symmetry of the occupied MO involved in the RXS process (see Fig. 32) [119]. Indeed, the polarization vector e of X-ray fluorescence is mainly parallel to the incoming X-ray photon polarization e if this occupied molecular orbital (MO) has p symmetry. When the occupied MO has p symmetry, the final X-ray photons are polarized mainly perpendicular to e (Fig. 32). The picture becomes the inverse if the frequency of the initial photon is tuned in resonance with a p-unoccupied MO (Fig. 32). So one can understand that the polarization measurements of RXS are very sensitive to the symmetry of molecular orbitals involved in the scattering process, and that the RXS spectral shape strongly depends on the polarization vectors of incident and scattered X-ray photons. The same polarization effect takes place for core excitation above the ionization threshold (I) in the vicinity of the shape resonances [119,136,137]. 5.1.4. Core excitation into the far continuum. “Atomic” limit When the excitation energy essentially exceeds the ionization potential, the physical picture of the discussed phenomenon is qualitatively different from the discrete resonant case. In this case the

Fig. 32. Space and symmetry selective resonant X-ray scattering by gas phase molecules.

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molecular field and the wave function are very close to being spherically symmetric. The oscillatory strengths for photoexcitation into continuum states of p ,p and p symmetry are the V W same (again using the diatomic molecule considered above as an example). This means that molecules of all orientations are core excited with the same probability, and that the emission occurs from a totally disordered ensemble of core excited molecules (see Fig. 32). As is the case for atoms, the spectral shape of K-emission does not depend on the orientations of the polarization vectors of the incident and final photons. It is necessary to note that some exceptions to this rule are possible (see for example RIXS scattering through L-shells discussed in Section 6.1). 5.2. Quantitative theory of RIXS polarization of free molecules The non-resonant contributions to the RIXS cross section (5) are negligibly small in the vicinity of a photoabsorption threshold, and can be omitted in the present context. For samples in the gas phase it is necessary to average the RIXS cross section (16) over all molecular orientations; eHe e eH"  +d d (4!a!b)#d d (4b!a!1)#d d (4a!b!1),,  GH IJ GI HJ GJ HI G H I J a""(eH ) e)", b""(e ) e)" .

(63)

The complex unit vectors of elliptical polarization e and e can be represented as sums of two orthogonal unit vectors of linear polarization x#iyi x#iyi , e" , (x ) y)"(x ) y)"0, x#y"x#y"1 . e" (1#i) (1#i)

(64)

The ellipticities of the incident and final light beams are characterized by the parameters i (!14i41) and i (!14i41). If the general theory is applied to systems without interference of different scattering channels, the averaged cross section (5) and (16) of K-shell scattering for linear polarized incident and final photons looks like p(u,u)"ruu "D ""D "f (h)D(u!u ,C )U(u!u#u ,c) . M DG GM DG GM G DM GD The function f (h)"  [2!cos h#cos u (3cos h!1)]"(1#R (3cos h!1)) DG  DG DG 

(65)

(66)

shows the dependence of the RIXS cross section on the angle h between polarization vectors e and e of incident and scattered photons and on the angle u "arccos(D ) D ) between the emission DG DG GM and absorption oscillators. Both the polarization P and the polarization anisotropy R 3R p(u,u) °!p(u,u) ° p(u,u) °!p(u,u) °  "    , R" P" p(u,u) °#p(u,u) ° 2#R p(u,u) °#2p(u,u) °    

(67)

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are used to describe results of polarization measurements [138—140]. If the overlap of the RXS resonances is small Eqs. (65) and (67) reduce to 3cos u !1 DG , R"R ,(3cos u !1) . (68) P" DG  DG cos u #3 DG Eq. (66) describes experiments which measure the angle h between polarization vectors, as is the case using the polarization selective spectrometers described in [46,121]. A different type of measurement is carried out in the soft X-ray region when the spectrometer is not polarization selective [68]. In such experiments only the angle s between the polarization vector e of incident photons and the propagation direction p of final photons is measured. Averaging of Eq. (65) over all polarization directions of emission photons yields the following result for the f -function (66) DG (69) f (s)"  (2!R #R (3cos s!1)), cos hPcos s . DG DG  DG  Eq. (68) is valid only when the RIXS resonances do not overlap. Very often the photoabsorption and RIXS lines are close lying and the more general Eq. (67) must be used. Because of this fact, the polarization and the polarization anisotropy depend on the excitation energy. It can be noted that the integral RIXS cross section coincides with the scattering cross section at the so-called “magic” angle h (or s ) K K 1 K54.74° . (70) h "s "arccos K K (3 5.3. Point group symmetry and polarization anisotropy It is possible to give a general symmetry formulation for the RIXS cross section of randomly oriented molecules which applies to any type of polarization for the absorbed and emitted photons, i.e. linear, circular, or elliptical polarization, see recent work of Luo et al. [141]. This formulation can be used to derive symmetry properties of the molecular levels participating in RIXS under any given experimental situation. As shown in Ref. [141] the information concerning polarization, molecular orientation, and intrinsic molecular properties can be separated in three different factors, leading to a tabulation relating symmetries and polarization ratios that cover all 32 molecular point groups and the 2 groups of linear molecules; this table is here recapitulated as Table 2. The theory has been applied to molecular systems as different as diatomic molecules [142,143], substituted benzenes [144,143] and fullerenes [145,143]. We consider the following RXS process. An incoming X-ray photon excites a core electron (c) into an unoccupied MO t . This core excited state decays as a result of a spontaneous transition J from the occupied MO t to the core shell t . As a prerequisite for the the general formulation it is H A necessary to connect the dipole moments d and d in terms of laboratory coordinates axes in JA AH which the polarization vectors and the photon directions are expressed. This can be accomplished by a directional cosine transformation ¹6"¹Kt #¹Et #¹Dt "¹?t , where the repeated A A K6 A E6 A D6 A ?6 Greek index a implies summation over the values m, g and f. t is the cosine of the angle between K6 the m axis of the molecular coordinate system and the X axis of the laboratory coordinate system. The eight other directional cosines t , t , t ,2 are named similarly; the general direction cosine K7 K8 E6

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t has a Greek letter for the molecular axis and a capital italic letter for the laboratory axis, see Fig. ? 130. Putting this into the Eq. (7) and taking into account Eq. (9) one obtains d@ t eHdA (l)t e F " u u (l) JA @  AH A JH JA HA u!u (l)#iC HA A or rearranging as

(71)

d@ dA (l) JA AH F "(t t )(eHe ) u u (l) "(t t )(eHe )F@A , (72) JH @ A  JA HA u!u (l)#iC @ A  JH HA A where we denote F@A JH d@ dA (l) JA AH (73) F@A" u u (l) JH JA HA u!u (l)#iC HA A as the RIXS transition element. We use the following notations: u "E(c\l)!E , u "u (l)" JA M HA HA E(c\l)!E( j\l), u "E( j\l)!E , d "10"d"c\l2, and d "d (l)"1c\l"d" j\l2 are resJH M JA AH AH onant frequencies and dipole matrix elements of X-ray absorption (cPl) and emission (jPc) transitions, respectively. It can be seen that F@A is a tensor of second rank. The quantity "F " is written as JH JH "F ""(eHe e eH)[t t t t ](F@AFMNH) . (74) JH  0 1 @ A M0 N1 JH JH All polarization information is collected in the first factor, all molecular information in the last factor, and all orientational information in the middle factor. Both the first and last factors are Cartesian tensors of fourth rank. For molecules with fixed orientation, such as surface adsorbates, one can use two of these factors to derive information on the third, e.g. knowing the polarization and the sample orientation, the RIXS spectrum allows a symmetry assignment of electronic states, or vice versa, from the knowledge of the polarization of radiation and the symmetries of the states involved in RIXS, the orientation of the sample can be derived. In the case of gas-phase samples, treated here, averaging must be employed. Only the orientational factor needs averaging; the polarization and symmetry factors are independent of orientation. A detailed description for the averaging procedure can be found in the work of Monson and McClain [146]. It gives 1t t t t 2"(1/30)[d d (4d d !d d !d d ) @ A M0 N1  01 @A MN @M AN @N AM #d d (!d d #4d d !d d ) 0 1 @A MN @M AN @N AM #d d (!d d !d d #4d d )] . (75) 1 0 @A MN @M AN @N AM Inserting Eq. (75) into Eq. (74) and contracting over Greek and italic indices, the following orientational average is obtained: 1"F "2"(1/30)[eHe e eH(4F@@FAAH!F@AF@AH!F@AFA@H) JH   JH JH JH JH JH JH # eHe e eH(!F@@FAAH#4F@AF@AH!F@AFA@H)   JH JH JH JH JH JH # eHeHe eH(!F@@FAAH#4F@AF@AH!F@AFA@H) .   JH JH JH JL JH JH

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This can be written as 1"F "2"j "Fj$ #Gj% #Hj& JH JH JH JH JH

(77)

j$ " F@@ FAAH , JH JH JH @ A

(78)

j% " F@AF@AH , JH JH JL @A

(79)

with

j& " F@AFA@H . JH JH JH @A Here the summation is put back and

(80)

F"!"e ) eH"#4"e ) e"!1 ,

(81)

G"!"e ) eH"!"e ) e"#4 ,

(82)

H"4"e ) eH"!"e ) e"!1 .

(83)

The averaged cross section is then given by u 1p(u,u)2"r j U(u#u !u,c) M u JH JH JH u "r (Fj$ #Gj% #Hj& )U(u#u !u,c) . (84) M u JH JH JH JH JH The preceding formulas reveal the strong dependence of the RIXS cross section on the polarization vectors of absorbed and emitted photons, and of the symmetries of the unoccupied and occupied MOs. Eq. (84) is perfectly general for photons of any polarization (linear-, circular-, or elliptical-). It can be noticed that Eq. (77) has the same form as the one for the optical two-photon transitions [146], but has different physical meaning. 5.3.1. Polarization dependence — general The molecular parameters j$ , j% and j& are dependent on the symmetries of the unoccupied, t , JH JH JH J and occupied, t , MOs. For orbitals with different symmetries, the behavior of the polarization H dependence will not be the same. Polarized RIXS spectroscopy thus provides a useful tool for assignments of the occupied and unoccupied MOs. The polarization vectors may be real, representing linearly polarized photons, or complex, representing circularly or elliptically polarized photons. If we denote the angle between the linear polarization vectors of absorbed and emitted photons by h, the square of the scattering amplitudes will be expressed as j "(!j$ #4j% !j& )#(3j$ !2j% #3j& ) cos h . (85) JH JH JH JH JH JH JH Such a formula is an exact prediction for the linear polarization behavior of the RIXS cross section. However, as shown in Eq. (85) the three molecular parameters j$, j% and j& cannot be

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independently determined by using only linear polarization vectors; circular polarization is also required. The polarization parameter P is defined as the ratio between different RIXS cross sections for different combinations of photon polarization vectors, and can be directly measured experimentally. Many different combinations of polarization vectors of absorbed and emitted photons can be formed. However, basically one needs at least three of these combinations, two of them involving the circularly polarization vectors. We label the RIXS cross sections as j (lp), j (ln) and JH JH j (cp) for two absorbed and emitted photons having parallel linear, perpendicular linear and JH circular polarization, respectively. The latter can be arbitrarily clockwise or counterclockwise. The polarization ratios can then be written as j (lj) !j$ #4j% !j& JH JH JH , (86) P " JH "  j (lp) 2j$ #2j% #2j& JH JH JH JH j (cp) !2j$ #3j% #3j& JH JH JH , (87) P " JH "  j (lp) 2j$ #2j% #2j& JH JH JH JH j (cp) !2j$ #3j% #3j& JH JH JH . P " JH " (88)  j (ln) !j$ #4j% !j& JH JH JH JH By measuring the polarization ratios, P , P and P , it is possible to assign the symmetries of    occupied or unoccupied MOs. Table 2 (see Section 9.6) collects the information concerning symmetry assignments of electronic states of molecules for any point group in terms of the polarization ratios. 5.3.2. Polarization dependence — isolated core holes An important special case is when there is only a single core orbital involved in the absorption (cPl). The symmetry assignments can then be obtained without knowledge of circular polarization vectors, and the relations for the j factors are the considerably simplified [141], in particular one has j$"j&. If u is the angle between the dipole moments d and d (l), then JA AH 2!cos u j (ln) . (89) P " JH "  j (lp) 2 cos u#1 JH The angle u is equal to zero when the occupied and unoccupied MOs have the same symmetries, then P ". When transitions to MOs of xy, yz, or zx symmetry, u"p/2, we have P "2. This    shows the possibility of making these symmetry assignments without using circular polarization vectors, as for instance exemplified in the investigation of the K-shell RIXS spectrum of the H S  molecule in Section 5.4.2. The use of Table 2 to derive symmetry of occupied or unoccupied MO levels is further commented in Section 9.6, and is exemplified in Section 9.6.1 for the assignment the occupied orbitals of benzene. 5.4. Measurements of polarization anisotropy and angular distributions Experimental measurements of polarization of RIXS by gas-phase molecules began in 1988 [147]. State sensitive RIXS anisotropy has been observed in the hard X-ray region (beamline X-24A of NSLS, polarization-selective spectrometer) for the CH Cl [147], H S [139], CF Cl [71],   

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CH Cl, CF Cl, CF Cl and CFCl [120] molecules (see also corresponding review papers by      Cowan [46] and Southworth [121]), and in the soft X-ray region (beamline 7.0 of ALS) angular distributions with a non polarization-selective spectrometer for the following molecules; C [148],  C [145], CO [68] and CO [149]. Contrary to other gas-phase molecules the C and    C samples were obtained as disordered films made in situ by vacuum evaporation on a surface.  Three of these experimental results are briefly reviewed below. 5.4.1. CF Cl-polarization anisotropy  Fig. 33 shows the Cl K-RIXS spectra by Lindle et al. [120] from CF Cl for emission photons  polarized parallel (h"0°) and perpendicular (h"90°) to the incident-beam polarization direction and with the incident-beam energy tuned to the major Cl K subthreshold resonance at u"2823.5 eV. This energy corresponds to the core excitation to the 11a antibonding MO. The assignment of 

Fig. 33. Cl K RIXS spectra from CF Cl following Cl 1sP11a excitation with 2823.5 eV photon energy[120]. The labels   "" and N refer to parallel and orthogonal oreintations of the measured fluorescence polarization relative to the incident e-vector. The two spectra have been scaled so that the areas of peak C are identical. The peak near 2823 eV is due to elastic scattering of the incident radiation.

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the emission peaks is the following: C(7e), B(10a and 5e) and A(9a and 4e). The experimental   scheme was described in Section 2.2 (see also Fig. 6). Results of the measurements presented in Fig. 33 indicate the strong dependence of the RIXS spectral profile on the final photon polarization. For excitation above the Cl K ionization threshold, u"2880 eV, the spectral shapes of the emission profile for the two polarization angles were identical in accordance with the theoretical prediction [119]. 5.4.2. H S-polarization anisotropy  Fig. 34 shows the SK-RIXS spectra from H S [139] for emitted photons polarized parallel  (h"0°) and perpendicular (h"90°) to the incident beam polarization direction and with the incident beam energy tuned to the major subthreshold resonance. The main features in the RIXS spectrum for H S are understood as peaks labeled A, B, and C corresponding to the 2b P1a ,    5a P1a , and 2b P1a transitions, respectively, where 1a is the SK-shell MO. The highest      energy peak in Fig. 34 is due to the elastically scattered beam and is suppressed as expected when the polarizer is tuned to h"90°. It should be evident from the experimental RIXS spectrum that the relative intensities of the emission peaks labeled C (A and B) are suppressed (enhanced) when the spectrometer is aligned to detect h"90° polarization relative to the h"0° polarization. The major X-ray photoabsorption line consists of two electronic transitions: a low-energy and strong contribution 1a P3b and a weak transition 1a P6a . So one can define the directions of the     absorption D and emission D dipole moments: According to the assignments of occupied and GM DM unoccupied MOs the emission dipole moments for peaks A(2b P1a ), B(5a P1a ), and     C(2b P1a ) point along the x-, z-, and y-axis, respectively. The 1a P3b and 1a P6a absorp      tion dipole moments point along the y and z axes, respectively (it is assumed here that the H S  molecule lies in the yz plane with the H-bond along the y-axis). Eq. (68) yields the following values of the polarization P "!, P "!, and P " for the strongest 1a P3b transition, while  !      P "!, P ", and P "! for the weaker 1a P6a transition. It is seen in Fig. 35 that the  !      polarization measurement made using excitation energies at or below the peak of the absorption resonance are consistent with the assumption of a 1a P3b excitation. However, as the excitation   energy is increased to the high-energy side of the absorption maximum, the emission peak polarizations vary in a way that is consistent with the additional presence of the 1a P6a   excitation. That is, P remains large and negative, P becomes less negative, and P becomes less  ! positive. So this example shows that polarization RIXS measurements can be used for the assignment of MOs. A recent confirmation of this statement is given by the work of Miyanao et al. [150] in which the resonant SKb X-ray-emission spectra of OCS at the SK threshold were reported. Significant variations in line shape and energy position, as well as substantial polarization dependence, were observed and interpreted. 5.4.3. CO-angular distributions The angular dependence of the X-ray emission from scattering theory applied to randomly oriented samples is obtained as [122] (see Section 5.2) I(s)"I (s)#I (s) , ,  I (s)"I [1!R] , ,   I (s)"I [1#R(3 sin s!1)] ,   

(90)

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Fig. 34. Sulphur K RIXS from H S with the polarimeter set to detect (a) parallel (h"0°) or (b) perpendicular (h"90°) @  polarized X-rays with respect to the incident-beam polarization. The solid curves are fits using Voigt peaks. The inset shows absorption near threshold. The arrow denotes incident-beam energy, on resonance. The extra markings on the absorption resonance denote the excitation energies used to measure the K polarizations in Fig. 35 [139]. @

where s is the angle between the polarization vector of the incident photon and the propagation direction of the outgoing photon, and R is the anisotropy parameter. I is proportional to the total  intensity emitted in all directions and summed over all polarization vectors, and R is the polarization anisotropy parameter for each final state. The notations N and "" refer to the polarization direction of the emitted photon relative to a plane which is perpendicular to the

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Fig. 35. Polarization of peaks A, B and C in Fig. 34 following varitation of frequency close to the first K absorption resonance of H S [139]. 

Poynting vector of the incoming radiation. With a spectrometer which is not polarization selective it is not possible to measure I or I directly, but it is still possible to measure the total intensity I(s) ,  as a function of emission angle s. In the investigation on CO reported in Ref. [69] RXE spectra were measured at s"0° and 90°. Here r"I(90°)/I(0°)"(2#R)/(2!2R) .

(91)

For a molecule like CO in which only one core-excited electronic state at a time is involved in the process, the full scattering model including electronic interference effects and a two-step model, where the excitation and emission steps are separated from each other, give the same result. Using the group Table 2 combined with the general formula for RXE transition elements in orientationally averaged molecules (84) the following R values for different final states of a C molecule T with R> ground state symmetry is obtained RP"! , RR>";   RR\"! . (92) RD"  ;   In the case of resonant excitation to a p unoccupied level the 5p and 1n bands correspond to R> and P final states, respectively. The r values for the 5p and 1n bands are given by rN "2; rN " . N L  For excitation to a n unoccupied level those r values are given by rL ", N 

rL " L 

(93)

(94)

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(in the latter case with D and R\ final multiplet states included, R> excluded [69]). The simulated spectra with these values for the angular distributions were found to reproduce the experimental CO spectra well [69].

6. Interference effects in resonant X-ray Raman scattering So far this review has concerned the main consequences of two resonant features of the RXS process, namely the energy conservation law — or Raman-Stokes law — (9), (25) and the condition for ordinary emission resonances (24). Some of the fundamental properties of RXS could be understood in the previous sections without prior knowledge of the structure of the scattering amplitude. It is evident, however, that a detailed description of the spectral peculiarities of X-ray scattering must account for the actual structure of the amplitude, as will be reviewed in the following. The focus in this section is put on the fact that when the intermediate core excited states (Fig. 1) are close-lying, the RXS process is strongly influenced by the interference among the several scattering channels. Channel interference seems to be the most important as well as the most conspicuous concept associated with the X-ray Raman process. 6.1. Direct and interference contributions. Coherence between core-excited states A common assumption, and a suitable starting point for elucidating the role of interference, is that the RXS cross sections are dominated by the resonant contribution. This holds, for example, in the case of radiative inelastic RXS in the soft X-ray region with a negligibly small Thomson—Waller term (8). One finds even in this simplified case that the RXS band profiles (16), (7) and (12), (13) differ strongly from the pattern that would be formed by a superposition of a set of displaced Lorentzians, characteristic for the ordinary theory of X-ray emission and the Auger effect [151,40,44,12]. In contrast to ordinary emission, the shape of the RXS bands is given by a sum p(u,u)"p (u,u)#p (u,u)   

(95)

of direct p (u,u) and interference p (u,u) terms    u p (u,u)"r "F (u)"U(u#u !u,c) , DG DM   Mu D G u p (u,u)"r Re F (u)FH (u)U(u#u !u,c) . (96)  Mu DM DG DG  D G $G Here the index of the core excited atom n (see Eq. (7)) is included for brevity in the index i. As seen from these equations the interference terms have the same order of magnitude as the direct terms when the energy difference u  of the intermediate core-excited states i and i does not exceed  GG (C#X "u "((C#X . GG G

(97)

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Here X is the detuning of the excitation energy from the photoabsorption band. A requirement for interference between the scattering channels is that there are two or more degenerate or quasidegenerate intermediate core excited states that can be excited coherently, and which can decay to the same final state. As reviewed in this section, interference effects strongly influence the spectral shape of RXS. The coherence among different core states simultaneously excited by the same X-ray photon is described by an off-diagonal matrix element of the density matrix (i Oi) [100] 






G G 1 1 . "eiSGGR GM MG ! , GG X  #iC  u #iC  X #iC G GM GM G GG GG

1 G " ED GM 2 GM

(98)

with C "(C #C )/2 and E as the electric field strength of the incoming radiation. Here, the Rabi G G GG frequency G gives the rate of absorption oPi. Coherence and interference between the coreGM excited states i and i is small when the energy gap u  between them satisfies the condition of Eq.  GG (97). The interference effects can be divided into two different classes. The first type is connected with close lying core-excited electronic states, while the second type originates in the coherence between sublevels — e.g. vibrational sublevels — of the intermediate states induced by the incident X-ray photons. The intermediate interferring states can belong also to a continuous spectrum, as is the case for solids, for excitation above ionization threshold or when core excitation refers to dissociative states. The interference between direct and resonant channels in non-radiative RXS (resonant photoemission) (12), (13) is also a well-known effect. The role of interference in all these cases can be very significant, and associated with important physical consequences which are discussed here and in the next sections. Condition (97) defines the situation for which the simplest two-step picture — the cross section p (u,u) (96) — breaks down and for which the correct   description is given by the one-step scattering model (cross section p(u,u) (95)). In this section we review some aspects of RXS that are particularly dependent on the channel interference effect. However, it is emphasized that being a central concept for RXS it is manifested, one way or the other, in most processes connected with RXS. Thus, for example, it is central in the description of elastic scattering (Section 7), symmetry selection and restoration (Section 9), and the Doppler effects (Section 14), and this is also the case for the concept of the RXS duration time, reviewed already in Section 4. 6.2. One-versus two-step models for the polarization anisotropy and the total cross section From the foregoing description it is clear that channel interference may lead to a strong dependence of the shape of the X-ray fluorescence spectrum on the frequency u and on the polarization vectors of initial and final photons. These predictions were quantified in [122] in terms of the one- and two-step formulations for the polarization anisotropy. A set of molecules were used to illustrate the following four different situations: 1. When the two-step model is applicable for both the unpolarized intensity I and the polariza tion anisotropy R; 2. When the two-step model is applicable for I but not for R; 

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3. When the two-step model is applicable for R but not for I ;  4. When the two-step model can not be applied neither for I nor for R.  Case (i): The first case covers the large class of molecules with energy isolated and symmetry non-adapted core hole states. For this case the classical formula for the polarization anisotropy is also applicable [139,71,120] and is identical to the two-step model. The analysis then becomes much simpler than in the other three cases. In case (i) (as well as in case (iii) below) the symmetry assignments can be made using information from linearly polarized light only, while the other cases in principle require circularly polarized light to achieve that goal [141]. Calculations in case (i) lead to simple rational numbers for the polarization anisotropy depending only on symmetry, and do not require prior knowledge of transition energies and core state lifetimes as is the case when interference effects are important. Case (ii): Cases (ii) and (iii) represent large sets of common molecules, for which the two-step model is found inadequate. Case (ii) represents the case when the X-ray absorption involves energy near-degenerate and symmetry adapted core orbitals. This leads to interference for the polarization features of the RXS experiment, but not necessarily for the total cross section; this depends on the actual symmetry point group. In a C molecule like CF Cl with the two Cl 1s core orbitals having T   different symmetries, the contribution from the interference to the total cross sections is absent, while it is non-absent for the polarization. The two-step model is thus adequate for K-emission of CF Cl but not for CF Cl .    Case (iii): This case represents many organic molecules, e.g. hyrdrocarbons, for which equal atoms have different chemical environments, i.e. when they are not symmetry related, and when the core ionization or excitation leads to small chemical shifts. Since these centers transform according to the totally symmetric representation, the two-step model can describe the polarization anisotropy, while in general there is a strong contribution from interference to the total cross section. This is nicely illustrated by benzene and the mono-substituted benzenes [122]; the smaller the chemical shifts among the ring carbons, the stronger is the interference of the RIXS transitions, and the more benzene-like is the outcome in the form of a spectrum. As shown in [122] the reason for the two-step model to be valid for I but not for R in case (ii), and vice versa for case (iii), could be  derived from the equivalence or inequivalence of the two-step model with the general one-step formulas for the cross section and polarization anisotropy, respectively. Case (iv). The final case represents a smaller class of molecules, like C , but are nevertheless  interesting from the point of view of the symmetry and parity selective character that they exhibit for the X-ray scattering process. It can be noted that it is just those molecules for which the symmetry and parity selective character of the X-ray scattering is most relevant, which are the most complex with respect to the interference effects and that require a full one-step implementation for both the polarization anisotropy and the integrated unpolarized cross sections. The simulated polarization features of C are shown in Fig. 36 [122]. As discussed in Sections 9 and 10 the  symmetry elements may be broken by vibronic (Jahn—Teller) coupling of the intermediate core excited states, as indeed seems to happen for core-excited C , but also that symmetry selection can  be restored by detuning the frequency. The findings reviewed above for randomly oriented species, are also of some relevance for the use of the polarization anisotropy for orientational probing of fixed, oriented molecules. Surface adsorbates are systems with fixed or partially fixed orientations that, when physisorbed, have the

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Fig. 36. The unpolarized intensity I and polarization anisotropy parameter R for the C molecule. (a) I and (b)    R calculated by one-step (solid line) and two-step (dashed line) models [122].

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simplicity in the organization of the unoccupied levels as that free molecules do, and should be relevant samples for measurements of polarization anisotropies, see Section 17. More complex molecules and solids have larger density of unoccupied states, which leads to stronger overlap of the core-excited resonances. This implies that interference will be more important also in case (i) despite that the bare core holes are isolated in energy. For the higher core-excited states there is little symmetry and polarization selectivity and the spectra turn progressively into their broadband excitation — and non-resonant analogues. It is therefore more difficult to extract symmetry information and to find simple relations between polarization ratios and symmetry or geometric properties for such samples. Solids forming energy bands are treated in Section 16. 6.3. Role of spin—orbit splitting for channel interference and polarization There can be several physical reasons behind the appearance of small splittings among core excited states which cause interference in the RXS spectra; open-shell electrostatic interaction leading to multiplet splitting, vibronic excitations, chemical shifts, spin—orbit and other splittings due to various relativistic interaction terms in the Hamiltonian. The common case of lifetimevibrational interference is discussed in Section 6.4; in this section we consider channel interference effects due to spin—orbit splitting. The ¸ -fluorescence of an atom (A) excited by an X-ray photon ''''' of a frequency (u) resonant with the unoccupied level nS constitutes a simple system for which channel interference due to the spin—orbit splitting appears. Due to this splitting (D) of the ¸-shell there exists two interfering inelastic scattering channels for emission from a closed nS-shell AH  PAH#c . AH  The RIXS cross section (95) of process (99) reads




c#AP

p(u,u)"p [(11#a#11b)"¸ "#(11#a#56b)"¸ " , M   # 2(34b!a!11)Re(¸ ¸H )]U(u#u !u,c) .   DM

(99)

(100)

Here



 2r p " M D D uu, D " dr rR (r)R (r) , (101) LQ M 5 ) 3 LQ LYQ N LQ  and u "u #D are the resonant frequences of emis¸ "(u!u #ιC )\ (i",), u    G GM H  sion transitions nSPp and nSPp , respectively; R (r) R (r) are the radial wave functions of the   LQ N ns and p electrons. To illustrate the polarization features of atomic RXS with this example two important special cases for applications of linear and circular polarized X-ray radiation are considered. 6.3.1. Linearly polarized incident and emitted radiation As one can see from Eqs. (63) and (100) X-ray emission is in the general case elliptically polarized when the incident X-ray beam has linear polarization. The X-ray spectrometer can select a certain linear polarization of final photons (see Section 2.2). It is thus relevant to consider the case when both incident and final X-ray photons have linear polarization, i.e. when a"b"cos h, where h is the angle between e and e. It is convenient to present the corresponding expression for the RXS

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Fig. 37. Polarization anisotropy R (103) and anisotropy ratio g (109) with interference (solid lines) and without interference (dashed lines).

cross section (100) as a sum of isotropic and anisotropic contributions p(u,u)"p(u,u)(1#R(u,h)), R(u,h)"R(3 cos h!1) .

(102)

The polarization anisotropy (67) 4"¸ "#19"¸ "#22Re(¸ ¸H )    R"  (103) 15("¸ "#2"¸ ")   shows that the degree of polarization of emission radiation depends on the frequency u (see Fig. 37a and b). Here p(u,u)"15p ("¸ "#2"¸ ")U(u#u !u,c) (104) M   DM is the isotropic RXS cross section which corresponds to scattering at the “magic” angle h (70). The K isotropic cross-section (104) is equal to the one averaged over scattering angles h. The anisotropy of polarization is also characterized by the parameter P defined by Eq. (67). The anisotropy of X-ray Raman scattering is strongly influenced by the interference term Re(¸ ¸H ) if the lifetime   broadening or detuning u!u is larger than the spin—orbit splitting D of the ¸ -shell [152] (see G ''''' Fig. 37a and b). The importance of the interference term Re(¸ ¸H ) can be seen in the limit   D/C ;1: R"1 with interference contribution and R" without interference. G  Eq. (100) demonstrates the possibility of controlling the phase of the interference contribution (last term at the right-hand side of Eq. (100)) by the polarizations of the incident and final photons. The interference term p (u,u)"6p Re(¸ ¸H )U(u#u !u,c)(3 cos h!1)  M   DM

(105)

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changes sign when the angle h between e and e goes through the magic angle (70). Because of this sign-changing polarization dependence, the interference contribution is equal to zero when the final photons from all directions and with all polarizations are collected by the spectrometer. Even X-ray scattering by this, simplest possible, system shows the principal role of the interference contribution. In the case of small spin—orbit splitting, the direct term (96) p (u,u)J22#69(e ) e) strongly differs from the total cross section p(u,u)J135(e ) e). For   example p(u,u)"0 while p (u,u)O0 for orthogonal polarizations eNe.   6.3.2. Circularly polarized radiation The RXS cross section for circular polarized initial and final X-ray photons depends on the phase shift between these waves. The cross section (100) averaged over these phases can, for the considered process, be presented as a sum of isotropic and anisotropic parts (102) but with a different meaning of the angle h. Now, h"arccos(L ) L)

(106)

is the angle between directions L"(x;y) and L"(x;y) of angular momenta of initial and final photons. The unit chiral vector L (L) is parallel or antiparallel to k (k) for right or left polarized radiation, respectively. The isotropic part p(u,u) coincides with the corresponding expression for linearly polarized light (104) while the anisotropic contribution (107) R(u,h)"R(3 cos h!1)#R cos h A  differs qualitatively from R(u,h)"R(3 cos h!1) (102). The value of R(u,h) averaged over h is equal to zero. However, contrary to (102), the polarization anisotropy R(u,h) is different from zero at the magic angle (70). The left or right incident X-ray photon transfers angular momenta to the gas, and the core excited gas now becomes chiral with a certain magnetic orientation, emitting X-ray photons of mostly a certain circular polarization (left or right) depending on the polarization of the incident photons. This strong correlation between polarizations of absorbed and emitted photons is described by the second term in Eq. (107) where "¸ "# "¸ "#7Re(¸ ¸H )   .   R "  A 3("¸ "#2"¸ ")  

(108)

6.3.3. Light-induced circular magnetic dichroism When the atomic gas is not a chiral target the circular polarization can be induced only by incident circularly polarized photons. The effect considered here can be understood as lightinduced circular magnetic dichroism (LICMD) or dichroism caused by the induced chirality of the core excited gas. This effect was recently investigated for molecules in Ref. [153]. A quantitative characteristic of LICMD is the anisotropy ratio (Fig. 37c and d). 4R 2(p (u,u)!p (u,u)) A * " (109) g" 0 2#R p (u,u)#p (u,u) 0 * which is given as a relative difference of the RXS cross sections for left p (u,u) and right p (u,u) * 0 circularly polarized emission photons. The importance of the ¸ -channel interference is ''''' displayed in the limit D/C ;1: g"2 with interference contribution and g" without interferG  ence, see also Fig. 37c and d.

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When the ground state molecule is chiral a natural circular magnetic dichroism (NCMD) is also possible. This effect has been investigated experimentally for RXS in solids [154,155] and theoretically for gas-phase molecules in [153]. Contrary to LICMD the natural magnetic dichroism takes place for a linearly polarized incident light beam. The NCMD effect is in practice measured as the difference of the RXS cross-sections for left and right circularly polarized incident X-ray photons. The NCMD signal is essentially smaller than the LICMD signal, and is investigated mainly by X-ray absorption measurements. It should also be noted that that the NCMD effect in magnetic solids is caused by the spin—orbit interaction as predicted by Erskine and Stern [156,157] (see also [158]). 6.4. Lifetime-vibrational interference Although core orbitals have traditionally been attributed a strictly non-bonding function owing to their localized and non-overlapping character, it is an established experimental fact that core ionized and excited molecular states are associated with vibrational structure. Monochromatized XPS spectra showed in the middle of the 1970s that core photoelectron bands can be asymmetric [159,160], and that at high resolution they even can exhibit fine structure. About the same time it was demonstrated that non-resonant X-ray spectra involving non-bonding valence orbitals show fine structure [33], which could only be described by potential energy surfaces of the core ionized states that are displaced from the corresponding ground state equilibria. The presence of excited vibrational levels in the core state entails interference effects of the RXS channels through these levels in cases for which the vibrational frequency is smaller or has the same order of magnitude as the lifetime width C [61,84,161—166]. The resonant term of the scattering amplitude (7) reads in the G BO approximation as follows: 1n"m21m"o2 . (110) F (u)"a D u!(E !E )#iC GK DL G K The constant a is different for radiative and non-radiative RXS (a is given by Eq. (62) for radiative RXS). Here "m2,E and "n2,E are the vibrational wave functions and the total molecular energies GK DL in the electro-vibrational intermediate "i,m2 and final " f,n2 states, respectively. For the sake of simplicity, the case of diatomic molecules is considered here with the additional assumption that only the lowest ground state vibrational level is populated. The generalization to the case of arbitrary temperatures and polyatomic molecules is doable and can be found in Refs. [61,164,104]. The condition of coherence for the nuclear wave functions in core hole decay is that the lifetime width of the short-lived intermediate state has to be comparable to the vibrational spacing. This condition is fulfilled for O1s hole states (2C K0.18 eV) of many oxygen containing G molecules, whereas for N1s (2C K0.13 eV), and C1s (2C K0.08 eV) core hole states these effects G G are not expected to be so dramatic because of the longer lifetimes. 6.4.1. Experimental observation of lifetime-vibrational interference effects Neeb et al. [165] investigated experimentally and theoretically the lifetime vibrational interference in several molecules by means of non-radiative RXS. The electronic participator decay as a function of the excitation energy within the bandwith of the core-excited intermediate state was studied using the X1B undulator beamline at NSLS. Among these molecules, O shows perhaps the 

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most dramatic interference effects, which can be understood by the fact that the lifetime broadening (CK0.18 eV) is only slightly larger than the vibrational spacing (0.16 eV). Fig. 38 shows the observed and calculated electron emission spectra produced in the decay of the 1p\1p P core S E S excited state to the 1p P ground state of the molecular ion. The theoretical simulations were E E carried out using the Rydberg—Klein—Rees (RKR) potentials [165] and a Gaussian spectral function with a FWHM of 0.5 eV. Neglect of the interference term (96) leads to spectra that have very little resemblance to the experimental results. For example, the lowest panel in Fig. 38 shows that the interference shifts the profile maxima as much as 4 eV.

Fig. 38. Experimental (triangles) and calculated (solid line) decay spectra of the 1p\1p state (C"0.18 eV) for O . The S E  excitation energies from top to bottom are u"529.8,530.4,531.0 and 531 eV. The dotted lines show the result when lifetime-vibrational interference is neglected. The curves are normalized arbitrarily [165]. Fig. 39. The calculated RKR potential curves for the states of O involved in the non-radiative RXS [165]. 

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In accordance with theoretical predictions the experimental profiles demonstrate an effective narrowing of the vibrational band when the excitation energy decreases. This narrowing can be explained by shortening of the RXS duration time (40) with increase of the detuning frequency [104]. A qualitative explanation of the strong dependence of the RXS profile can be given as follows: When the excitation energy is tuned in the region of strong photoabsorption (X&0) the duration of RXS (40) equals C\ which is longer than a period of vibration. In this case RXS is G “slow” and the nuclei have time to propagate to a second crossing point at the potential surface of the core excited state, see Fig. 39. The effective width of the RXS profile D in this adiabatic limit is  large (Fig. 39), in agreement with experimental data. An inverse picture takes place if the excitation energy goes below (XK!1 eV) the photoabsorption maxima, in which case the RXS duration is shorter than the vibrational period and the molecule has no time to change its geometry. In this “sudden” limit the RXS process looks like a direct (sudden) transition from the ground state potential surface to the final state potential. Fig. 39 shows that the effective width of the RXS profile D now is narrower than in the adiabatic limit. A more complete discussion of the narrowing  effects in RXS will be given below. 6.4.2. NO, N , O , CO, H O    The lifetime-vibrational interference effect in non-radiative RXS has been studied experimentally and theoretically by Carrol et al. [167], Poliakoff et al. [168], Carrol and Thomas [169—171], Murphy et al. [172], Osborne et al. [173], Neeb et al. [174], Rubensson et al. [175], and Eberhardt [48]. Cesar et al. [61] investigated theoretically the role of interference in the Auger [176] and X-ray emission spectra [177] of H O (Fig. 40). These workers found that the lifetime-vibrational  interference can lead to substantial distortions of the profiles of the X-ray emission and Auger bands and to shifts of the peak maxima of the order of &1 eV. The detailed experimental investigation of the lifetime-vibrational interference in radiative RXS of N [137], CO [68] and CO [103,117] was carried out quite recently. The scheme of all these   experiments is the same as outlined in Section 2.2. A detailed investigation of the lifetimevibrational interference in non-resonant X-ray emission spectra of gaseous O was performed by  Glans et al. [178]. The comparison of the CK- and OK-emission spectra (Figs. 41 and 42) of CO [68] shows that the interference has a stronger influence on the RXS profile in the case of oxygen spectra, which can be explained by the larger lifetime broadening for oxygen 2C"0.18 eV, than for carbon 2C"0.086 eV. 6.5. Integral interference The interference term (96) is a sign-changing function (Fig. 40). It is therefore natural to expect that the integral of the interference term over the final particle energy u is equal to zero. However, that is not so in the general case; for example, the integral is not equal to zero even in a simple 4-level system, consisting of initial, final and two intermediate states, for which interference takes place. Intuition suggests that the interference contribution to the total intensity of the RXS process should be equal to zero when integrated over a nearly complete set of scattering channels. Formally, this statement is associated with certain sum rules of non-trivial nature which can be of importance when analyzing data with prominent interference effects. The aim of this section is to review the integral properties of the interference term in the RXS cross section, and to explore

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Fig. 40. (a) Calculated non-resonant X-ray emission spectra of H O. Direct contribution is depicted as dotted line while  the total intensity is shown as solid line. The sign changing curve is the interference contribution. (b) Experimental spectra [61].

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Fig. 41. Radiative RXS CK-spectra of CO. The energies given in the figure are the excitation energies used in the experiments. The experimental spectra are compared with simulated spectra [68]. The spectral function was assumed to be of Gaussian shape with a FWHM of 0.8 eV. The simulated band profiles of the elastic peak (which follows to the excitation energy) were corrected for effects due to self-absorption. The topmost simulated band profiles of the elastic band shows the influence of self-absorption on the elastic scattering spectrum, in the case of an excitation energy of 287.92 eV. Fig. 42. Radiative RXS OK-spectra of CO. The spectral function was assumed to be of Gaussian shape with a FWHM of 0.75 eV. See caption to Fig. 41.

under which conditions this term integrated over final particle energies equals zero. The simplest case of narrowband excitation is considered here; one can, however, show that the final results for the interference contribution are valid for an arbitrary spectral function. 6.5.1. Integral lifetime vibrational interference The RXS amplitude in the BO approximation (110) leads directly to the following important sum rule: The integral of the interference term (96) over the final particle energy u is equal to zero for both radiative and non-radiative scattering channels [179] p0(u)"0, p,(u)"0 .  

(111)

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The evidence of this result is simple



p0,(u), du 

dp0,(u,u)  du dO

1o"m21m"m21m"o2 "p "0 . (112) M (u!D!u #iC)(u!D!u #iC) KYM KM K$KY This term is thus equal to zero due to the orthogonality of the different vibrational wave functions of the intermediate core excited state: 1m"m2"0 for mOm. Eqs. (111) imply that for a given electron transition, the integral RXS cross sections p0(u) and p,(u) follow the convolution of the photoabsorption profile with the spectral function of incident radiation. 6.5.2. Integral lifetime electronic state interference. Optical theorem A preliminary analysis of electronic state lifetime interference effects can be made from the outset of a strict result connected with an optical theorem for the scattering amplitude [180]. As is well known the total RXS cross section, which is a sum of radiative (R) and non-radiative (N) contributions, coincides with the X-ray absorption cross section:





dp0(u,u) dp,(e,u) # de dO , (113) du dO de dO Q where the summation runs over all Auger electron spin directions. The optical theorem reflects the particle conservation law or, more formally, the unitarity of the RXS ¹-matrix [180] expressed as p(u)" du dO de

4pr (114) p(u)"! M ImF "4pau "D "D(u!u ,C ) , M GM GM G p G where F is the forward elastic X-ray scattering amplitude (7). Taking into account Eqs. (113), (114) M and that a total decay rate of the core excited state C includes both radiative and non-radiative G contributions one can conclude that the interference between different X-ray scattering channels is absent in the total RXS cross section (113) [179]





dp0(u,u) dp,(e,u)   # de dO "0 . (115) du dO de dO Q Since the non-radiative RXS cross section (Section 3.1.2) is larger than the radiative one in the soft X-ray region, the contribution of the latter can be neglected in Eq. (115) p (u),p0(u)#p,(u), du dO de   

p,(u)K0 . 

(116)

6.5.3. Examples of integral vibrational and electronic state lifetime interference The integral interference terms (112) and (116) include a sum over the complete set of intermediate and final states. One can expect that to reach p (u)"0 in practice it is sufficient to take into  account only a finite set of these states. This assumption can be checked by introducing the partial integral cross sections p(u,n), p (u,n) and p (u,n) of scattering to the final state n and calculating   

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the partial sum over of the final states n lying below N N

p (u"N)" p (u,n) .   LM In accordance with Eqs. (111) and (116)

(117)

p (u"N)Kp (u)"0, N<1 . (118)   Vibrational interference in O . The non-radiative RXS scattering around the 1s\p excitation in  E molecular O is considered here in order to illustrate the vibrational interference effect. The FC  amplitudes 1o"m2 and 1m"n2 were calculated using Rydberg—Klein—Rees potentials with experimentally determined spectroscopic constants. The grey bold lines in panel (b) of Fig. 43 show the integral cross section p(u,n) as a function of energy position of different scattering channels oPn. The thin dark line shows the integral decay spectrum p (u,n) without taking interference into   account. A dramatic vibronic lifetime interference effect can be discerned, redistributing intensity from the low to the high kinetic energy side of the spectrum. The upper part of Fig. 43 shows the integral of the interference term as a function of energy position of the scattering channel oPN with lowest kinetic energy. One can see from this N-dependence that in spite of that p (u,n)O0 the integral interference term p (u"N) tends to   zero for large N in accordance with the optical theorem. This demonstrates that even in a case with substantial interference effects, the integral interference term summed over all vibrational excitations is zero. In a partial yield experiment, however, where only part of the excitations are monitored, the intensity would remain dependent on the vibronic interference. Electronic interference in the Auger spectrum of Ne. The situation is very similar for electronic state lifetime interference. Around the K edge in neon the absorption spectrum shows resonances assigned to 1s\np (n"3,4,5) transitions (Fig. 44a). In the electronic decay to the 2p\np states there is a large probability that nOn, that is all the excitations reach the same final states and one may observe interference. In Fig. 44c a perfectly monochromatized beam centered on the 1s\4p resonance is assumed. The spectrum consists of 2p\np configurations where n"3,4,5. The 2p electrons couple to D and S parental terms, giving rise to six final states in total. In this case the interference effects are small since the energy spacing is large relative to the lifetime broadening. In the upper part of the panel the integrated intensities p(u,n) and p (u,n) are shown. At inter  resonance excitation, however, the interference effects can be emphasized [181]. In Fig. 44 the monochromator energy is tuned between the 1s\3p and 1s\4p resonances, resulting in large interference effects. Also, in this case, one finds that the interference term integrated over the entire energy region and summed over all intermediate (m) and final (n) close lying electronic states with large oscillatory strengths is zero, and the partial yield experiment, that discriminates against some of the final states, gives also in this case a result at variance with the absorption probability. The result is clearly dependent on the lifetime of the intermediate states.

7. Elastic radiative X-ray scattering The investigations in the field of elastic RXS — REXS, or anomalous elastic X-ray scattering — have primarily focussed on diffractional problems, and, in contrast to the inelastic case, the fine

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Fig. 43. Panel (a) shows the 1s\1p vibronic excitation spectrum of O . The bold line shows the band shape using E  a realistic lifetime broadening (2C) of 0.18 eV. To make the vibrational substructure obvious, a spectrum (dotted line) has been simulated by using a lifetime width of only 0.05 eV. The decay spectrum in panel (b) has been excited at v "7 by K a perfectly monochromated photon beam. The grey bold lines in panel (b) show the X% final state transitions described E by p(u,n) using a lifetime broadening of 2C"0.18 eV for the intermediate 1s\1p state. The thin dark lines show the E decay spectrum given by p (u,n) taking no interference into account. The difference between the grey and dark lines   gives the pure interference contribution. The integral of the interference contributions p (u"N) is shown as a full curve  on the top panel (b) as the function of energy position “highest” scattering channel oPN [179].

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Fig. 44. The top panel shows the K-shell Rydberg excitation in neon. The lifetime broadening (2C) is set to 0.31 eV. The decay spectra in panels (b) and (c) have been calculated with a perfectly monochromatized beam tuned in between the 1s\3p and 1s\4p resonance (inter-resonance) and at top of the 1s\4p resonance (on resonance), respectively. The decay spectra show the spectator transitions at highest kinetic energy involving two holes in the 2p outer shell. Results with interference (p(u,n), grey broad lines) and without interference ((p (u,n), thin dark lines) are given. The dashed  dotted curve at the top of each decay spectrum shows the integral of the interference term p (u"N). Note that the  interference contribution for sufficiently large N sums to zero both for inter- and for on-resonance excitation [179].

structure of the elastic cross sections near the ionization threshold has been little attended, see however [55,2,182,183,89]. Furthermore, the theoretical endevour has notoriously favored atoms over molecules [55]. In spite of that RIXS and REXS can be measured in the same experimental setup, these processes differ qualitatively, and it is relevant to review the special aspects of REXS

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separately. The distinction is connected with the two large contributions to the REXS amplitude, namely the non-resonant form factor and the amplitude of scattering through the infinite set of intermediate states. The form factor is suppressed by a factor of R/j for soft RIXS, where R and j denote the molecular size and the X-ray wavelength, respectively. Far from the core ionization threshold, these two scattering processes constitute the main contributors to REXS, while close to the absorption threshold the resonant scattering can be comparable or even exceed the nonresonant scattering. One can then expect a strong dependence of the shape of the REXS spectral profile on the interference between the resonant and non-resonant scattering amplitudes. 7.1. Scattering amplitude. Thomson scattering The shape of the REXS spectral lines differs strongly from a Lorentzian shape due to the interference effect. The molecular scattering has polarization features that are completely different from the atomic case and that can lead to an elimination of the isotropic atomic-like scattering. The spectral shape of elastic scattering cross-sections is strongly influenced by the excitation energy and the functional shape of the incident radiation. Amplitude (7) for elastic scattering can be written as 1 1 K" ! , (119) F(u)"(e ) e)oq# u (e ) D )K (e ) D ), GM MG G GM G u!u #iC u#u GM G GM G when the incident X-ray photons are linearly polarized. The first term in Eq. (119) is the amplitude of the Thomson scattering with the molecular form factor [55]



8 oq"1o" eιq  rH"o2" dro(r)eιq  r, oo"Z (120) H given by the Fourier transform of the electron distribution o(r) in the molecule. The form factor is equal to the total number Z of electrons in the molecule for forward scattering (0"0) or in the soft X-ray region. Here q"2p sin(0/2) is the change of the photon momentum for elastic scattering at angle 0. The remainder of the REXS amplitude F consists of resonant, non-resonant (last term in K G (119)) and tail contributions. The latter are caused by the infinite set of the REXS channels oPjPo with "u !u"
p(u,0)"r((e ) e)q#q ) . M

(121)

The anisotropic q and isotropic q q "  (3Tr( fK fK R)!" f "), 

q""oo#f "#q 

(122)

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contributions are expressed through the symmetrical REXS tensor fK "fK and its trace f ?@ @? 1 1 f"Tr( fK )" u "D "K , (123) fK " u (D ) (D ) K , GM MG ? GM @ G GM MG G ?@ 3 3 G G where the indices a and b enumerate the Cartesian components of the vector D . MG 7.2. Subnatural linewidth resolution Eq. (121) shows that, in addition to the instrumental broadening, the spectral resolution of REXS is defined only by the width c of the spectral function. It is important to note that, contrary to RIXS, the spectral resolution of the REXS measurements is the same for discrete and continuum unoccupied states [136] and that this resolution is free from any lifetime broadening. So, contrary to X-ray absorption measurements, REXS spectroscopy can be considered as a spectroscopy of unoccupied states with a resolution better than the lifetime broadening [182,136]. One can show that for spherical symmetry (atomic limit) the anisotropic contribution of the target vanishes: q "0. Eq. (122) shows that REXS allows directly to measure this anisotropic contribution since (124) p(u,0)"rq if eNe . M This result implicates that the form factor oq does not contribute to the REXS profile when the polarization vectors of incident and scattered X-ray photons are orthogonal. So the spectral shape of REXS is in this case principally different from the case with non-orthogonal e and e vectors, as discussed further below. 7.2.1. Asymptotic limit When the excitation energy greatly exceeds the core ionization potential (u
(126)

7.3. Diatomic molecules It is relevant at this point to review the polarization and spectral features of REXS in some detail by applying the general theory to K-scattering of heteronuclear molecules. Molecules with axial symmetry are used for illustration in this subsection, thus with p and p discrete and continuum

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states. The scattering tensor is then diagonal in the molecular frame


 f

0 0 p (127) fK " 0 f 0 . p 0 0 f N The trace f"Tr( fK ) of the REXS tensor (127) and the partial scattering amplitudes corresponding to the p and p states can according to Eq. (123) be written as 1 (128) f "f !if " u "D "K , k"p,p GM MG G I I I 3 GZI with the z-axis defining the molecular axis. The sum over i assumes the summation over all discrete and continuum states of k symmetry ( P #de), where e is the photoelectron energy. The G G anisotropic contribution f"2f #f , p N

*f"f !f (129) q ""*f ", p N  shows directly that q is caused by the difference of the transition dipole moments and the resonant frequencies corresponding to the p and p states. According to the TRK sum rule q P0 and fP!2 since f ,f P! below the resonance region u;I. p N  7.3.1. REXS spectra of CO Several aspects of the theory outlined above can be illustrated by an application to the anomalous elastic X-ray scattering of carbon monoxide near the K-absorption threshold [89]. Input data for such an analysis, i.e. transition energies, oscillator strengths and the K-absorption cross sections are readily obtained from simulations (e.g. the static exchange method [186,187]), or from the Stobbe formula [79] for photoabsorption cross sections. From such data the contribution from high-energy continuum states (e'100 eV) to the real part of the scattering amplitudes f can I be derived. Results from calculations of the REXS cross section performed according to Eqs. (121), (122), and (129) are shown on panels (a) and (b) in Figs. 45 and 46 with the detailed Rydberg fine structure shown on panel (b). The real ( f ,*f ) and imaginary ( f ,*f ) parts of the scattering amplitudes (128) and (129) are depicted on panels (b) and (c). The scattering amplitudes in the Rydberg band is given on panel (c). As discussed above, one must realize several qualitative differences between inelastic and elastic scattering. The tail excitation of discrete and continuum states and the Thomson scattering (with a large form factor oq) can give significant contributions to the real part of the elastic scattering amplitude f . For example, these two contributions become significant close to the strong 1s\2p resonance in the OK spectrum of CO (see Fig. 45a). The interference between Thomson (oo), “tail” and resonant scattering induces Fano-like minima at the long wavelength wings of the 2p resonances in the OK spectra (Fig. 45a). The difference between frequencies of the 2p resonance and the Fano minimum are equal to 1.52 and 2.30 eV for the OK and CK spectra, respectively. Close to the Fano minimum, oo and f have the same order of magnitude. The shift of the Fano resonance from the REXS resonance increases when the magnitude of the ratio of the oscillatory strength of the REXS resonance and oo#f  increases.

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Fig. 45. Computed O1s REXS cross section (a,d) and the REXS scattering amplitudes f"2f #f (b,c) of the CO p N molecule. The angle between polarization vectors h"0°. Close to the O1sP2p-resonance the cross section and scattering amplitude f"f #ιf  have the following values: p(u,h) "19225r, ( f ) "126 for

 M

 u"534.49 eV, ( f ) "!69 for u"534.4 eV, ( f ) "58 for u"534.59 eV. The O1s ionization potential is



 indicated by the arrow. The thin solid line in panel (a) shows the REXS cross section reduced by a factor 100. The frequency of the Fano like minimum at the long wavelength side of the 2p resonance is equal to 532.97 eV [89].

The strength of the 2p resonance in the OK spectrum is an order of magnitude smaller than in the CK spectrum due to the smaller oscillator strength and larger lifetime broadening. This results in a significant deviation from a Lorentzian profile at both the short and the long wavelength wings of the O1s\2p elastic band (Fig. 45a). As one can see from Figs. 45 and 46 the interference of different scattering channels yields a very strong deviation from Lorentzian profiles of the REXS spectral bands near the Rydberg resonances. The influence of weak 1sPp shape resonances [186,187] lying near 310 eV (CK absorption) and 555 eV (OK absorption) is not essential in the particular case studied here (CO molecule, see Fig. 45a). However, this influence might be important for molecules where the shape resonances are strong, as for SL absorption of SF . '''''  A comparison of Figs. 45 and 46 demonstrates the strong qualitative difference between the parallel (h"0) and orthogonal (h"90°) orientations of initial and final polarization vectors. As mentioned above, the elastic scattering for h"90° is caused only by the molecular anisotropy ( f Of ) (see Eqs. (124) and (129)). In this case the REXS cross section is an order of magnitude p N smaller than in the case h"0 since the form factor and the isotropic part of the scattering amplitudes f vanish. I 7.4. Profile of REXS spectral bands versus profile of excitation function Since the double differential cross section (121) is a product of p(u,h) and U(u!u,c), the spectral function strongly disturbs the shape of p(u,h) when narrow band excitation is used. The dependence of the REXS profile on the functional shape of the exciting radiation and on the

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Fig. 46. The O1s REXS cross section (a,d) and the difference of the REXS scattering amplitudes *f"f !f (b,c) for p N p and p scattering channels in the CO molecule. The angle between polarization vectors h"90°. Close to the O1sP2p resonance the cross section and scattering amplitude difference *f"*f #ι* f  have the following values: p(u,h) "

 2399r, (* f ) "63 for u"534.49 eV, (* f ) "!31 for u"534.39 eV, (* f ) "32 for u"534.59 eV. The C1s M





 ionization potential is indicated by the arrow. The thin solid line in panel (a) shows the REXS cross section reduced by a factor 250 [89].

excitation energy is exemplified in Fig. 47, which shows the O1s spectrum of CO. The spectral function cuts out some part from the broad band spectral profile (thin line). The comparison of the REXS profiles for two different excitation energies (u"542 eV, Fig. 47a and u"538 eV, Fig. 47b) shows that the spectral profiles consist of two parts. The position of the first resonant feature (marked by symbol U) follows approximately the excitation energy u according to the Raman—Stokes law u"u while the position of the second spectral feature (marked by symbol A) does not depend on the excitation energy. These resonances are given by the cross section (121) corresponding to the case of broad-band excitation. The two qualitatively different spectral features are direct consequences of the Stokes doubling effect (Section 3.4). 7.5. Polarization of elastically scattered X-ray radiation X-ray emission spectroscopy provides information on energy levels and wave functions of occupied molecular orbitals (MOs) [12], whereas X-ray absorption spectroscopy allows the investigation of unoccupied MOs and quasistationary states, or shape resonances, lying near the ionization threshold [188]. Identification of the symmetries of occupied and unoccupied MOs has been achieved indirectly by using strict and effective selection rules imposed by the dipole approximation and by comparing energies and relative intensities observed in these spectroscopies with MO calculations. In many cases, however, a direct method for determining the symmetries of MOs would be preferable, especially for close-lying or disputed structures for which current computational schemes may not be of sufficient accuracy. As has been shown in Section 5, the

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Fig. 47. The effect of the spectral function of exciting radiation on the shape of the O1s REXS spectral profile (121) of the CO molecule with the Gaussian unnormalized spectral function: U(u!u,c)"exp(!((u!u)/c) ln 2). The thin solid line corresponds to broadband excitation (c"R) (121). The thick solid line corresponds to c"12.5 eV. The dotted line corresponds to c"0.8 eV. (a) The excitation energy is equal to: u"542 eV. (b) The excitation energy is equal to: u"538 eV [89].

RIXS spectral profiles of free molecules depend strongly on the frequency u and polarization e of the excitation photons near the X-ray absorption threshold. As a result, the symmetry of occupied MOs can be determined experimentally from the RIXS measurements. The use of group theory tabulations and RIXS polarization ratios [141] to derive the symmetry of occupied levels when the symmetry of the unoccupied levels are known, and vice versa, was described in Section 5.3. Corresponding direct experimental determination of the symmetries of unoccupied MOs of molecules in the gas phase is often not possible by traditional methods of X-ray absorption spectroscopy (however, see Section 5.4). The symmetries of unoccupied MO’s of randomly oriented molecules can also be defined through polarization measurements of REXS [182]. The polarization (67) can be written according to Eq. (121) as P"q/(q#2q ) .

(130)

It is then instructive to consider the REXS polarization (130) from the point of view of channel interference




MH  PM#c . c#MP $ MH G

(131)

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Here M denotes the molecule in an intermediate core excited state i. If, for simplicity, the Thomson G scattering is neglected, Eqs. (130), (128) and (129) show directly the qualitative difference between pand p-polarizations; P " . (132) P ", N  p  The result for P coincides with the classical absorbing-emitting picture (68) since absorbing and N emitting dipoles then have the same orientation. The large deviation of P " from the classical p  value  is a pure quantum interference effect, namely, a direct consequence of the interference of  1sPp P1s and 1sPp P1s scattering channels. This interference is strong since the "1s\p 2 V W V and "1s\p 2 intermediate states have the same energy. Since only the spectral region in the vicinity W of strong photoabsorption is considered the Thomson and non-resonant contributions can be neglected for the polarization anisotropy. The direct REXS cross sections, p (u,u) (96) for free molecules have identical polarization   dependence, whereas the interference (cross) terms have the polarization dependences [182] (133) f (h)"  [3cos h!1#cos u (3#cos h)] GGY GGY  different from the direct terms (i"i) when the symmetries of the interfering core-excited states, i and i, are different (compare this equation with corresponding result (66) for RIXS). This is due to the dependence of the interference term on the angle u between transition dipole moments GGY D and D [182]. These facts imply that the part of the spectral shape due to the sum of direct GM GYM terms does not depend on the polarization of initial and final photons, but rather that the spectral and polarization dependence of the total REXS cross section will vary for different combinations of symmetries of the interfering states. This specific dependence actually allows an experimental definition of the symmetries of unoccupied states, even when these give overlapping bands in the corresponding X-ray absorption spectrum [182]. 7.5.1. REXS polarization of H S  The findings described above are well illustrated by computing the polarization dependence of REXS for the H S molecule, because of the organization of its lowest core-excited spectrum with  three close-lying MO levels, 3b , 6a and 7a . The following 3-level model for the sulphur    K-excited states of the H S molecule is used [139]: C "C "C "0.5 eV: (d /d )"       0.2; (d /d )"0.5: and D "0.7 eV; D "3.3 eV, where D is the energy separation between the     G lowest state (i"1) and the next states (i"2,3). The often maintained, but still disputed, identification of the SK X-ray absorption resonances is the following [139,189]: 1"SKP3b , @  2"SKP6a , 3"SKP7a .   Fig. 48a displays the computations of P in a 2-level approximation. The polarization is constant, P", when the interference is neglected or when the levels 1 and 2 have identical symmetry. This  fact is easy to understand, see Eq. (133), because in both cases absorbing and emitting dipoles are oriented in the same direction (u "0), and in accordance with the classical picture the polarizaGGY tion P would be . As derived above, different symmetries of two interfering levels impose  a frequency dependence for P (Fig. 48) and a deviation from the classical result P". The third  core-excited state in the H S molecule is not far from the first (D "3.3 eV) and should therefore be   included in a three-level model for the frequency dependence of polarization. As seen in Fig. 48b, the close-lying third level makes the frequency dependence of P very sensitive to an alteration of

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Fig. 48. Polarization P of elastically scattered K X-ray photons of H S as a function of the relative final photon energy  (u!u ). u is the frequency of a transition between the K-shell and the first unoccupied MO level (level 1). Positions   and relative intensities of the K X-ray absorption resonances are displayed by vertical bars. (a) Two-level approximation. Dashed-dotted line: Same symmetry of levels 1 and 2 or no interference. (P"). Solid line: Different symmetries of levels  1 and 2. (b). Three-level approximation. Dashed-dotted line: Same symmetry of levels 1 and 2 or no interfence, (P").  Solid line: Different symmetries of levels 1 and 2. (b) Three-level approximation. Dashed-dotted line: Same symmetry of levels 1,2, and 3 or no interference (P"). Dashed line: The symmetry of levels 2 and 3 is the same but different from the  symmetry of level 1. Dotted line: The symmetry of levels 1 and 3 is the same but different from the symmetry of level 2.

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symmetries of the unoccupied MOs 1 and 2. For example, the spectral dependence of P differs significantly between the three MO assignments (3b 6a 7a ), (6a 3b 7a ) and (3b 6a 3b ) (see          dashed, dotted and solid lines, respectively, in Fig. 48b). From this example one can conclude that interference of REXS channels through close lying unoccupied levels with different symmetries leads to a strong frequency dependence of the polarization P and as a result to a deviation from the classical two-step absorption-emission picture with P".  8. Profiles of RXS spectral bands 8.1. Moments of RXS profiles. Many-level systems The basic principles for the formation of spectral profiles for X-ray Raman scattering were discussed in Section 3.4 from the outset of a simple 3-level model. The line-shape distortions and the Stokes doubling could be understood from the energy conservation law applied to this model. It was shown that even in the simple 3-level system RXS can exhibit a complicated spectral shape, differing from a single resonance. In the case of real atoms the spectral shape can also be complicated by multiplet structure. In molecules the vibrational structure and lifetime-vibrational interference effects are evidently additional sources of complication for the RXS profile. It is therefore natural to simplify the description of the RXS spectra of many-level systems, and to use the spectral band moments for this purpose. This concept is useful for the analysis of experimental RXS spectral features and for deriving information on potential energy surfaces of states involved in the RXS process. The three first moments are considered in this section: The area or zero moment, the center of gravity or the first moment, and the spectral width or second moment of the RXS profile. 8.2. Center of gravity of vibronically broadened RXS resonances For narrowband excitation the center of gravity (CG) follows a linear dispersion (33), while a finite spectral width of incident radiation leads to a non-linear dispersion (see Section 3.5.1). One can expect strong differences between the dispersion of a real many-level system and the simple 3-level system of Section 3.5.1. Here, this problem is reviewed for the special case of the RXS profiles with vibrational structure. 8.2.1. Asymptotic behaviour of center of gravity The main spectral features of the center of gravity (28) can be understood in the limiting case of fast RXS (short RXS duration q (40)) A q ;D\ . (134) A 
The RXS amplitude (110) leads to the following asymptotic expression for the CG (28) [101] e(u)+u!e1 !Xu /(X#C)o . (135) DM G G When RXS is fast (59) the CG follows very closely the Raman—Stokes law e(u)+u!e1 . The DM center of gravity e1 of a sudden or vertical “absorption” transition from the ground electronic state DM

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o to the final one f can be written in the harmonic approximation as e1 "E (R )!E (R )#(u!u)/4u . (136) DM D M M M D M M Here E (R), u and R are the total electronic energy at the point R, the vibrational frequency and H H H the equilibrium internuclear distance of the states j"o,i,f, respectively. The nuclear dynamics strongly influences the asymptotic behavior of the CG through the parameter kuu (u!u)(u!u) G M D , o" D G(R !R )(R !R )# M (137) M G M D u 4uu M M G where k is the reduced mass. This parameter can be positive, negative and equal to zero depending on the precise relation between frequencies and positions of the potential surfaces of the electronic states involved. The asymptote of the CG (135) is antisymmetric relative to the Raman—Stokes law (33) if oO0. When o"0 one should take into account a higher order correction to this law. One can understand that this correction is proportional to 1/(X#C), and, therefore, that the asymptote of the CG defines a symmetrical function of detuning for the case o"0. Fig. 49 shows that knowledge of the asymptotic behaviour allows a prediction of the behaviour of the CG in the photoabsorption

Fig. 49. Influence of nuclear dynamics on the center of gravity. Thick solid curves show exact calculations according to Eqs. (28) and (110). The Raman—Stokes law (33) is depicted by dashed lines. Thin solid curves show asymptotical behaviour of the center of gravity (135). C"0.09 eV. (a) o'0. Data for O K-emission of CO (Table 1). (b) o"0. Input data are the same as for (a) except data for the final state which coincide with the ground state data u "u "0.27 eV, D M R "R "1.128 As . (c) o(0. u "u "0.27 eV, u "0.22 eV, R "1.128 As , R "1.05 As , R "1.399 As . c"0 [101]. D M M D G M G D

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Fig. 50. Influence of nuclear dynamics on the center of gravity of elastic RXS. Data for O K-emission of CO (Table 1). Potential surfaces of initial ground and final states coincide. c"0 [101].

region using a very simple analysis of the o function (137). The change of sign of the o function (137) leads to inversion of the CG frequency dependence relative to the Raman dispersion line (compare Fig. 49a and b). Two important cases for which o"0 can be singled out. This function is equal to zero if the potentials of ground and core excited states coincide (R "R , u "u ) or if the M G M G potentials of ground and final states coincide (R "R , u "u ) (Fig. 50). The latter case takes M D M D place for elastic scattering as further discussed below. 8.2.2. Role of interference for center of gravity Figs. 38, 40, 42 display the strong influence of the lifetime-vibrational interference on the position of the center of gravity. Fig. 51 shows that the lifetime-vibrational interference effect also plays a very important role in the dispersion relation for the X-ray Raman effect. 8.2.3. Elastic scattering The dispersion relation for elastic RXS is depicted in Fig. 49 and Fig. 50 using data for the CO molecule. Since the initial and final states possess identical potential surfaces for elastic RXS, and since here u 'u , the dispersion relation shows anti-Raman behavior [101] (see Eq. (138)). The M G convergence of the dispersion (135) with respect to the Raman law (33) when the detuning tends to R is presented in Fig. 52 and Fig. 49. As can be seen in these figures the convergence is slow (J1/X) in the general case (Fig. 52) but quite fast (J1/X) in the case of elastic scattering (Fig. 49).

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Fig. 51. Influence of lifetime-vibrational interference on center of gravity. Input data for O K-emission of CO (Table 1). c"0 [101].

Fig. 52. Influence of vibrational structure on CG slope. X-ray absorption profile is given for C"0.001 eV. (a) Data for O K-emission of CO molecule (Table 1). Slope (138) is positive. (b) Anti-Raman behavior (u 'u ). Input data are the D G same as for (a) except u "0.23 eV. Slope (138) is negative. c"0 [101]. D

8.2.4. Small lifetime broadening When the lifetime broadening is small, C ;u ,u ,u , the CG oscillates in the region of G M G D photoabsorption but follows closely the Raman law outside this region (Fig. 52) [101]. Comparison of these oscillations with the photoabsorption profile (Fig. 49) shows that the minima of the

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CG positions correspond approximately to the maxima of the absorption cross section. When the lifetime broadening C increases, the oscillations of the CG disappear and the deviation from the G Raman—Stokes law also disappears in the limit (X#C
(139)

u!u D, e1 "E (R )!E (R )# G GD G M D M 4u M u u!u u u!u G #ku(R !R )(R !R ) M G G ku(R !R )# G e" " D G M G D M G G D GD u#C 2u u#4C 4u G G M G G G (140)





then contribute to the center of gravity. The first term e1 refers to the vertical or sudden transition, GD while the second term e" is purely of dynamical origin [190]. Indeed, the dynamical contribution GD e" tends to zero when the core excited state is short-lived (C
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Fig. 53. A pictorial representation of the RXS processes for on-resonance and for detuned excitation. (a) Potential surfaces of ground and final states are the same. (b) Potential surfaces of intermediate core excited and final states are the same. Fig. 54. Spectral width (second moment (29)) of inelastic RXS. Inelastic scattering to the final state ( f ):1p\2pD. Data for oxygen K inelastic RXS of the CO molecule (Table 1). (a) C"0.09 eV, c"0.1 eV, C(R)K0.308 eV. (b) C"0.01 eV, c"0.1 eV, C(R)K0.308 eV. (c) C"0.01 eV, c"0.01 eV, C(R)K0.3 eV, C(u )K0.21 eV [101]. 

(q (40)) of the RXS process (Fig. 53a) [103,117], c.f. Section 4. If the duration time is much smaller A than the time of deformation of the nuclear wave packet (134) at the intermediate state energy potential surface, the molecular frame does not adapt to this potential, but will be determined directly by the final state potential (Fig. 53a). Off resonance, the RXS process is prompt while on-resonance the lifetime of the resonant state determines the interaction time. Detuning from resonance shortens the duration of the RXS process (Fig. 21) and one then probes preferentially transitions associated with shorter duration times, for which the influence from the core excited state has diminished. The corresponding transition amplitude is then proportional to the Franck—Condon amplitude between ground state and final state vibrational wave functions (62). So if these two potential surfaces are close to each other, the RXS spectral profile (62) collapses to a single line (Fig. 53a) F "a/(X#iC )d . (141) D G LM A diatomic molecule treated in the harmonic approximation constitutes a suitable system for illustrating the main spectral features of the second moment C(u). Numerical simulations for carbon monoxide depicted in Figs. 54 and 55 are based on Eqs. (29), (110), (18) and Table 1. Both figures show narrowing of the RXS profile under detuning from photoabsorption maxima. The strength of this narrowing is larger for REXS than for RIXS.

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Fig. 55. Spectral width (second moment (29)) of elastic RXS. Data for oxygen K elastic RXS emission of the CO molecule (Table 1). (a) C"0.09 eV, c"0.1 eV, C(R)"c/(2K0.071 eV. (b) C"0.01 eV, c"0.1 eV, C(R)K0.071 eV. (c) C"0.09 eV, c"0.01 eV, C(R)K0.071 eV [89].

Table 1 Vibrational frequencies u , equilibrium internuclear distances R and lifetime broadenings C for ground (o), core-excited H H H (i) and final ( f ) states of the CO molecule ( j"o, i, f ) [69] State

u (eV) H

R (As ) H

C (eV) H

Ground (o): XR> Core-excited (i): O1s\2n P Final ( f ):1n\2n"'D

0.27 0.18 0.14

1.128 1.280 1.399

0 0.09 K0

8.3.2. Second band moment for excitation at the wings of the X-ray absorption The RXS amplitude (110) leads to the following asymptotic expression for the second moment (29) [104]



C(u)KC(R)"




 

ku 1 u!u  D c# D (R !R )# M M D u 2 2u M M



,

(142)

in the case of fast RXS (large detuning "X" or large C) (134). It is reminded that c is expressed through the HWHM c of the spectral function by Eq. (31). Eq. (142) shows that in the sudden limit (134) the spectral width C(R) is larger than the width c/(2 of the spectral function C(R)5c/(2. The additional broadening (to the width c/(2) is caused by vibrational structure. The collapse of this vibrational structure to a single resonance with the width C(R)"c/(2 takes place when the interatomic potentials of ground and final states coincide: u "u , R "R (Fig. 55c). M G M G

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8.3.3. Second band moment for adiabatic X-ray absorption Fig. 54 shows that some suppression of the vibrational broadening also takes place for inelastic RXS of the CO molecule. Contrary to elastic RXS (Fig. 55), this suppression is not complete when exciting at the wings of the X-ray absorption profile (since the potential surfaces of ground and final states do not coincide as for REXS, see Eq. (142) and Table 1). The shape of the second moment C(u) has an additional feature, namely some suppression of vibrational broadening near the adiabatic 0—0 absorption transition for small detuning (X"!0.043 eV was used in Fig. 54c). The numerical experiment (Fig. 54c) prompts one to look for a minimum of C(u) for small widths (c,C;u ) and for the 0—0 absorption transition with X"(u !u )/2. With a small lifetime G G M broadening, the interference between different vibrational levels can be neglected, whereas a small width of the Gaussian spectral function makes it possible to tune exactly to resonance with the 0—0 absorption transition. By means of these two assumptions an expression for the RXS second moment C(u ) of the 0—0 absorption transition can be given. Likewise, Eq. (142) indicates that this  expression can be separated into atomic C and molecular C contributions: 

 C(u )K(C (u)#C ) . (143)  

 The RXS second moment for a free atom (without vibrational structure)





 

cC  (cC) !C , if C;c , " C (w)"  p (pRe w(z) cC  c " , if C
(144)

depends only on the interplay between c and C through the parameter z"iC/c and coincides with the corresponding expression of C(u) of the three-level atom (31) for X"0. The molecular contribution caused by vibrational broadening is






 

1 ku u!u   D (R !R )# G D . (145) C " G D

 2 u 2u G G Contrary to Eq. (142) the vibrational broadening C(u ) (143) is defined by the difference of the  potential surfaces º (R) and º (R) of the core excited and final states. The comparison of this G D expression with the corresponding molecular contribution for fast RXS (142) indicates that this spectral width refers to a sudden emission transition: iPf. Physical intuition does not contradict this statement. Indeed, the choice of detuning frequency X"(u !u )/2 and small c and C leads to G M core excitation exclusively to the lowest vibrational level m"0 of the intermediate state. The accompanying vertical emission transition iPf yields the vibronic broadening of Eq. (145). Fig. 54c shows that the narrowing of the inelastic RXS profile in the region of strong photoabsorption (XK!0.043 eV) can be stronger (C(u )K0.16 eV for c,CP0) than at the wings of the X-ray  absorption profile (C(R)K0.3 eV). It is necessary to note that the decrease of C(u) for small X takes place in the elastic case too (see Fig. 55b and c). But for elastic RXS this effect is smaller in comparison with the narrowing C(R)"c/(2 obtained by excitation at the wings of the absorption band.

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8.3.4. Energy dependence of center of gravity and second moment Both the center of gravity e(u) and the second moment C(u) quantities have asymmetrical dependencies on the excitation energy. The reason for this can be found in the different FC amplitudes for the absorption transition below and above the crossing point, see Figs. 52, 54, 55. This asymmetry becomes stronger if the anharmonicity of the interatomic potential is taken into account. An increase of density of core excited electronic states with an increase of excitation energy also enhances this asymmetry. Related narrowing effects of the RXS profiles can be anticipated also for many-level systems, where fine structures may appear due to different types of perturbing interactions, e.g. spin—orbit and open-shell electrostatic interactions leading to spin-sublevel and multiplet splittings. One can thus foresee quenching also of such structures as an effect of frequency detuning. 8.3.5. Experimental evidence of collapse effect Experiments giving evidence of the collapse effect [105] have been performed at beamline 51 at the MAX I laboratory in Lund, Sweden (see Section 3.2.2). The electron spectrometer was set in the plane of the polarization of almost completely linearly polarized light. The Auger resonant Raman spectrum of the CO molecule was measured at several energies close to the C1s\pH(l"0) resonance. The collapse of the vibrational fine structure of the bands associated to the singly ionized X (5p\2R>) and B (4p\2R>) final states was thereby observed (Fig. 56).

Fig. 56. Frequency detuned C1s\pH resonant Auger participator spectra of CO leading to the three main final states of CO>; the X(5p\ R>), A(1p\ P). and B(4p\ R>) states. Left columns display the measured spectra; right columns the results of the simulations [105].

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While the A state of CO> shows complex vibrational profiles at both resonant and non-resonant frequencies, the X and B states show very simple features, which can be used to illustrate the vibrational collapse. As seen in Fig. 56, the X state has a second vibrational level excited at resonance, but collapses to a single line for large detuning. The B state shows the reverse behavior, i.e. a single line at resonance but additional vibrational levels off-resonance. These remarkable features can be traced to the short and long resonant duration times and to the matching of potential energy surfaces. The final X state refers to ionization of the non-bonding 5p orbital, so the minimum of its potential curve is very close to that of the ground state (GS), whereas the potential energy curve of the B state is very close to that of the core excited state [162,193]. As easily inferred from the FC principle the latter situation leads to only one strong band for the GS-core-B resonant process (the 0—0 and 1—1 transitions between the core and B states have almost the same energies), while the GS-core-X band still shows a distinct vibrational excitation at the resonance energy. A dynamic, wave packet, illustration is included in Fig. 53, which also shows the case for detuned frequency (or short duration times). For the X band a short duration time q means that the wave packet has no time to propagate on A the shifted core excited state potential, but is transferred instantaneously on the unshifted final X state potential. It will accordingly not deform at all and the band profile collapses to a single line. For the B state, detuning has the implication that the wave packet deforms to a new equilibrium on the shifted B state potential, and a vibrational envelope will develop. In the resonant B case the wave packet was already deformed by the shifted core excited potential and no further deformation occurs for emission to the equally shifted B state. To study the resonant contribution corresponding to a finite detuning frequency, numerical simulations based upon Eqs. (96) and (110) (which neglect direct photoemission) were also presented in Ref. [105]. An intermediate state lifetime width (FWHM) of 85 meV [194,173] and a photon band width (FWHM) of 100 meV were used in these simulations. As can be seen in Fig. 56, the simulated spectra resemble the experimentally obtained spectra well. This indicates that the direct contribution has a negligible impact on the decay spectra and that the collapse of vibrational structure observed for a small detuning frequency in bands B and X is due to the resonant contribution only.

9. Role of symmetry in radiative X-ray Raman scattering 9.1. General symmetry analysis A salient feature of X-ray Raman scattering is the strong dependence of the spectral shape on the frequency of incoming radiation in the vicinity of a core level photoabsorption threshold [92,195,196,95,197—200]. Although there are several more or less subtle reasons for this frequency dependence it is safe to say that the symmetry of a target constitutes the main grounds for this feature for radiative RXS [95,141,201]. To explore the symmetry related frequency dependence, it is relevant to consider RXS with a narrowband X-ray beam at a frequency near the core ionization threshold and with the spectral width c smaller than the gaps between different core excited states. One can then speak about RXS as a process with symmetry selective excitation simply because the frequency u can be tuned to an

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exact resonance with core excited states "i2 of certain symmetries. The symmetry of an intermediate core excited state is defined by the direct product of irreducible representations CM;CB;CG

(146)

of the ground state CM, the core-excited state CG and of the of dipole moment operator CB. The dipole selection rules then dictate which symmetries for the final states " f 2 are allowed. It is important to note that these rules can be violated even in the soft X-ray region if the photon wavelength is comparable or smaller than the size of the system [195,95,202,203] (see Section 16). In analogy to Eq. (146) the symmetry of the RXS final state can be derived from the direct product CG;CB;CD .

(147)

This shows that the shape of the X-ray emission spectral band depends on which excited state is involved, and it is easy to understand that the spectral shape of X-ray fluorescence must depend on the frequency of exciting radiation u which selects an intermediade state "i2 of certain symmetry. As indicated above, there can be reasons other than the symmetry selection rules for frequency dependence of the resonant X-ray emission spectra. Different transitions are obviously associated with different transition moments, different radiative yields, and many-electron effects [196,97] may play different roles in the various frequency regions. The picture is changed qualitatively when the excitation energy exceeds the core ionization threshold. In this case, the spectral shape of radiative RXS does not depend on the excitation energy since the intermediate continuum states reached by the core excitation are infinitely degenerate and different symmetries cannot be selected by tuning the frequency. A complete analysis of the symmetry selection rules for the radiative RXS tensor has been given by Luo et al. [141]. 9.1.1. Molecules with a center of symmetry The general rules given by Eqs. (146), (147) can also be applied to the class of molecules with a center of symmetry. Indices g (gerade) or u (ungerade) are used for electronic states which are symmetric or antisymmetric to inversion through the center of symmetry. According to dipole selection rules (146) and (147) only core excitations to intermediate states of different parity than the ground states are allowed, and only emission to final states with the same parity as the ground state are allowed. The parity of the electronic state does thus not change under resonant X-ray scattering, and the RXS tensor has the following selection rule [95,141]: gPg or uPu .

(148)

which, however, is strict for electronic transitions only in diatomic molecules and which for polyatomic molecules can be violated due to vibronic coupling of the electronical states of opposite parity (see Section 10). 9.1.2. The fullerene molecule Fullerene with its high I symmetry constitutes a most conspicuous example of the symmetry F selectivity in RIXS. Fig. 57 shows the RIXS spectra of C computed in the frozen orbital  approximation. The relevant unoccupied molecular orbitals governing the absorption spectrum are of t , t , t , h , h , a and g symmetries [204], corresponding to four relatively strong S E S E S E E

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Fig. 57. Calculated non-resonant and resonant X-ray emission spectra of C . The lifetime of core-excited states is  C"0.15 eV and the linewidth of incoming photons is c"0.05 eV.

absorption peaks in the X-ray absorption spectrum. The first three have contributions from the first four “LUMO” orbitals (in the following denoted as the LUMO, LUMO#1, LUMO#2, , orbitals), and correspond, in order, to the t , t , t and h molecular orbital symmetries. The S E S E two latter levels are energetically nearly degenerate, with h providing the relatively larger part of E the intensity to the corresponding, unresolved, band. The fourth absorption band corresponds to the unoccupied orbitals with symmetries h , t , a and g , where h dominates. S S E E S The RIXS spectra in Fig. 57 are calculated with a small width, 0.05 eV, of the spectral function for the incoming photons, and are shown together with the non-resonant, X-ray emission spectrum. With this small width, the effect of tail excitation is effectively excluded, i.e. the emission lines in each

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of the RIXS spectra are provided by the decay from one sole unoccupied orbital. Under the condition of narrow excitation it is evident that for a molecule with such a high symmetry as C the resonant (RIXS) spectra are much more sparse than the non-resonant spectrum. The first  selection for the RIXS transitions is dictated by parity, i.e. RIXS from an unoccupied orbital of ungerade (gerade) symmetry displays only occupied orbitals of ungerade (gerade) symmetry. Further symmetry selection is given by the fact that the symmetries of allowed occupied orbitals for the RIXS transition should have symmetries C and C;h , where the symbol C represents the E symmetry of the unoccupied orbital involved in the RIXS transition. For instance, for an unoccupied orbital with symmetry a , only occupied orbitals with symmetries a and h are E E E allowed, while orbitals with t , t or g symmetries are forbidden. For unoccupied orbitals with E E E symmetries t , t or g , the occupied orbitals with a symmetry are forbidden. If the unoccupied E E E E orbital has h symmetry, no occupied orbitals with same parity as the unoccupied orbital are E forbidden, etc. 9.2. Selection rules and core hole localization The question of core hole localization and symmetry breaking has constituted a classical problem in X-ray spectroscopy of symmetrical targets like crystalline solids and molecules with equivalent atoms [205—208]. The analysis of symmetry selection in the resonant X-ray emission spectra has an obvious close tie with this problem. There are both “physical” and “computational” aspects of the core hole localization problem. The latter aspect stems from the one-particle picture and the instability of the Hartree—Fock method (implementing this picture) towards symmetry breaking and hole localization. The broken symmetry solutions describing the core hole are thus more stable than the symmetry adapted ones. “The apple of discord” was introduced in this region by Bagus and Schaeffer [205], who showed that core excited state of O with a delocalized core  hole is unstable since the core-excited state with core hole localized at one of the oxygens has lower energy. The symmetry breaking can be seen as a relaxation effect in the lower point group, which corresponds to a correlation contribution in the higher point group. The “relaxation” which localizes the core hole thus corresponds to different symmetry adapted mixings in the higher symmetry point group [209,207]. Beyond the one-particle (one-configuration) approximation, symmetry adapted and relaxed solutions are easily retained, without energy change, by superposing the relaxed configurations with core holes localized at different sites. Thus a limited manyparticle approximation to the RIXS process solves the “computational” part of the localization problem, in that a localized hole is replaced by a number of symmetrized combinations of the localized holes. From this point of view the localization problem is only pedagogic; one has to go beyond the one-particle approximation to restore symmetry, and symmetry breaking does not refer to the exact wave function, but merely to an approximation of it. It is relevant to point out that in the family of core electron spectroscopies resonant X-ray (radiative) emission spectroscopy is the best suited member for analyzing the symmetry dilemma; radiative because the non-radiative counterpart does not contain the appropriate selection rules, resonant because the selection of transitions is independent of whether the core hole state is symmetry broken or not in the non-resonant case, emission because the symmetry adapted level splittings most often are smaller than the lifetime width of the levels and can therefore not be distinguished by absorption spectra (see, however, recent NEXAFS spectra of acetylene, [210]).

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The fundamental problem of core hole localization for RXS is also closely connected with the interference phenomenon. As shown below both localized (L) and delocalized (DL) representations lead to the same result in strict one-step theory of RXS. The analogy with the two slit experiments shows that the localization problem in reality is a fundamental quantum mechanical problem of particle-wave dualism which depends strongly on the measurement procedure. Section 9.2.1 describes the conditions for obtaining knowledge of core hole localization, and when this knowledge principally is absent. 9.2.1. Qualitative picture A qualitative picture of the localization problem is most easily obtained by considering X-ray fluorescence of a homonuclear diatomic molecule (M ). The process of resonance inelastic scatter ing of an X-ray photon c by this molecule can proceed along the following two indistinguishable channels:






(MHM) J P(M ) #c . (149) c#M P  HJ  (MMH) J This process consists of an absorption of a core-electron into an unoccupied molecular orbital (MO) t and the subsequent emission of a final X-ray photon c caused by the transition of an J electron from the occupied MO t to the core—hole localized at one of the atoms of the molecule. H Here (MHM) and (MMH) denote molecules in intermediate excited states with the core-electron J J promoted from the left and right atoms, respectively, into an unoccupied molecular orbital t ; (M ) denotes the molecule at a final optically excited state jPl. The excited molecules J  HJ (MHM) and (MMH) have the same energy, and one can therefore not distinguish between the two J J channels of the RIXS process given by Eq. (149) (the channels (a) and (b) in Fig. 58(I)). As a result, these channels will interfere, and a phase correlation between them leads to a strong excitation energy dependence for the spectral shape of the fluorescence spectra [195,95]. Another approach is to assume delocalization of the core holes. In this case the channel interference depends on selection rules of the high symmetry point group (here D ). The F intermediate core excited states "1s\t 2 and "1s\t 2 have the same energy, and one can  J  J obtain delocalized core hole wave functions t(1p )t(1p ) by orthogonal transformations E S (1s ,1s P1p ,1p );   E S (150) t (1p)"(1s $1s )/(2 , ES   neglecting for simplicity the vanishing overlap integral between core orbitals 1s and 1s . In this   case RIXS proceeds through the 1p\t or 1p\t delocalized core—hole states: E J S J (M ) \  NE RJ P(M ) #c (151) c#M P  HJ  (M ) \  NS RJ The equivalence of the localized (149) and delocalized pictures (151) is the special case of the general assertion: The RIXS amplitude F is invariant relative to the choice of representation for the degenerated core excited states [95,211,212] due to invariance of




"t (1p)21t (1p)", ? ? ?



t (1p)"a 1s #a 1s ? ?  ? 

(152)

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Fig. 58. Qualitative picture of symmetry selective resonant inelastic X-ray scattering. Only the scattering channels with the same final states can interfere.

relative to the unitary transform aRa"1. A symmetry selective excitation of inner-shell electrons can be considered by tuning the incoming photon frequency of a strict resonance transition to an unoccupied MO t of certain symmetry dictated by the dipole selection rules. If the unfilled MO J t has g or u symmetry only transitions from, respectively, 1p or 1p core states are allowed by J S E dipole selection rules and only electrons from an occupied MO t with g(u) symmetry undergo H X-ray emission to an inner hole of u(g) symmetry (Fig. 58(II)). Apparently, dipole emission from an occupied MO of u(g) type is forbidden for the same reason. To emphasize the important role of symmetry, this process is commonly addressed as symmetry selective RIXS.

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Only a part of the MOs assigned in the normal X-ray emission spectrum generated by broadband excitation can be displayed in symmetry selective RIXS, for which the symmetry of unfilled MOs t dictates a one-to-one correspondence to the symmetry of the occupied MOs t . J L The symmetry of t thus forbids one out of two scattering channels in (151) and therefore, and J contrary to (149), interference between the two inelastic scattering channels in process (151) will be absent. It is necessary to note that absence of interference of the channels (151) is caused by the particular (high) symmetry of the point group D . The interference of the scattering channels F through delocalized core hole states can of course take place for molecules with other point groups. 9.3. Parity selection rules for fixed and randomly oriented molecules The degree of order for the scattering target, which can be 1-, 2- or 3-dimensional, is relevant for the possibility to measure parity selection rules for the RIXS cross section. The case of complete, 1-dimensional order, with a fixed-in-space molecular orientation, has special consequences, although it is never realized exactly even at low temperature, due to zero-point librations. The cross section of RIXS by a diatomic molecule M (16) is obtained as  p "aRK "U(X !X ,c) H J . p(u,u)"(1#P cos(q ) R)) M X#C

(153)

To emphasize the resonant character of the cross section the frequency detunings X "u!u (l) H HL and X "u!u from the X-ray emission, u (l), and absorption, u , resonances are introduced. J JL HL JL Here R"R !R is the internuclear radius vector with R as the radius-vector of the nth atom,   L all unessential quantities are collected into the constant p , u "E(n\l)!E and u (l)" M JL M HL E(n\l)!E( j\l), u "E(n\l)!E , u (l)"E(n\l)!E( j\l) are resonant frequencies of JL M HL X-ray absorption (nPl) and emission ( jPn) transitions, respectively. We introduced the parity of the final state P (P"#1 or !1 if the final state " f 2 is gerade or ungerade, respectively). We consider here for conceptual simplicity only the p MO’s t and t . In this case aRK "(eYH ) RK )(e ) RK ) H J depends on the orientation of e and e relative to the molecular axis RK "R/R. In the general (a,b)-representation (152) the RIXS cross section is naturally divided into two contributions typical for quantum mechanics p(u,u)"p (u,u)#p (u,u)#p?@ (u,u) , ? @ 

(154)

referring to the direct term p "p #p and to the interference contribution p?@ . Contrary to the   ? @  total cross section p, the interference contribution p?@ and hence the notion of coherence is not  invariant. Moreover, it is possible to suppress the coherence p?@ between scattering channels a and  b by the an appropriate unitary transformation to the new core states [95,211]. 9.3.1. Dark and bright states Among the infinite set of the states (152) two special states — or basis sets — exist [211] (a,b"#,!) t (1p)"(1/(2)(1s $1s eιE)   !

(155)

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with the phase g"!(p ) R) or g"!(p ) R). We call one of the states (155) “dark” and the other one “bright” because the RIXS amplitude is equal to zero for one of these states [211]. Thus only one scattering channel is open while the other, the dark one, is closed, which one depends on the parity P of the final state " f 2. Due to the absence of the cross term p>\ one can say that the cross  section is “diagonal” in the representation (155). This allows to conclude that this representation is an eigenfunction of the RIXS process. Thus RIXS by an ideal oriented system “measures” directly the scattering through the “bright” core state, since scattering through the “dark” core state is strictly absent. Contrary to the general (a,b)-representation (152) for a two-channel scattering problem, the presence of “dark” and “bright” states reduces the RIXS scattering to a one-channel problem which is free from the notion of interference. We will see below that the concept of non-interfering states has the advantage not only of generating the simplest possible description, but that it also allows some conclusions about the ¸ and DL natures of core excited states in both the hard and the soft X-ray regions. 9.3.2. Parity selection rules and Young’s double-slit experiment Eqs. (16) and (153) show immediately that RIXS is absent if the Bragg parameter q ) R/p is an integer: p(u,u)"0 if q)R "m"odd if f"g , p q)R "m"even if f"u . p

(156)

This equation demonstrates nothing else than the parity selection rules for the gerade (g) and ungerade (u) final states [211]. One can here find the connection between these parity selection rules and the Young’s doubleslit experiment (YDSE). To begin with, we recall that the dipole approximation for the intraatomic transitions is valid only when the size a of the core shell is small compared to the wave length of Q radiation j. Fig. 59 shows the geometry for the resonant X-ray Raman scattering by a homonuclear diatomic molecule in this limit (a ;j), and the Young’s double-slit experiment with the slit width Q smaller than j. As one moves across the screen, a whole series of alternating bright and dark regions is observed, forming an interference pattern, which is a clear evidence that light behaves wave-like in this experiment. A consideration of the YDSE gives a qualitative picture of the selection rules (156): A constructive intensity maximum when the path difference D"AB#BC (see Fig. 59, I) is an integral number of wavelengths (secondary waves arriving in-phase), and a destructive minimum when D is an odd number of half-wavelengths (secondary waves arriving 180° out-of-phase). The path difference D"(q ) R)/k

(157)

is equal to zero for forward scattering, q"0, and leads directly to the selection rules (156). One can easily understand that the YDSE interference fringe can have a phase opposite to that shown in Fig. 59, I. To realize this we have to shift the phases by 180° of the wave incident to one of the slits.

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Fig. 59. Parity selection rules and Bragg conditions. The Huygens’ principle yields the alternating bright and dark pattern of interference fringes in RIXS and YDSE. The compensating plate P shifts the phase of the incident wave by 180°. The interference fringes I and II correspond to YDSE without and with plate P, respectively.

The interference pattern for this case is depicted in Fig. 59, II. The picture for the YDSE is the same as for RIXS (Fig. 59). One can say that the switch of parity observed in RIXS is obtained in the YDSE by means of the phase shifting plate P, see Fig. 59. 9.3.3. Loss of coherence due to orientational dephasing Gas-phase and surface adsorbed molecules represent disordered and partially ordered systems, respectively, the latter with an alignment axis close to which the molecules librate even at zero temperature. In these cases the RIXS cross section (153) must be averaged over molecular orientations, and the Bragg condition (156) cannot be valid simultaneously for molecules with different space orientations. Hence, the parity selection rules are violated in real systems in accordance with the wave length of radiation and with the degree of molecular disorder [95,211,212]. We consider now the orientational dephasing of the scattering channels in the ¸ representation. As one can see directly from the resonant part of the RIXS amplitude (7), the partial scattering amplitudes F and F through the atoms 1 and 2 differ by the phase factor   ιq R (158) e  , which is orientationally sensitive. The RIXS cross section in the ¸ representation reads p(u,u)"p (u,u)#p (u,u)#p(u,u) ,    p(u,u)"Pp "aRK "cos(q ) R) .  M

(159)

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Fig. 60. Orientational dephasing in RIXS. A suppression of coherence in YDSE when the incident light consists of a few plane waves with different incident angles. Fig. 61. The dependence of the partial RIXS cross sections p (u,u) and p (u,u) on the scattering angle 0, q"2psin(0/2) S E [211]. 3D disorder: Chaotical orientation of the molecular axes in 3D space j"l"p. pRK2.8 for K-spectrum of Cl .  (eNe, eNq, (e ) q)"cos(0/2). The polarization vector of the incident photon e is parallel to the ½-axis, e is parallel to the X-axis, p is parallel to Z-axis, while p is lying in the ½Z plane.

The partial RIXS cross section corresponding to the scattering channels through atoms 1 and 2 are the same; p (u,u)"p (u,u)""aRK "p /2. The interference of the paths 1 and 2 leads to the   M interference term p. This cross or interference term caused by the phase factor (158) is different for  molecules with different orientation. The interference fringes (Fig. 59) become blurred due to the dephasing, as qualitatively illustrated for YDSE and for RIXS in Fig. 60. Let us illustrate first the orientational dephasing for the gas phase molecules. The average interference term demonstrates quenching of the channel coherence in the hard X-ray region [95,211] (see Fig. 61) qR<1 . (160) pKPp "aqˆ "sin(qR)/qRP0,  M Since the coherence between scattering channels 1 and 2 is absent one can say that the hard X-ray photon “sees” an individual atom (slit). Due to the independent scattering through the localized core excited states W and W , p(u,u)"p (u,u)#p (u,u) and the parity selection rules for the     RIXS process are violated as was confirmed in the recent experiment reported in Ref. [213] (see Section 10.1.1). Eq. (160) is valid only in the limit qR<1. The strict expression for p can be found  in Ref. [214]. The orientational dephasing for the aligned physisorbed molecules will be considered below.

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In the soft X-ray region pR;1 ,

(161)

and the states (155) which “diagonalize” the cross section are nothing else than the DL ungerade and gerade core shell states (150). Orientational dephasing is then negligibly small, due to (161). We remind that the scattering channels 1 and 2 in the ¸ representation now become strongly coherent, pO0. Hence in the ¸ representation the photon “cannot distinguish” atoms (slits) and one  cannot say that scattering through the ¸ states 1s and 1s are independent events [95,142,137]. So   the ¸ representation is not an appropriate physical notion in the soft X-ray region. However, the RIXS channels through the DL states (150) do not interfere pES "0 in the soft X-ray region.  Moreover, as was mentioned above, the DL representation reduces from a two-channel problem in the ¸ representation to a one-channel problem. This makes the scattering through the DL states (150) totally independent and leads to the strict parity selection rules: p"0 if the final state is ungerade. The measurable quantity is now the direct DL contribution [95,215,142,137,211,212] p(u,u)"p (u,u) , (162) E corresponding to one-channel RIXS through the DL ungerade core excited state [95]. The experimental evidence of these parity selection rules for O [142] and N [137] shows directly that   the soft X-ray photons scatter through DL independent channels. 9.4. RXS of diatomic molecules in the soft X-ray region In the present section we review the diatomic case in the soft X-ray region (qR;1) in some detail. For that purpose it is pertinent to use Eqs. (63) and (7) and to rewrite the expression for the orientationally averaged RXS cross section (16) as fLLYU(u#u !u,c) JH JH p(u,u)"p M (u!u (l)#iC )(u!u (l)!iC ) HL HL HLY HLY HJ LLY

(163)

with 1 fLLY" [(D ) D )(D (l) ) D (l))(4!a!b)#((D ) D (l))(D ) D (l))(4b!a!1) JH LYH JL LH JLY LYH 30 JL JLY LH # (D ) D (l))(D ) D (l)))(4a!b!1)] . (164) JL LYH JLY LH The following notations for the dipole matrix elements of X-ray absorption (nPl) and emission ( jPn) transitions are used D "10"D"n\l2 and D (l)"1n\l"D" j\l2. C is the lifetime JL LH HL broadening of the X-ray emission line jPn. Other notations are the same as those defined in Section 9.3. This expression can be seen as a simplification in the soft X-ray region of the general diatomic formula given by Eq. (153). The electron excited to the vacant MO t screens differently J the subsequent decay of electrons from various occupied levels j to the inner shell n. This specific screening effect leads to a dependence on l of the frequencies u (l) and the dipole matrix elements HL D (l). LH

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9.4.1. Diatomic molecules with delocalized core holes Formula (163) is applied in this subsection to RIXS K-spectra of a diatomic molecule M with  delocalized core holes. The localized representation yields the same results (see Section 9.3). In this case the f function, showing the polarization dependence of RIXS, will depend on the symmetry group for the molecule through the particular form of the symmetry adapted LCAO expansions of the molecular orbitals. For simplicity the LCAO expansion can be restricted to atomic s and p orbitals. Normalized p and p molecular orbitals and doubly degenerate p and p MOs as E S E S composed of the normalized p and p atomic orbitals (AOs) are considered:   t "C p $C p #2 , (165) H H  H  where only contributions from the p AOs are retained. The dipole matrix element D in the DL GI representation can be expressed in the frozen orbital approximation through the atomic dipole matrix element d and the coefficient C ,C at the MO (165) H H D "0, i"k , GI (166) D "(2C dn, iOk . H GI Here i"g,u and k"g,u are the parities of MO t and core shell 1p , respectively; the unit vector H I n is directed along or perpendicular to the molecular axis for p and p MOs, respectively. Formula (166) implies that gu transitions are only allowed and that gg or uu are parity forbidden. The selection rules (166) lead directly to the following expression for the RIXS cross section of Eq. (163) p(u,u)"p D(X , C )U(X !X ,c)f (h) , (167) M H HL H J JH JH where summation over j"(p ,p ) and l"(p ,p ) over all occupied (p ,p ) and unoccupied (p ,p ) G G G G G G G G orbitals is assumed. Linearly polarized incident and final X-ray photons with the angle h between e and e are considered. The symmetry selective interaction is then described by the functions: f  H G(h)"  N G(C GCI H)(3#cos h)d ,  p p p GH pp f  H G(h)"  N G(C GCI H)(2!cos h)d , pN  N N p GH (168) f  H G(h)"  N G(C GCI H)(2!cos h)d ,  p p N GH Np f  H G(h)"  N G(C GCI H)(1#2cos h)d ,  N N N GH NN where f (h)" fLLY; i, j"(u,g). The population numbers of the ground state can be equal to JH LLY JH N G"0,1,2 and N G"0,1,2,3,4 in the general case. Eqs. (167) and (168) demonstrate directly that p N only occupied MOs of g(u) type are represented in the K-emission spectrum when the frequency u of the incoming photon is tuned into an exact resonance with respect to a vacant MO of g(u) symmetry. 9.4.2. Core excitation to Rydberg states or into the far continuum With modern synchrotron beams, the frequency u can be tuned to exact resonance with some Rydberg levels. A Rydberg level l is defined by quasidegenerate orbitals of p , p or higher SE SE

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189

symmetries, over which the f (h) functions (168) can be summed. This partial summation JH f  (h)" f (h), yields an isotropic f-function J JH JH f  (h)"N (C CI  ) , (169) JH  H H J where the CI  coefficient refers to the lp-atomic orbital in the Rydberg type MO. This result shows J the principal difference between core excitation to valence and Rydberg states. The selection rules (168) are absent in the latter case and apparently the same result is valid for core excitation into the far continuum. Eq. (169) and the selection rules (148), (168) for the scattering tensor F (u) connected with the JL scattering amplitude, F (u)"eHF e, explain the dependence of the spectral shape of fluorescence JH JH (167) on the symmetry of the unoccupied MO t and hence on the frequency of the incoming J radiation u. The angular dependence (168) of the X-ray emission spectrum (167) is also symmetryspecific. This effect is caused by the orientation-selective nature of the interaction of X-rays with molecules (see Section 5), for which the unoccupied MOs oriented along the polarization vector e are excited. 9.4.3. Symmetry selection in a (2#2) model The formalism reviewed above can be illustrated by means of a (2#2) model with (n ,p ) as S E occupied MOs and (p ,n ) as unoccupied MOs. Closed shells (N E"2, N S"4) and unit MO L E S N coefficients are thus assumed. In the broadband excitation case, both unoccupied levels, p and n , E S are excited, resulting in an emission spectrum with two lines, n P1s, p P1s (line 3, Fig. 62). The S E same spectral shape is observed by “tail excitation” with frequency u quite far from a K-ionization threshold. In the opposite limit of narrowband excitation, individual absorption lines can be resonantly (and exclusively) populated; 1sPp or with 1sPnJ (lines 1 and 2, respectively, Fig. 62). E S That only one resonance emission band is obtained in any of these two cases can be seen as an effect

Fig. 62. K-fluorescence spectrum of an M -molecule for a (2#2)-model under symmetry selective excitation. The  positions of emission lines n P1p and p P1p reside at 0 and 2 eV, respectively. The X-ray emission photon energy is S E E S given relative to the n P1p line. h"90°. D"u E S!u S E"2 eV. C"0.1 eV. (1) Excitation of K-electron into S E N N L N p unfilled MO only (c"0.1 eV). (2) Excitation of K-electron into n unfilled MO only (c"0.1 eV). (3) Broadband E S excitation of K-electron into the unoccupied MOs p and n . E S

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of the symmetry selection rules for the scattering tensor in the delocalized core—hole case or as an effect due to interference of the scattering channels (149) in the localized core—hole case. 9.5. Symmetry selection involving electronic continuum resonances Conventionally, X-ray scattering spectra are divided into resonant and non-resonant counterparts, referring to below- and above-threshold excitation ((u(I and u'I), respectively. However, X-ray photoabsorption spectra often also show resonant features above the ionization threshold. In the case of solid state spectra, these resonant features are mostly caused by the multiple scattering of photoelectrons from surrounding atoms or from local potential barriers. Indeed, the structures of these electronic continuum resonances, which differ qualitatively both from the discrete states and from the non-resonant continuum, are manifested as enhanced features in the X-ray absorption spectra just above threshold, 10—30 eV. These so-called shape resonances were found experimentally by Zimkina et al. [216] and were theoretically explained by Mazalov et al. [217], Sachenko et al. [218] and by Dehmer and Dill [219]. Modern investigations can be found in Refs. [220,184,188,221,188,222,187]. One can expect that RXS of molecules under interaction with the continuum resonances poses novel features not present in discrete state RXS or non-resonant continuum RXS. The theory of radiative RXS involving electronic continuum resonances was developed in Ref. [136], and is demonstrated here for the simple case of a diatomic homonuclear molecule and narrowband excitation above the core ionization threshold I. Due to the integration over continuum photoelectron energies, the RIXS cross section reads p(u,u)J p (u)D(X , C )fI (h), e"u!u!I , (170) J H HL HJ H JH where I is the ionization potential of the molecular level j, p (u) is the partial X-ray photoH J absorption cross section to the continuum state t with the photoelectron energy e and CJ l"+n ,n ,p ,p , denoting quantum numbers of the continuum wave function. The fI -functions in E S E S Eq. (170) are given by Eqs. (168) in which CI P1. The expression for the photoelectron energy J e shows that only close to a strong and narrow shape resonance (e ) one can expect a linear M dispersion relation u"u!(e #I ), while in the opposite case the emission frequency is constant M H u"u . HL Close to shape resonances the branching ratio p (u)/p(u) depends strongly on the excitation J energy (Fig. 63). Hence the shape of the RIXS profile must also depend on the excitation energy. This is illustrated by the energy dependence of the partial absorption cross sections of the N molecule [219,220], see Fig. 63b. A strong enhancement of the p partial cross section takes  S place at &0.8 Ry of photoelectron energy. In this broad region the p excitation channel S dominates in comparison with the photoexcitation to the p , n and n continuum states. Thus E S E according to the selection rules and Fig. 64 only two resonances 1n P1p and 2p P1p will be S E S E observed in the X-ray emission spectrum, of N (see below), and the two resonances 3p P1p and  E S 2p P1p are forbidden. All four resonances will be seen when u departs from the absorption E S resonance with the p continuum state since the p , n and n continuum states (except p ) are S E S E S excited with comparable probabilities. The emission spectrum is reminiscent, in this case, of the

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Fig. 63. (a) Total absorption cross section measured by ordinary X-ray absorption spectroscopy. (b) The partial absorption cross sections [219,220]. Fig. 64. Qualitative illustration of selection rules for inelastic X-ray scattering with core excitation to continuum states.

ordinary non-resonant emission spectrum. It can be noted that the measurement of the RIXS profile versus polarization and excitation energy allows one to measure the partial X-ray absorption cross section above the core ionization threshold [136]. 9.6. Symmetry analysis and polarization anisotropy As discussed in Section 9 the RIXS transitions are forbidden when occupied and unoccupied MOs have opposite parity: ug or gu, and only the final states "g\g2 or "u\u2 can be reached from the ground state. This can be easily understood from the fact that the selection rule is controlled by the parameter D D . For instance, if occupied and unoccupied MOs have opposite parity: ug or JL LH gu, then D D "0, and the transition is forbidden. As commented in Section 9 it is possible to JL LH obtain the general symmetry selection rules for the RIXS process by means of group theory; the product of irreducible representations C ;C ;C ;C , where l and n denote unoccupied and J ? @ L occupied orbitals, and a and b dipole moment components, must contain the totally symmetric representation. Due to the similarity between the RIXS transitions and optical two-photon transitions, the symmetry selection rules of the latter can be used to analyze the RIXS transitions. Following group theory it is possible to give the RIXS selection rules for any symmetries of the MOs involved in the RIXS transitions in a compact form as described by [141], and briefly recapitulated here in Section 5.3.1. Table 2 collects the results for different point groups. The allowed occupied MOs for the RIXS transitions are listed together with the features of the

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Table 2 Symmetry assignments of occupied (unoccupied) MOs for the RIXS transitions, when the unoccupied (occupied) MO is of symmetry CM. For the inversion symmetry CM can be gerade (g) or ungerade (u). All other elements should be labeled gerade (g). The symbol (x) shows the cases where the value of P is not possible to find by means of the group theory G Polarization ratios

Groups

Case

P 

P 

P 

C  (C ) G

C  (C ,C ) F Q

1 2

x x

x 3/2

x x

CM

CM b;CM

3

3/4

3/2

2

4

R

3/2

0

C T

D  (D ) F

C  (S ,C )  F

C T (D ,D ,D ) B  F

C  (S ) 

CM a ;CM  b ;CM  b ;CM 

CM b ;CM  b ;CM  b ;CM 

CM e;CM

CM e;CM

CM e;CM

b;CM

b ;CM  b ;CM  a ;CM 

Polarization ratios

Groups

Case

P 

P 

P 

C T (D ,D )  B

C  (C ,C ) F F

C T (D ,D ,D ) F F 

C T (D ) F

¹ (¹ ) F

Q (O ,¹ ) F B

1 2 3

x x 3/4

x 3/2 3/2

x x 2

CM e;CM

CM e ;CM  e ;CM 

CM e ;CM  e ;CM 

CM n;CM d;CM

CM t;CM e;CM

CM

4

R

3/2

0

a ;CM 

a ;CM 

e;CM t ;CM  t ;CM 

u"exp(2p/3).

polarization ratios. The symmetry selection rules are certainly reflected in the polarization of scattered radiation, which can be defined as the ratio between different RIXS cross sections for different combinations of photon polarization vectors, as given by Eqs. (88). However, basically one needs at least three of these combinations of polarization vectors of absorbed and emitted photons; P , P , and P , Eqs. (88). The ratios include RIXS the cross sections p (lp), p (ln) and p (cp),    JH JH JH referring to absorbed and emitted photons having parallel linear, perpendicular linear, and parallel circular polarizations, respectively. By measuring the polarization ratios, P , P and P it is    possible to assign the symmetries of occupied or unoccupied MOs, see Section 5. Table 2 shows that the appearances of P are different for different symmetries of allowed occupied MOs. A closer G inspection of the table reveals that the symmetry assignments are nearly always unique for all the groups. Obviously, Table 2 can also be used to assign the symmetries of unoccupied MOs if the occupied MOs are known. In that case C in Table 2 denotes the symmetry of an occupied MO. The table is general; it covers selection rules for polarized resonance inelastic X-ray scattering from randomly oriented molecules belonging to any of the 32 crystallographic point groups or to the two groups of linear molecules [141].

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9.6.1. Symmetry assignments of benzene MOs from RIXS polarization Benzene, with its high D point group symmetry, serves as a good example for the use of the F polarization ratios, Eqs. (88) and Table 2, to assign symmetries of occupied or unoccupied levels. For instance, according to the parity selection rule, the allowed occupied MOs for the unoccupied a MO are of a , a , b , b , e and e symmetries. However, if the other symmetry elements E E E E E E E are considered, only MOs of a , a , e and e symmetries are allowed, while b and b are E E E E E E forbidden for MO transitions. If the polarization ratios are found to be R, 0 and 0 for P , P and   P , respectively, the occupied MOs must have a symmetry. If P ", P " and P "2, the   E     occupied MOs can be assigned as e . For e , one has P ", P O and P O2. Then the  E E     remaining MOs must be of symmetry a . If the unoccupied MO has b symmetry, then the E S allowed occupied MOs should be b , e , e and b . Table 3 summarizes the assignment S S S S procedure. Of course this table should be interpreted in a general sense; as shown in next section electronic selectron rules are often broken — so indeed also for most core-excited levels of benzene [223]. When such symmetry breaking is due to vibronic coupling the Table 3 should be applied to the symmetries of the various vibronic levels involved in the RIXS process. 9.7. Experimental observation of parity selection rules for RIXS Homonuclear diatomics lack nontotally symmetric vibrational modes, so the selection rules refer directly to their electronic state symmetries. The first experimental evidence of the parity selection Table 3 Symmetry assignments of occupied MOs for the benzene molecule. C denotes symmetries of unoccupied orbitals, Pi the computed polarization ratios, and 1,2,2,10 denote the occupied valence molecular orbitals C

Pi

1

a E

P1 P2 P3 Assign.

    2 a E

e S

P1 P2 P3 Assign.

2     H

2     H

2     H

2     H

e S

P1 P2 P3 Assign.

3.2 0.89 0.27 e S

    2 HH

    2 HH

0.25 0.37 1.48 e S

b S

P1 P2 P3 Assign.

    2 e S

    2 b S

R   0 b S

    2 e S

2

*Possible symmetries are eS,bS and bS. **Possible symmetries are bS and bS.

3

4

    2 e E

    2 a E

5

6

7

8

2     a S

9

10

    2 e E

2     e E

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rules for the RIXS process was given by Glans et al. [142] for molecular oxygen with the excitation energy tuned below core ionization threshold. The same experimental program was realized more recently for the N molecule but with the core excitation both below and above the core ionization  threshold [137]. The details of these experiments can be found in the original papers and in Section 2.2, the main results are reviewed as follows. 9.7.1. RXS for core excitation below the ionization threshold: N  The measurements were made at beamline 7.0 at the ALS. All spectra were recorded in a direction parallel to the polarization vector of the linear polarized synchrotron radiation (at s"0°, see Eq. (69)). The non-resonant X-ray emission spectrum of N (Fig. 65) was obtained by  tuning the excitation energy a few eV above the ionization threshold. The spectrum therefore lacks “initial-state” shake-up and shake-off satellites, which are present in high-energy excited spectra. The assignment of the non-resonant spectrum is shown in Fig. 65. The relative intensities for the

Fig. 65. Non-resonant X-ray emission spectrum of N . The thin solid lines correspond to simulated contributions from  the three different final states, and the thick solid line is the broadened sum of those contributions [137]. Fig. 66. The non-resonant X-ray emission spectrum and RXS spectra of N recorded using resonant excitation to the  1n (v"0), 3p and 3pn orbitals, respectively. The solid lines are simulated band profiles obtained using the lifetimeE E S vibrational interference formulas (96), (110) [137].

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2p , 1n and 3p emission bands obtained from the spectrum in this figure are 15, 100 and 80, where S S E the intensity of the 1n emission band was normalized to 100. S RXS spectra, obtained by making selective core excitation to the 1n unoccupied valence orbital E and the 3sp and 3pn Rydberg orbitals, are presented in Fig. 66. These spectra are compared to the E S non-resonant spectrum shown in Fig. 65. The selection of spectator bands in the RXS spectra is obviously determined by whether the core electron is excited to a gerade (g) or an ungerade (u) orbital. In the first two cases (excitation to 1n or 3sp MOs) the core hole must be in the E E 1p orbital, see Fig. 58(II), since when a photon is absorbed or emitted the parity must change, S and only electrons from gerade orbitals are allowed to fill the core hole. In the second case (excitation to 3pn ), the core hole is in the 1p orbital and only electrons from ungerade orbitals are S E allowed to fill the core hole. The inversion symmetry of the orbital to which the core electron is promoted can thus be determined by which emission bands appear in the RXS spectrum. The results presented in Fig. 66 demonstrate directly the parity selection rules. The shift of peak positions in the RXS spectrum is caused by the different screening effects for different unoccupied orbitals t . J 9.7.2. RXS for core excitation above the ionization threshold: N  As it was shown in Section 9.5 the production of one of the core—hole types (1p or 1p ) may be E S substantially enhanced if the excitation energy is tuned to a shape resonance. Fig. 67 presents spectra obtained by tuning the excitation energies to different parts of the p shape resonance S region (see Fig. 63) in N . Because the shape resonance is embedded in the ionization continuum,  both ungerade and gerade final states are possible, but the ungerade symmetry of the shape resonance leads to a propensity for creating a 1p core hole (Fig. 63b) and hence to a propensity for E decay involving ungerade orbitals. Compared to the non-resonant spectrum the relative intensities of the 3p emission band in the “shape” spectra are considerably smaller, which is in agreement E with the theoretical prediction [136].

10. Breaking and restoration of electronic selection rules The problem of symmetry breaking and particle and hole localization has relevance for a great many phenomena, but is most conspicuous for core level spectroscopies for which it has been debated for a long time [205,207] (see also Section 9). Molecular oxygen and the other homonuclear diatomics serve as the best demonstration cases for parity selection; they do not possess non-totally symmetric vibrational modes that can break the electronic symmetry and the selection rules therefore operate directly on the electronic symmetry of the states. For polyatomic molecules one has observed cases when the symmetry selection is maintained in resonant X-ray emission [145], but also cases when it is broken, e.g. in spectra of benzene [223] and fullerene [145]. Actually, the latter situation seems to be the more common. The lack of apparent symmetry selection for resonant spectra pertaining to higher core-excited states make them rather nonresonant like, while the lowest unoccupied molecular orbital level (LUMO level) can provide a resonance, which is sufficiently isolated to observe symmetry selection in RXS. However, even in resonant X-ray emission spectra of LUMO resonances spectral features have been observed that could be traced to symmetry breaking [145].

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Fig. 67. The non-resonant X-ray emission spectrum and RXS spectra of N obtained by tuning the excitation energy to  different parts of the p shape resonance [137]. S

In the context of the radiative RXS process, the violation of electronic dipole selection rules can be effectuated by two qualitatively different mechanisms. The first mechanism refers to the photon wave function itself; contrary to X-ray photoabsorption or non-resonant X-ray emission, the photon wave functionJexp(ιp ) r) can lead to breaking of the selection rules for the RXS scattering tensor [195,95]. Electronic—vibrational (or electron—phonon) interaction constitutes the second important mechanism for violation of the electronic selection rules [224,225,95,226,117]. Both these mechanisms are briefly reviewed below. The role of electron—phonon coupling for solid state spectra is described in Section 16.4. 10.1. Breaking of electronic selection rules due to orientational dephasing The previous analysis of interference effects highlighted the crucial role of the phase between different scattering channels. When the size of scatterer is comparable with the wavelength of the photon (j) the electronic selection rules break down due to the orientational dephasing (see Section 9.3).

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Taking into account the phase factor exp(i(q ) R )) (158) in the scattering amplitude (7) and L making use of the ¸ (localized) representation, the fLLY-function in Eq. (163) must be replaced by JH (171) 1eiq  RLLY(eH ) D )(e ) D )(e ) D )(eH ) D )2"fLLYsLLY , LH JL HLY LYJ JH JH where the angular brackets denote averaging over random molecular orientations, and R " LLY R !R is the internuclear radius vector. The function sLLY describes the deviation of the expression L LY JH (171) from Eq. (164), as caused by the interference of X-ray photons at equivalent atoms of the molecule. The explicit expression for sLLY is quite formidable in the case of a K-spectrum [227,214]. JH However, the limiting dependences of these functions in terms of the interference parameter qR LLY are easily derived [95]: sLLY"1 if qR ;1 , JH LLY (172) 1 if qR <1 . sLLY"& LLY JH qR LLY The long wave limit is evident because exp(ιq ) R )K1 when qR ;1. In the opposite limit, LLY LLY qR <1, this exponent oscillates strongly as a function of the angle between q and R . An LLY LLY averaging of the RIXS cross section over molecular orientations yields the factor 1/(qR ), thus LLY decreasing the nondiagonal f-function by (qR ) and hence breaking the interference between LLY scattering channels (149). As shown in Section 9.3 and below the role of the interference factor exp(ιq ) R ) is quite different for ordered samples like aligned molecules or crystalline solids. The LLY case of ¸-spectra with the ¸-electron excited into s-orbitals, yields a quite simple expression for the ratio [95] sin(qR ) LLY , (173) sLLY" JH qR LLY which also exposes directly the interference character of the discussed terms. An intriguing possibility for structure determination by means of RXS [95,227] can be envisaged considering the dependence of the RXS spectral shape on the scattering angle through the diffractional factor sLLY (173). JH The RXS spectral shape given by Eqs. (168) looks completely different when the photon wavelength and the size of the scatterer are comparable. In this case the Kronecker d symbol at GH the right-hand side of Eqs. (168) must be replaced by (174) d P(1#Ps) JH GH  with the final state parity P"#1 if the parities of MOs t and t are the same and with P"!1 J H in the opposite case. One can see from this equation that the selection rules (148), (168) are totally broken for large scatterers or for hard X-ray scattering (d P, since sP0) [95]. The sLLYGH  JH JH function is changed from 1 to 1/pR;1 in the hard X-ray region when the scattering angle 0 is changed from 0° to 180°. This strong angular dependence of sLLY allows an active operation of the JH selection rules by changing the scattering angle. 10.1.1. Orientational dephasing in the hard X-ray region The breakdown of the parity selection rules in the hard X-ray region [95] was recently confirmed experimentally [213]. The measurements with the gas-phase Cl molecule were made at beamline 

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Fig. 68. The RIXS spectra for Cl detected in parallel (circles) and perpendicular (squares) polarization following  excitation of the pre-edge resonance (uK2821 eV). Calculated integrated intensities for perpendicular polarization are shown as stick heights [213].

X-24A of the National Synchrotron Light Source (see Section 2.2). The Raman emission spectra at the K edge of Cl were detected in parallel (e""e) and (eNe) perpendicular polarization (see Fig. 6).  The spectra (Fig. 68) include allowed contributions XR>PK PX and XPK PBP as well E S S E as strong forbidden transitions XPK , K PAP and XPK , K PCR>. The calculated S E S S E S integrated intensities shown as stick heights in Fig. 68 are in good accord with the measured perpendicular polarized spectrum [213]. The observed strong violation of the parity selection rules is due to a large value of the interference parameter: qRK4. 10.2. Breaking of selection rules due to vibronic coupling Electro-vibrational (vibronic) coupling can be considered as the second important effect leading to violation of electronic selection rules for the RXS tensor. The theory outlined above is based essentially on the Born—Oppenheimer approximation. As follows directly from first order perturbation theory this approximation breaks down when the spacing between adjacent electronic states is comparable or smaller than the electro-vibrational interaction. For such quasi-degenerate systems the vibronic coupling of the electronic states leads to a strong mixing of the electronic states of different symmetry with the final result of a breaking of the electronic symmetry selection rules [228]. This effect is absent in diatomic molecules since the electronic states of opposite parities cannot be mixed by the totally symmetric vibrations. Such couplings, often referred to as quasi-Jahn—Teller couplings, have been shown to perturb the line shape or fine structure observed in X-ray absorption and photoelectron spectra. The theoretical basis for symmetry breaking of core hole states in non-resonant spectra of polyatomics in general and of CO in particular was originally demonstrated by Domcke and 

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Cederbaum in 1977, when they showed that the O1s\ R and the R states of CO> couple over S E  the antisymmetric stretching modes, thereby lowering the molecular symmetry from d to F c [206]. They proved the relation between the symmetry breaking and core—hole localization T and indicated that such effects could be the rule rather than the exception for molecules with an element of symmetry. This is because core—hole states are quasi-degenerate in the high symmetry point group and therefore couple over one (or more) non-totally symmetric vibrational modes leading to a lowering of the point group. CO is thus an excellent example of such “quasi Jahn—Teller” effects, and later work with geometry optimization [229] indeed seemed to verify the predictions of Ref. [206]. Several theoretical works along this theme have subsequently been elaborated and reviewed [230—232,226]. The experimental verification of core—hole symmetry breaking has, however, only been indirect, mostly through analysis of band profiles and Franck—Condon progressions of non-resonant X-ray emission [208] spectra and photoelectron and photoabsorption spectra [229,233—240]. In resonant X-ray emission the vibronic coupling and the accompanying core—hole localization leads to the appearance of “forbidden” transitions [149,117]. This makes RXS a powerful method for detecting and even quantifying vibronic coupling and symmetry breaking. The violation of electronic selection rules due to a vibronic coupling has been observed in the RIXS spectra of crystalline diamond and of graphite [225], of the evaporated disordered films of C [241,145], C [148], and   of the gas-phase benzene [223] and CO molecules [103,117]. All of these experiments were  performed at beamline 7.0 of ALS (see Section 2.2). It is emphasized that the vibronic-coupling breaks down only for the electronic selection rules, not the selection rules for the total electron—vibrational wave function of the molecule. 10.2.1. Theory of RXS beyond the BO approximation In the same way that O and N are good test cases for demonstration of parity selection in   resonant X-ray emission, carbon dioxide is a natural candidate to examine symmetry breaking in RXS. The introduction of a carbon atom in between the two oxygen atoms adds an antisymmetric stretch mode which can couple the gerade and ungerade oxygen core-excited states and therefore break the parity selection rule. A general theory of RXS beyond the BO approximation based on the Green’s function technique was developed by Cesar et al. [117]. The special case of three-atom molecules can be more simply described, and the most transparent model of this particular feature of RXS is thus given by the CO molecule [242].  The O1s transitions to the unoccupied 2n molecular level in CO are followed by decay S  transitions to the ungerade and gerade final states, resulting in the high-energy and low-energy peaks, respectively. Initially, the molecule is in the gerade ground-state. Without vibronic coupling the two intermediate core-excited states, W ""1p\2n 2 and W ""1p\2n 2, are eigenstates of E S S S E S the unperturbed electronic Hamiltonian. These two states can be assumed degenerate (E,E +E ) with negligible energy splitting (K0.005 eV). Of the two final states, " f (u)2" E S "1n\2n 2P and " f (g)2""1n\2n 2P , only the latter can be reached by dipole transitions E S S S S E because the absorption transition to the dark, gerade, state is forbidden by the dipole selection rules. The core excited electronic states W and W of different parity are coupled by the E S antisymmetric stretch mode Q . The coupling potential S »"iQ (175) S

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can be expanded up to the linear term Q . This linear model has been applied to describe the S Jahn—Teller effect [243] and to different problems in X-ray spectroscopy [224,226]. The total Hamiltonian of the core excited molecule is, in a two-level approximation, the sum of » and the adiabatic Hamiltonian H "E#h M H » . (176) H" M » H M The nuclear Hamiltonian h accounts for two normal stretch modes: the symmetrical and antisymmetrical modes with the normal coordinates Q , Q and vibrational frequency u , u , respecE S E S tively. The orthogonal transformation º diagonalizes the Hamiltonian (176)
















H 0 1 1 !1  , H "E#h$», º" , (177)  0 H 1 1 (2  and removes the degeneracy of “delocalized” states W and W and carries out the transition to E S localized electronic states [226] (Fig. 69) HI "ºRHº"

W "(1/(2)(W !W ), W "(1/(2)(W #W ) , (178)  E S  E S which are the eigenfunctions of the hamiltonians H and H , respectively. The results (177), (178)   have a deeper physical meaning: According to the Jahn—Teller theorem [243] the perturbation removes the degeneracy of the adiabatic states W and W by lowering the molecular symmetry. E S Apparently, the new Hamiltonians H (177) are reduced to the sum of two harmonic Hamil tonians hI "hI #hI with the shifted normal coordinate of the antisymmetrical mode   Q"Q $i/M u (179) S S S S and M as the effective mass of the antisymmetric vibration [242]. Thus, the total eigenvectors for S core excited states are the products of electronic and vibrational wave functions W "1,m2 and  W "2,m2 with m"(m ,m ) as the vector of the vibronic quantum numbers for the symmetric and  E S antisymmetric modes. The equilibrium internuclear distances of the core excited states W and  W are shifted relative to the unperturbed value by Gi/M u (see Eq. (179)). The final expressions  S S

Fig. 69. Symmetry selective RIXS.

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for the RIXS scattering amplitudes (7) of the allowed (F E) and “forbidden” (F S) scattering channels D D read [242] 1 F ?" uuDR ? D ($1o"1,m2¸m11,m"n2#1o"2,m2¸m12,m"n2) , @D gu m D 2 (180) ¸m"(X!m u !m u #iC )\, X"u!(l #(u !u #u !u )) . E E S S G M  E ME E ME Here l "E!E is the difference of the electronic energies of intermediate and ground states, M M u is the ground state vibrational frequency, 1o"i,m2 and 1i,m"n2 are the FC factors between M? vibrational wave functions of the ground and core excited states and between vibrational wave functions of the core excited and final states. The signs $ correspond to the gerade and ungerade final states, respectively. Due to the dipole selection rules the parity b of the core excited electronic state in D ? is opposite to the final state parity a (if a"g,u then b"u,g). The scattering amplitude @D is written here as a function of excitation energy. The dependence of F ? on the final photon D frequency u can be obtained with help of the Raman—Stokes law (9). 10.2.2. Bond length-dependent symmetry breaking As discussed in Section 10.2 RIXS can be used for detecting and even quantifying vibronic coupling and symmetry breaking in core excited states. The benzene and fullerene molecules are examples for which both symmetry breaking and selection are observed in the same spectra, while, for instance, the O1s carbon dioxide spectrum shows an almost complete symmetry breaking at resonance. The question is if symmetry breaking in RIXS spectra is the rule rather than exception for molecules with an element of symmetry, that is when it contains delocalized core orbitals near in energy. The problem was addressed in Ref. [244] (Fig. 70), by analyzing soft X-ray emission spectra from three related hydrocarbon molecules; acetylene, ethylene and ethane. A clear trend was observed going from acetylene to ethane. The acetylene spectra are strongly dependent on the excitation energy, with the relative intensities of the peaks changing in a simple manner. Ethylene also shows some intensity variations, while in contrast, the ethane spectra appear to be very similar. Taking difference spectra between nonresonant and resonant spectra an estimate of the amount of symmetry breaking could be achieved, with the conclusion that acetylene is largely symmetry selective, ethane not at all, while ethylene is a complex intermediate in this respect. A simple rationalization of these observations was made in terms of the the C—C bond length in the C H , C H , C H series, using the vibronic coupling (VC) [244] parameter       (181) j"1u"º"g2/*E&e0C? , upon which the amount of symmetry breaking or the degree of the core hole localization is dependent. Here "u2 and "g2 are the nearly degenerate symmetry related core orbitals with energies e and e , respectively. *E"e !e and º"(jº/jQ ) with º being the VC operator and Q the S E S E SM S normal coordinate. The denominator of the expression for j contains the small energy difference of the core orbitals which depends exponentially on the separation R of the carbons and on the C spatial extent a of the core orbitals. Slow variations in the numerator in Eq. (181) can be neglected. This simple expression thus predicts that a single bonded species like ethane, which has a comparatively long bond distance, is more likely to break symmetry than a triple bonded species like acetylene, with a smaller bond distance, and that oxygen containing symmetric molecules are more

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Fig. 70. RIXS spectra of acetylene, ethylene and ethane [244]. Top panels: Absorption spectrum from Ref. [236] (C H ),   [371] (C H ) and [234] (C H ). Panels a—g: Full lines: Emission spectra recorded at the excitation energies marked by     arrows in the top panel. The transitions are labeled by the valence orbital from which the deexcitation takes place. Dotted lines: Difference spectra between resonant and nonresonant spectra (described in Ref. [244]). The non-resonant spectra contain bars determined by intensities obtained from Hartree—Fock calculations and vertical energies obtained from photoelectron spectra [244].

likely to break symmetry than carbon containing ones of the same size. This is also reflected by the amount of near degeneracy, *E, between the gerade and ungerade C1s orbitals which differs by as much as an order of magnitude due to the different C—C bond lengths for the three molecules. That symmetry is preserved for core-excited acetylene was also proved very recently through highresolution NEXAFS, which showed band widths smaller than the energy splitting between gerade and ungerade C1s orbitals [210]. 10.3. Restoration of selection rules through frequency detuning 10.3.1. Qualitative picture The symmetry breaking caused by vibronic coupling of electronic states of opposite parity (Fig. 69) “takes time”, that is roughly a vibrational time period q (60). If the RXS duration time q (40) 
 is much smaller than q , the molecule has no time to execute the antisymmetric vibration that 


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Fig. 71. (a) Dependence of symmetry breaking parameter s (ratio of intensities of “forbidden” and “allowed” transitions) on detuning u of incoming photon frequency from absorption resonance. (b) Dependence of symmetry breaking parameter s on effective broadening C"D of the X-ray absorption electro-vibronic band for broadband excitation. 


introduces forbidden parity in the electronic wave function through the vibronic coupling. In the case of resonance excitation the RXS process is prompt (see Fig. 21), and the lifetime of the resonant state determines the duration time (40). Tuning away from resonance shortens the duration of the RXS process, and one probes preferentially transitions associated with shorter lifetimes. Thus, vibronic symmetry breaking will be less and less effective [103,117,242,192] (see Fig. 71). This result follows also directly from the expression for the fast RXS amplitude F &1 f "DRD"o2 (59). The D RXS process thus becomes independent of the (symmetry broken) intermediate state, and restores the “quadrupole selection rules” of the RXS scattering tensor. 10.3.2. Excitation with small detuning Eq. (180) clearly displays the restoration of the parity selection rule for the RIXS tensor (gPg, F SP0) when the detuning frequency exceeds the vibrational broadening of the photoD absorption resonance: X'D . It is convenient to characterize the strengths of the “forbidden” 
and “allowed” transition by the ratio [103,117,242] s(u)"p (u)/p (u) S E of the corresponding integral partial RIXS cross sections [242]

(182)

1 p (u)" ruu(DR ?D ) +("1o"1,m2"#"1o"2,m2")"¸m" @D ES m ? 2 M $2Re 1o"1,m2¸m11,m"2,m 2¸H 12,m "o2, .  m  m

(183)

This equation is written for a narrowband excitation. Because of the dependence of the duration time (40) on the lifetime broadening C it is easy to understand that the relative intensity s of the “forbidden” band tends to zero with increasing C , G

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Fig. 72. Dependence of the symmetry breaking parameter s (182) on the excitation energy for a narrowband excitation and identical adiabatic potential surfaces of ground and core excited states. Here u ,u is the ground state vibrational M MS frequency. The strength of vibronic coupling is assumed equal to unity: i/(2M u"1. S S

Fig. 71b. In the case of broadband excitation (U(X,c)"const) the C-dependence of the s-function was investigated in Ref. [226]. Fig. 72 [242] shows the non-monotonic and asymmetric dependence of the symmetry breaking parameter s (182) on the excitation energy and the restoration of selection rules for detuning larger than the effective width of the electro-vibronic photoabsorption band D . This parameter has a smooth frequency dependence only when the lifetime broadening is 
comparable with the vibrational frequency u . S The breakdown and restoration of the RIXS selection rules caused by detuning is demonstrated by the R> and R\ RIXS bands of CO (Fig. 73) calculated for different excitation energies [117]. S S  The calculations are carried out for the ground XR , core-excited 1p\2p; P and final valence E E S S excited 1p\2p; BR\, 1p\2p; DR\, and 1p\2p; ER> electronic states of the CO E S S S S E E S S  molecule. 10.3.3. Excitation with very large detuning The s-function given by Eq. (182) is in practice valid only fairly close to the absorption resonance being considered. When the incident photon frequency is tuned very far from this resonance many other core-excited states, centered at different energies and of different symmetries, will simultaneously be excited with comparable probabilities due to tail excitation. When these core-excited states are of different symmetry the symmetry character of the emission will also be mixed, and the emission will become more like non-resonant X-ray emission [95]. So the symmetry breaking parameter s (182) will in practice have a non-monotonic behaviour. At first, the s-function decreases (solid line, Fig. 71a) and then it increases (dashed line, Fig. 71) as the absolute value of the detuning increases. Fig. 71a shows also the behaviour of the s-function for positive detuning frequencies. The frequency dependence of s is qualitatively the same if the detuning is negative but is for “real molecules” different due to the non-homogeneous distribution of core-excited states

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Fig. 73. Calculated (2p ) resonant X-ray emission spectra of CO assuming narrow-bandpass (c"0.1 eV) excitation. S  Excitation energies: (a) u"536.00 eV, (b) u"536.30 eV, (c) u"536.43 eV, (d) u"536.59 eV, (e) u"536.75 eV, (f) u"536.91 eV, (g) u"537.22 eV, (h) u"538.00 eV. The intensities of the vibronic bands R\ and R> have to be E S summed together for the final total spectrum. The D\(D\) and R\(R\) final states are assumed degenerate [117]. S E S E

above and below the resonance and due to the difference in the FC factors above and below the turning point. 10.3.4. Experimental investigations of breakdown and restoration of electronic selection rules The experiments with the CO molecule were made at the undulator beamline 7.0 at ALS in  Berkeley (see Section 2.2), see Figs. 74 and 75. A comparison between Figs. 73 and 74 shows that theory reproduces all main observed spectral features [117]. Moreover, both theory and experiment demonstrate the breakdown of the electronic selection rules close to the photoabsorption resonance and their restoration with the detuning away from the vertical photoabsorption energy. The effect of the selection rule restoration is seen directly in Fig. 75. A list of other experimental

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Fig. 74. Oxygen K2p resonant X-ray emission spectra of CO with different detuning energies below the 2p resonance. S  S The resonant spectra show two spectator peaks and a weak high-energy elastic peak due to a participator transition [103,117]. Fig. 75. Ratio between high-energy (“forbidden”) and low-energy (“allowed”) spectator peaks (Fig. 74) of the 2p S resonant oxygen K emission spectra as a function of detuning energy measured from the vertical O1sP2p absorption S energy. i+D +1.4 eV. Monochromator resolution is estimated to 0.65 eV. Solid line shows results of theoretical 
simulations [103,117].

investigations of non-adiabatic effects in RIXS of more complex systems was presented at the beginning of Section 10.3.

11. X-ray resonant scattering involving dissociative states With the development of tunable, narrowband, synchrotron-radiation sources, the studies of the resonant X-ray scattering process are no longer limited to systems with discrete bound states, but can and do now also involve systems with states that are unbound along the nuclear degrees of freedom. The diatomic hydrides served as the original prototypes for non-radiative RXS spectra involving dissociative states. Decay channels with dissociation preceding electronic decay were first identified in the spectrum of HBr recorded at the 3dPpH excitation energy [245]. The HCl

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2pPpH resonance was also found to decay predominantly by dissociation followed by the electronic decay [246]. The time scales of the dissociation and the Auger decay were estimated to be of the same order of magnitude [245]. The H S molecule served as the first polyatomic species  showing similar features [247,248]. Studies of this species clearly indicated that the character of the core-excited state determines the relaxation path, and that dissociation before decay indeed is possible even for short-lived core hole states. The 2p absorption spectra in this molecule, as in HCl, exhibits a pre-edge structure [249,247] consisting of a broad band due to excitations to the first unoccupied molecular orbital — 6a and 3b in the case of H S, 6p or “pH” for HCl — followed by    a series of sharp peaks corresponding to excitations to Rydberg orbitals. The identification of the Auger spectra for the various excitation energies indicated that the first type of excited states relaxed through Auger transitions in dissociative fragments, while excitations to the bound Rydberg states showed resonant Auger decay in the molecular environment. Calculations on core-excited adiabatic interatomic potentials of different molecules, for example O [250], HBr  [251], HCl [246], H S [247], have confirmed that intermediate or final states with dissociative  character are indeed relevant to consider for the RXS process. The experimental conclusions about the relaxation paths of the core-excited states thus followed from energy assignments of the Auger decay spectra. These spectra were interpretable in terms of diagram levels of the fragments. The assignments, the excitation energy dependences as well as mass spectroscopic data gave hints of a mechanism in which dissociation is faster than the electronic decay of the excited fragment. From further experimental progress with synchrotron radiation it has also been possible to use line shapes and the Auger resonance Raman effect [24,42,86] to draw conclusions on the character of the intermediate and also of the final states [252,253,88,102]. Only the bound states showed the expected resonance narrowing of the bands (Raman effect), while the Auger transitions to final dissociative states lacked such narrowing and were determined by their lifetime broadening only. In this section the theory for RXS involving dissociative states is reviewed with emphasis on the time-independent description. The time-dependent theory for the RXS process is further elaborated in the following section, Section 12.

11.1. Decay channels involving continuum and bound states Many, if not most, molecular core-excited states are dissociative or predissociative, and it is desirable to include these in a general theoretical treatment. Interference effects will be a central concept also in such a treatment. The main peculiarities of the problem can be understood for the special case when the incoming X-ray photon excites the molecule to the adiabatic interatomic potential º (R) of a dissociative intermediate state, or above the dissociation threshold if º (R) has A A minimum. As is shown in Fig. 76 two qualitatively different channels for the radiative decay exist. One is the decay from an intermediate continuum nuclear state "c2"uA A(R) to a bound vibrational # state " f 2"uD (R) of the final internuclear potential º (R). Here E is the molecular energy of the K D A continuum state uA A(R). This channel is thus a “continuum—bound” channel. The second type of # channel is given by the decay into final dissociative states " f 2"uDD(R) with the molecular energy # E , the “continuum—continuum” channel. In the considered case, the cross section (95) of the D resonant scattering of a narrowband X-ray beam is given by the sum of cross sections p (u,u) and A@

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Fig. 76. Excitation and decay schemes. The ground (o), intermediate (c) and final (f) states are displaced relative to each other. All notations are explained in the text.

p (u,u) for the continuum—bound and continuum—continuum decay channels: AA p(u,u)"p (u,u)#p (u,u) , A@ AA p (u,u)"(u/u) "F "d(u!u#u ) , (184) A@ DK KM K p (u,u)"(u/u)"F ", E "u!u#E . AA D D M The RXS cross section (184) has two qualitatively different contributions, one of them, p , is sharp A@ in frequency, while the other, p , has a smooth frequency dependence. The spectral width of the AA incoherent part of the cross section p is defined by the width of the continuum scattering AA amplitude F . The spectral distribution of F is given by the spectral distribution of the D D Franck—Condon amplitudes as discussed further in the next subsection. Eqs. (184) show directly that for the continuum final state the positions of the emission resonances do not follow the Raman—Stokes law (9) and that the width of the emission peak cannot be made smaller than lifetime broadening [91]. In the common Born—Oppenheimer and Condon approximations, the electronic transition matrix elements are treated as constants instead of as functions of the nuclear coordinates, and the scattering amplitudes (110) for the continuum—continuum F and for the continuum—bound D F channels read: DK 1uD "uA 21uA A"u 2 # M , (185) F "a dE #DK #A A u!u #iC DDK AM where u "E !E . The continuum nuclear wave functions of states i"c, f are here normalized AM A M to a d-energy function. The sum on the right-hand side of Eq. (185) implies that for a bound intermediate state "c2 one needs to integrate over the energy E or to sum over the vibrational states A uA if the incoming photon frequency is tuned above or below dissociation threshold of the L intermediate state, respectively. The scattering amplitude (185) is defined by Franck—Condon (FC) amplitudes (the overlap integrals) between the vibronic wave function u (R) of the ground state and M the continuum nuclear wave functions uA A(R), and between uA A(R) and the final state nuclear wave # # functions uDD(R). #



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11.2. Space correlation between absorption and emission The interference between different intermediate continuum states plays a fundamental role in the damping of emission as the internuclear distance increases. To explicitly show this fact it is convenient to rewrite the scattering amplitude (185) F "1uDD"u 2 D # A

(186)

in terms of the stationary wave packet u (R) and the time-independent Green’s function A u "aG u , A # M



uAH(R)uA A(R) # G (R,R)" dE #A # A E!E #iC A

(187)

with E"u#E . The stationary lifetime broadened Green’s function G (R,R) describes the M # propagation of nuclei on a decaying potential surface, º (R), from the internuclear distance R, A where the molecule was core excited, up to R, where the emission transition took place. When the duration of RXS q (40) is short (61), the Green’s function [102] A d(R!R) G (R,R)+ # E!º (R )#iC A M

(188)

shows that the emission and absorption transitions take place at the same point (*R"R!R"0) (Fig. 77). This result also follows directly from Eq. (59). Thus, in the limit of a small lifetime C\ or large absolute value of detuning X"u!u (R ), the molecule has no time to spread from the AM M point of absorption. Here R is the equilibrium internuclear distance in the ground state, M u (R )"º (R )!E . The finite RXS duration time means that the core excitation cuts off the GM M G M M coherent superposition u (187) of the core excited states residing in an energy band width given by A the inverse duration time "E !E"4q\ (Fig. 77). When q is short all intermediate states A A A ("E !E"4R) give coherent contributions to the wave packet u (maximum interference between A A core excited states). In this case the point of emission is known exactly (R"R, *RP0) according to Eq. (188). The X-ray excitation cuts off only a small part ("E !E"4q\P0) of the continuum A A intermediate states if q is long (Fig. 77). Hence, according to the uncertainty principle the A d(R!R)-function in Eq. (188) is broadened and the explicit information about the emission point R is lost (*RPR) (Fig. 77). A deeper understanding of the case of finite q can be obtained with help of the lifetime A broadened Green’s function (187). In the relevant region, the nuclei move with an energy E larger A than the potential height º (R). So the criterion of applicability of the quasiclassical approximation A is fulfilled everywhere and the quasi-classical wave function can be written as






0 A exp !i p (R) dR . uA A(R)" A # (p (R) A A

(189)

Here p (R)"(2k(E !º (R)) is the momentum, c+R is the classical turning point where M A A A p (c)"0, and A is the normalization constant. The small correction term containing the wave A reflected by inhomogeneities of the potential is neglected in Eq. (189). In the classically accessible

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Fig. 77. Illustration of E-interference (upper panel) and t-interference (lower panel). *R"R!R is the distance between absorption (R) and emission (R) points, v is the characteristic nuclear velocity, and q is the duration of RXS. All A notations are explained in the text.

region (R,R5c and R5R), the lifetime broadened Green’s function shows strong space correlation between the absorption and emission processes G (R,R)"GM(R,R)e\CO0Y0, GM(R,R)"!2ipA # # #

exp(!i0Yp(R) dR) 0 , (p (R)p (R)

(190)

where p (R)"p(R)!iC/v(R), p(R)"(2k(E!º (R), and v(R)"p(R)/k is the relative velocity of A the nuclei at the point R. The lifetime broadening C is assumed here to be small in comparison with (E!º (R)). As follows from the factor exp(!Cq(R,R)) in Eq. (190) the emission intensity is A negligible if the time of propagation between the absorption point (R) and the emission point (R)



0Y dR (191) v(R) 0 exceeds the lifetime; q(R,R)5C\. Indeed, the emission takes place only as long as the population of the core excited state remains unexhausted. A second reason for small contributions from far “emission points” R to the scattering amplitude is given by large detuning frequencies X"u!(º (R )!º (R )). This contribution is A M M M small for large positive detunings (X'0) because of the strong oscillation of the Green’s function G (R,R)JeιN0Y\0 in the integral (186), where p"("2kX". When detuning is negative (X(0) the # amplitude of excitation is exponentially small GM(R,R)Je\N0Y\0 for core excitation in the # q(R,R)"

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classically forbidden region (E(º (c)). One can conclude that both positive and negative detunA ings “cut off” the far emission transitions with the sudden limit (188) for large "X" (61). It can be pointed out that the scattering amplitude F tends to the fast (or vertical) limit (188) for different D reasons for positive and negative detunings. It is easy to understand that the internuclear distance "R!R" between absorption and emission points cannot exceed the distance (vq ) passed by the nuclei during the RXS duration (Fig. 77): A "R!R"4vq . (192) A The small contribution of the “far” emission transitions is caused by the interference between the core excited continuum states uA A(R) coherently excited into the band "E !E"4q\. So this A A # interference between continuum states and the finite value of q plays the key role in the damping of A X-ray emission at large internuclear distances. It should be noted that the time-dependent representation gives an alternative view on this problem [102], see Section 4.8, and also Section 12. 11.3. Franck—Condon amplitudes The measured X-ray lineshape results from the interplay between the shapes of several functions; the photon function, the lifetime broadening function, and the vibrational (discrete or continuous) distribution function. This interplay will in turn be dependent on the character of the participating states, if they are bound or dissociative. The spectral shape of the RXS cross section is defined by the spectral shape of the FC factors. Only three types of the FC factors can enter the scattering amplitude, namely, the bound—bound, the bound—continuum and the continuum—continuum FC factors. The properties of the bound—bound FC factors are well known and they do not connect directly with the discussed dissociative problem, and therefore only the properties of the latter two kinds of the FC factors will be reviewed in the following. 11.3.1. Bound—continuum Franck—Condon amplitudes Consider the transitions between a dissociative state uG G(x) and a bound state uH (y), where K # x"R!R and y"R!R are the deviations of the internuclear distance R from an equilibrium G H position R and R of the electronic states i and j, respectively. The characteristic length a of the G H G bound wave function and characteristic scale a of oscillations of the continuum wave function are GH (193) a "( /ku , a "( /2kF ) GH GH H H with i, j"+o, c, f , and F "!(dº (R)/dR) H as the interatomic force at the equilibrium point GH G 0 R of bound state j. Very often the scale of oscillations of the continuum wave function is larger H than the size of the bound state a /a <1 (the opposite limiting case was considered in Ref. [102]). GH H One can show [102] that in this case the bound—continuum FC factor is proportional to the vibrational wave function of the bound state 1uH "uG G2K(2ka uH (!*e /F ), *e "E !º (R ) . K # GH K G GH G G G H An important characteristic of this FC factor is its spectral width c "F a . GH GH H

(194)

(195)

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A thorough analysis of bound—continuum FC factors has been given by Tellinghuisen [254]. Eq. (194) can also be obtained in the reflection approximation [255] where the continuum wave function is replaced by d(x!x ) with x as the classical turning point. One notes that the spectral M M shape of the FC factors (194) copies the space distribution of the bound state nuclear wave function uH (x). This important peculiarity of the FC factors leads to the principal possibility of mapping the K vibrational wave function. 11.3.2. Continuum—continuum Franck—Condon amplitudes The next important decay channel is formed by transitions into final continuum states uDD # lying above the dissociation threshold of the final state potential º (R) (continuum— D continuum decay channel). To estimate the scattering amplitude for this channel it is necessary to assume that the internuclear distance remains unaltered during the electron transition cPf (vertical approximation). Suppose that this transition takes place at points R near the crossing point R [76] AD u"u (R ), AD AD

(196)

of the potential surfaces º (R) and u#º (R) (u (R)"º (R)!º (R)). Only such points can give A D GH G H significant contributions to the FC factors. The solutions of the Schro¨dinger equations for the intermediate and final nuclear states close to this stationary point R are given by Airy functions AD Ai(x) [98]. Taking into account the properties of the Airy functions [98,256] one receives the following expression for the continuum—continuum overlap integral 1uDD"uA A2"d(e !e ), if F "F D A D A # #




1 1 e e D! A 1uDD"uA A2" Ai $ # # c a F F D AD A



, if F OF D A

(197)

with c "a F""( /2k)(F !F )F", a "(a a "(F !F )/F" , AD D A D A D A F"(F F , A D

a "( /2kF ) . G G

(198)

Here the slope F "!(dº /dR) and the kinetic nuclear energy e "E !º (R) of states i"c, f are G G G G G calculated at the stationary point R"R . Eq. (197) is written for F '0. The # and ! signs in AD G the argument of the Airy function should be used for the cases F 'F and F (F , respectD A D A ively. 11.4. RXS cross sections for dissociative potentials 11.4.1. Continuum—continuum decay channels. Fast RXS in the molecular region Results reviewed in Section 4 showed that the shape of the RXS profile depends strongly on the duration q of the scattering process which can be changed by a variation of the excitation energy, A Fig. 22. When the RXS duration is short (61) the spectral shape of RXS is defined according to

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Eq. (59) by the projection of the ground state vibrational function onto the continuum wave function of the final state [102] a 1uD "u 2, q\





11.4.2. Continuum—continuum decay channels. Slow RXS in the dissociative region In the other important limiting region, the dissociative region, the spectral width of the continuum—continuum FC is small and c ;q\, c , (200) AD A AM F PF . According to lim Ai(x/a)"ad(x) the FC factor (197) is now equal to A D ? 1uDD"uA A2"d(e !e ). From Eqs. (184) and (185) we thus have the following expression for the RXS D A # # cross section in the harmonic approximation and in the limit (200) and C(c [91, 102, 106] AM u!u (R )#*e  F AM M p (u,u)"p DM *(u!u (R ),C)exp ! , (201) AA AD AD c F AM AM where *e"F a and all unessential quantities being collected in the constant p. AM AM It is necessary to complete the special important case when the solution (201) describes the emission in the dissociative region R PR. Indeed, this limit agrees with the limitations of Eq. AD (200), since the spectral width of the continuum—continuum FC factors (197) tends to zero (c P0) AD due to the the slope F P0 when R PR (Fig. 78). Contrary to the molecular contribution, Eq. G AD (199), the position of the atomic-like resonance u"u (R), Eq. (201), does not depend on the AD excitation energy. One thus arrives at the conclusion that for the structures in an RXS experiment involving dissociative potentials, the Raman—Stokes law holds for the “molecular” part but not for the “atomic” part of the spectral structures. This piece of information can be useful in assigning the spectra.






11.4.3. Intermediate region If the spectrometer is tuned in the intermediate region with the crossing point (196) lying in the intermediate molecular region (R (R (R), the spectral width c (198) of the M AD AD

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Fig. 78. Qualitative separation of the R-space into molecular and dissociative regions (upper panel). The qualitative dependence of the spectral width of the continuum—continuum FC factor 1 f "c2"1u D"uA A2 (197) on the internuclear # # distance R (middle panel). The qualitative dependence of the continuum—continuum FC factor (197) on the energy *E"F(e /F !e /F ) in the molecular and dissociative regions (lowest panel). The FC factor amplitude (J1/c ) is D D A A AD large in the dissociative region but small in the molecular region (the oscillatory character of the FC factor is not depicted).

continuum—continuum FC factor depends strongly on the difference between the slopes F and A F governed by R (Fig. 3). There are two important limiting cases in this respect. In the first case, D AD the spectral width c is small in comparison with the widths of the bound—continuum FC factor AD c and the lifetime broadening. In this limit (200) the RXS cross section (201) is described by AM a product of a Gaussian and a Lorentzian with the spectral widths c and C. In the opposite AM limiting case q\;c , c , A AM AD

(202)

the RXS cross section reads








X#F a  u!u!u (R ) nC F F DM D exp ! AM AM DM AD Ai ! p (u,u)"p AA c c c F F AM AD AD AM A



e\CO , (203)

where X"u!u (R ), u (R )"º (R )!E , AM M DM AD D AD M






F

 F F   A D c "c . " AD AD "F !F " 2k F !F D A A D

(204)

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The damping factor exp(!2Cq) on the right-hand side of Eq. (203) shows that the time q"q(R ,c) AD (191) of propagation on the core excited potential surface U (R) to the crossing point R where the A AD emission transition takes place cannot exceed the lifetime C\ of the core excited state. Here c is the classical turning point lying near R . M Comparison of Eqs. (201) and (199), (203) (see also Fig. 78 and Fig. 79) shows that the RXS profile consists of two qualitatively different contributions. The first contribution, (201), represent decay transitions in the product of dissociation. The second contribution, caused by the spectral transitions in the molecular region, is as a rule broader (with the width min+c ,c ,) than the AM AD first one.

Fig. 79. The upper panel shows the qualitative dependence of the spectral shape of the conitnuum—continuum RXS cross section on the emission frequency u. When the lifetime broadening is sufficiently small the decay transitions in the molecular region form the wing of the spectral band with weak intensity in comparison with the narrow resonance caused by decay transitions in the dissociative region. According to Eq. (203) the spectral shape of the wing has oscillatory character. The lower panel illustrates the “molecular-like” background and “atomic-like” resonances in the RXS spectrum. Fig. 80. Formation of the atomic-like resonance with “blue” and “red” wings.

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The ratio of the integral “molecular-like” (199) and “atomic-like” (201) contributions can be estimated as the following [106]: c C D p  (u) + DM m(X), m(X)" eXAA , z" AA (X!F a )#D p (u) b D(p A A AA D

(205)

where the dimensionless constant b is of the order of 1. Eq. (205) shows that the relative intensity of D the atomic-like dissociative resonance (201) decreases (up to zero) when the excitation energy is tuned far from the resonant frequency u (R ) of the vertical photoabsorption transition. This effect AM M was observed recently in the resonant Auger spectra of HCl [106] (see also below). Expression (203) shows that the Airy function leads to slowly damped oscillations of the cross section in the “molecular” region (R (R (R). The smooth part of the cross section (203) plays M AD the role of a background for the resonance contribution (201) in the dissociative region as illustrated in Fig. 79. As one can see from the lower panel of this figure the background has an asymmetrical spectral shape due to the non-linear potential of the final state. It should be noted that the spectral shape of the background also can have a resonant structure if the final state is bound, Fig. 80. The atomic-like resonance has in the general case both blue and red wings [102,106,113,109]. 11.4.4. Evidence of the molecular- and atomic-like contributions in RXS spectra Experiments giving evidence of the “molecular-” and “atomic-like” contributions in non-radiative RXS have been performed at beamline 51 at the MAX laboratory in Lund [106], Fig. 81. The spectrum of the HCl molecule was measured at several energies close to the Cl 2p PpH X-ray  photoabsorption transitions. The spectrum shows a three-band feature with a broad background (&6 eV). The narrow atomic-like resonances with HWHM (C"70 meV) can be assigned to the final states of the Cl> ion: A"S, B"D, and C"P, Fig. 81. The following final molecular states of the HCl> ion correspond to these atomic states, S:R>, D:*,R>,P, and P:R\,R\,P [257]. To give the semiquantitative description of these spectra a simplified model was used with only three final molecular states [106]. The “molecular” part of the RXS cross section was approximated by the asymptotic formula (199). The result of the simulations (Fig. 81) show that the total RXS profile is the sum of the three bands, each consisting of the narrow atomic-like contribution with the lifetime width C"70 meV and a “molecular” background with the full-width at half-maximum (FWHM) 2c (ln 2+1.8 eV. The ab initio time-dependent calculations of the DM Auger resonant Raman spectrum of the HCl molecule (see Section 12.2.3) confirm the main spectral features of the experimental spectra shown in Fig. 81. The quantity g"p  (u)/p (u)"z/(1#z) refers to the relative contribution of the molecular AA AA part defined as the ratio of integral cross sections p  (u) and p (u)"p  (u)#p (u), see Fig. 82. AA AA AA AA Both theory and experiment demonstrate the enchancement of a molecular-like broad background by detuning the excitation frequency, and that, in contrast to the atomic-like resonances, it follows approximately a Raman—Stokes dispersion law [101]. A recent experiment by Magnuson et al. [258] at the sulphur ¸-edge of OCS clearly demonstrated the possibility to observe atomic fragmentation also in resonant X-ray emission (radiative RXS). A very strong emission band from atomic sulphur was observed following excitation to the

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Fig. 81. Experimental resonant Auger spectra, showing the 2p 3pP3p, 2p 3pP3p region of the atomic Auger   decay transitions in HCl as function of detuning [106]. Fig. 82. Comparison of the experimental and theoretical relative integral contribution of the molecular background [106].

lowest core excited resonances in OCS, while higher excitations lead to molecular like emission spectra. The dipole selection acts to simplify the atomic emission features compared to the non-radiative case. It seems that strong coupling among many possible close-lying dissociation channels also lead to a simplification of the spectral outcome. 11.4.5. Bound—continuum and continuum—bound channels. Mapping of vibrational wave functions Only the bound vibrational states uA (x) are populated if the excitation energy u is tuned below K the dissociation threshold º (R) of the core excited state, see Fig. 76. In this case both the A

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bound—bound and bound—continuum decay channels are open, and the total cross section becomes the sum of the corresponding cross sections; p(u,u)"p (u,u)#p (u,u). The properties of @@ @A p (u,u) were described above in Sections 6.4 and 8.1. @@ In the general case of several coherently excited intermediate vibrational states, the scattering channels will interfere through these discrete states, Section 6.4. In accordance with Eqs. (185) and (194) the contribution of the bound—continuum decay channel to the total RXS cross section is equal to





 uA (x) K . (206) p (u,u)" q Ku!u (R )!e #iC @A AM M K K when q\;c ,c . Here A DM DA (207) x"(1/F )(u!u!u (R )#F a ), q "a(2ka u/u 1uA "u 2 . K M DA DM A DA A K DA It is remarkable that the dependence of the cross section (206) on the emission frequency u copies the space distribution of the vibrational wave function uA (x) of the core-excited state. Indeed, when K C(u one can tune the frequency into exact resonance with some vibrational state m. In this case A the RXS cross section (206) simply becomes p (u,u)J(uA (x)). This equation thus shows how the @A K vibrational wave function can be mapped, and that the cross section is equal to zero in the m points where uA (x) is equal to zero. The latter expression for p (u,u) leads to the simple geometrical K @A consideration given in Fig. 83. This consideration is based on the physical meaning of x (207) as the classical turning point for propagation on the potential surface º of the final dissociative state (see D Eq. (194)). According to this physical meaning the spectral shape of the RXS cross section reflects the square of the vibrational wave function (uA (x)) of the core excited state by the linearized K potential of the final state (Fig. 83). It is important to note that Eq. (194) and its geometrical interpretation (Fig. 83a) are based on the linear approximation of the potential º near RA . Thus D 

Fig. 83. A geometrical illustration of the proportionality of the bound—continuum RXS cross section p (u,u) (206) to @A the square of the vibrational wave function uA (x) of the core excited state at the classical turning point x (207). The linear K approximation of the final state potential º is depicted with a thick dashed line; the exact potential º is depicted with D D a thin solid line. The cross sections are obtained by reflection on (a) a linear potential and (b) a nonlinear potential.

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only in the case of such a linear potential, the mapping p (u,u)J(uA (x)) is linear (see Fig. 83a). In @A K the general case of a non-linear potential the mapping is evidently non-linear (see Fig. 83b), and the application of the reflection method will not produce a direct copy of the squared wave function as given by p (u,u)J(uA (x)), but a deformation of this wave function depending on the particular @A K shape of the potential asymptote. The description of the continuum—bound decay channel is very similar to the case reviewed here and can be found in Ref. [102].

12. Time-dependent theory of resonant X-ray Raman scattering As stated in the previous text there are two basic representations to describe resonant Raman scattering; the time-independent and the time-dependent representations. These give different interpretational content to the RXS process despite that they obviously lead to identical results. For example, although RXS through dissociative states has an obvious “dynamic flavor”, a fully time-independent derivation could explain all known characteristics, as reviewed in the previous section. Despite this fact and despite the fact that studies of RXS at the present time have involved only stationary experiments, time-dependent treatments have gained an increasing popularity on the theoretical side owing to their inherent interpretability and our inclination to relate spectral features to processes rather than to states. The time-dependent representation allows a penetration into the physics of the scattering process because of to two important notions; the time evolution of the electro-vibrational wave packet and the duration of the scattering process. The results reviewed in Section 4 demonstrated that the duration time is a main physical concept which can be exploited for an active manipulation by the RXS spectral shape. This concept also forms a basis for the evaluation of the time-dependent RXS cross section on which the present section focuses. 12.1. Time-dependent representation of the RXS cross section The time-dependent representation for the RXS cross section in the case of narrowband excitation was considered for the first time in Refs. [83,84] (see also [101,102]). Lee and Heller [259] introduced the concept of time-dependent wave packets in the time-dependent representation for the amplitude of optical Raman scattering. Further development of this technique has occurred for many problems connected with RXS [260,190,102,117,261,114,113,110,262] (see also Section 4). As in the time-independent case we account for the situation when an incoming X-ray photon with the frequency u is absorbed, core exciting the molecule to the state "c2. Due to the Coulomb interaction and vacuum fluctuations this intermediate core excited state decays by emitting Auger electrons and X-ray photons with the energy E to the final state " f 2. The radiative and nonradiative RXS amplitudes have the same structure near the resonant region [42] 1 f "Q"c21c"D"o2 . F" E!u #iC AD A

(208)

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The operator D describes the interaction of the target with the incident X-ray photon. In the case of non-radiative RXS, Q is the Coulomb operator, and Q"DYH when the emitted particle is the final X-ray photon [42]. The half-Fourier transform of the denominator on the right-hand side of Eq. (208) yields the time-dependent representation for the scattering amplitude [263,102]



O F"F(R), F(q)"#i dtei#>#D>iCR1 f " (t)2 . 

(209)

The wave packet reads

(t)"Qe\i&ARD"o2.

(210)

Let us note that in strict theory the molecular Hamiltonian H is the same for all electronic states j (except the final state in non-radiative RXS). However, we will use the notation H with index j due H to two reasons; firstly to identify the electronic shell in which the wave packet evolves, and, secondly, to apply directly the general theory to nuclear degrees of freedom with the nuclear Hamiltonian depending on the electronic state j. A corresponding time-dependent representation for the RXS cross section can be obtained by a Fourier transform of the spectral function





  1 duU(u,c)e\iSR . U(u,c)" Re dtu(t,c)eiSR, u(t,c)" p  \

(211)

Since the U-function is real, its Fourier transform has the property: uH(t,c)"u(!t,c). We note that u(t,c)"d(t) and u(t,c)"const correspond to the cases of having white and monochromatic incident light beams, respectively. Very often the spectral function U is approximated by a Gaussian (18). In this case






tc . u(t,c)"exp ! (2 ln 2)

(212)

To receive the time-dependent representation for the RXS cross section we review the method outlined in Refs. [84,102,113]. Substituting (209) and (211) in Eq. (16) the following dynamical representation for the RXS cross section is obtained



 1 p(E,u)" Re dqp(q)u(q,c)eiS\#>#MO p 

(213)

in terms of the autocorrelation function p(q)"1t (0)"t (q)2 . # #

(214)

Here



t (q)"e\i&DOt (0), t (0)" # # #





dtei#\CRt(t) .

(215)

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The wave packet t (q) with the initial value t (0) is the solution of the non-stationary Schro¨dinger # # equation with the final state Hamiltonian H , whereas the wave packet D (216) t(t)"ei&DRQe\i&ARD"o2 admits two different interpretations and computational strategies. 12.1.1. Two-step evolution of the wave packet The two-propagator representation (216) prompts that t(t)"t(0,t), t(q,t)"ei&DR\OQe\i&ARD"o2

(217)

is the result of a two-step evolution: The initial wave packet D"o2 propagates after core excitation in the core excited state from time equal to 0 up to t. After the decay transition at moment t the wave packet evolves in the final state in the opposite time direction from moment t up to 0. The evolution of the wave packet t(q,t) is given by two coupled Schro¨dinger equations i(j/jq)t(q,t)"H t(q,t), t(t,t)"Qt (t) , D A i(j/jt)t (t)"H t (t), t (0)"D"o2 , A A A A

(218)

12.1.2. One-step dynamics A conceptually different interpretation of the wave packet (216) is based on the one-step dynamics with one effective time-dependent hamiltonian *»(t). By differentiation of Eq. (216) with respect to t the following Schro¨dinger equation is obtained: (219) i(j/jt)t(t)"*»(t)t(t), *»(t)"ei&DR*»e\i&DR, *»"QH Q\!H A D with the initial condition t(0)"QD"o2. The solution of the latter equation is straightforward






R t(t)"U(t,t )t(t ), U(t,t )"¹exp !i dt *»(t ) (220) M M M   RM where ¹ is the time-ordering operator. At this point it is worth pointing out the following striking property of the wave packet t(t): The evolution of t(t) is completely halted in the dissociative region (RPR) where *»Pu (R), AD º (R)!º (R) since the evolution operator U(t,t ) (220) then becomes equal to the c-number: A D M (221) *º(R)Ku (R), U(t,0)Ke\iSADR, t(t)Ke\iSADR"o2 . AD The same result follows immediately also from the two-step representation (216), since in the dissociative region H KH #u (R) and hence again t(t)Kexp(!ιu (R)t)"o2. This result is A D AD AD important also from the viewpoint of numerical simulations since it makes it possible to avoid the integration of Eq. (219) as well of Eqs. (218) in the region of dissociation. The two-step and one-step approaches for the evaluation of the wave packet t(t) (216) leads to two qualitatively different numerical techniques. The two step technique requires more computational time because the solution of the time-dependent Schro¨dinger equation (218) for t(q,t) requires in advance the solution of the time-dependent Schro¨dinger equation for t (t). The one-step A method is free from this disadvantage, but contains on the other hand the complicated operator *»(t) which needs to be diagonalized.

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We review also a third method for a time-dependent evaluation of the RXS cross section, namely one which is based on the evolution of the RXS cross section for monochromatic excitation with the forthcoming convolution of this cross section with the spectral function. This version can be preferred over the one- and two-step techniques. 12.1.3. Convolution of the cross section for narrowband excitation The main characteristic features of RXS can be unravelled when the spectral width c of the incident light beam is small. The spectral function U for the narrowband incident beam (c;C) can be replaced in Eq. (16) by the Dirac d-function. Hence, the denominator in Eq. (208) becomes equal to (u!(E !E )#ιC). The autocorrelation function then takes the form A M p(q)Pp (q)"1W(0)"W(q)2 . (222) M Here



W(q)"e\i&DOW(0), W(0)"



dt eiS>#M\CRQt (t) . A

(223)  Once more a two-step technique is retained; (1) the solution of the Schro¨dinger equation (218) for t (t)"exp(!iH t)D"o2 with the initial condition t (0)"D"o2, and (2) the solution of the A A A Schro¨dinger equation for W(t) with the final state Hamiltonian H and with the initial condition D W(0). However, this two-step technique is advantageous over the first one commented above, since the initial condition W(0) does not depend on time. Having settled the question of evaluation of the RXS cross section p (E,u) for the monochroM matic incident X-ray beam, one can evaluate the RXS cross section p(E,u) for arbitrary spectral distribution of incoming radiation as the following convolution [95,113]:





1  (224) p(E,u)" du p (E,u )U(u!u ,c), p (E,u)" Re dq p (q)eiS\#>#MO .  M   M M p  From the computational point of view the method given by these equations is both simpler and faster in comparison with the two techniques described above (see also below). One of the computational advantages of this approach is the possibility to use a parallel algorithm. 12.2. One-step nuclear dynamics We turn again to the important special case in RXS when only the nuclear degrees of freedom need to be taken into account — which often is motivated when the different electronic transitions in the RXS spectra are well separated — and assume the validity of the Born—Oppenheimer (BO) approximation, separating the nuclear and electronic degrees of freedom. We will also neglect the dependence of the electronic transition matrix elements D and Q on the nuclear coordinates. This approximation is sufficiently good for the photoabsorption matrix element D since the ground state nuclear wave function is localized close to the equilibrium molecular geometry. The same assumption concerning the amplitude of the decay transition Q can be justified for the bound—bound decay transitions. However, this dependence of Q can be more essential for the continuum—continuum decay transitions since a large span of internuclear distances then is involved, and this dependence

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might influence the ratio of the molecular and atomic-like contributions. On the other hand, for hydrides with one heavy atom, like the HCl molecule, numerically evaluated in a following subsection (Section 12.2.3), the total rate is probably dominated by the heavy atom. Investigations on the partial and the total Auger decay rates for the H O molecule indeed indicates that most of  the internuclear dependence of these rates is allocated at distances shorter than the equilibrium, so that the “constant resonance width” approximation holds [264]. When the R dependence of the transition matrix elements is weak, D and Q can be factored out, and one can thus simplify the RXS amplitude by putting D"Q"1 .

(225)

(We notice that the approximation D"const, Q"const most often is used in the numerical simulations and in the expressions for the Franck—Condon (FC) factors. All other results reviewed here are free from this assumption). The question of interest is posed by the pure nuclear problem with the Hamiltonians H "K#º (R), a"o,c, f , (226) ? ? where K is the nuclear kinetic energy operator. It is relevant to note the physical meaning of the wave packet t(t) (216) in the Q"const approximation. The one-step (219) evolution of this wave packet is given exactly by the interaction picture with H as the unperturbed Hamiltonian and D *»"*º(R) (227) as the perturbation. One obtains the remarkable result that the one-step dynamics is determined by the difference *º(R) between the potentials of the core exited º (R) and final º (R) states. A D 12.2.1. Mapping of the core excited wave packets and potentials. The reflection technique Another physical interpretation of the wave packet W(0) is obtained by first considering RXS with a monochromatic incident X-ray beam. In this case the RXS amplitude (208) becomes the projection of the coherent superposition of the core excited states W(0) on the final state Q"c21c"D"o2 . (228) F "#i1 f "W(0)2, W(0)"i D u!u #iC AM A This representation immediately shows that W(0) (223) is the coherent superposition or “wave train” of the eigenstates of the core excited Hamiltonian H created due to the core excitation. A The ground is now prepared for expressing an entirely different representation for the RXS cross section, namely p (E,u)"1W(0)"d(E!H )"W(0)2"tr d(E!H )o, o""W(0)21W(0)" (229) M D D with E"u#E !E. Through this expression a mapping of the squared wave packet "W(0)" and M internuclear potentials can be obtained. The easiest path to the desired semiclassical result is given by the Wigner transform (see Ref. [113]). To proceed further we use the conventional separation of the internuclear domain into two regions, the “molecular” (R(R ) and the “dissociative” (R'R ) B B regions. Here R is the minimal internuclear distance where º (R) and º (R) are close to the B A D

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corresponding dissociation limits. Using this partitioning the RXS cross section becomes the sum of the “molecular” and “atomic-like” contributions p(E,u)"p

(E,u)#p (E,u) .

 

(230)

Here

 

0B dR d(E!X!*º(R))"W(0)" ,   (231) p (E,u)K dR d(E!*º(R))"W(0)", *º(R)"º (R)!º (R) . A D  0B This general reflection technique contains an important special case which allows to map *º(R)"º (R)!º (R). When the slope *º(R)"d *º(R)/dR is not equal to zero one can write A D "W(0)" , p(E,u)K "*º(R)" p (E,u)K



*º(R)"E!X if R(R , B *º(R)"E if R'R , (232) B where the prime denotes a derivative with respect to position. What strikes the eye here is the simple relation between the RXS cross section and the slope *º(R). This relation is important for the inverse problem of finding potentials from the spectroscopic measurements. However, the simplicity is ephemeral in the molecular region where the wave packet W(0) is a complicated function of R. The situation is more promising in the dissociative region where the R-dependence of "W(0)" "W(0)"K"W(0,R )"e\ODO, q"1/C B is caused [102] only by the time of flight



q " D

0 dR R K , v(R) v




v(R)"

(233)



2  , v"v(R) [º (R )!º (R)] A k A M

(234)

and by the lifetime q"1/C. Though "W(0)" is a smooth function of R, W(0) shows the fast oscillations typical for continuum states (Fig. 85). The atomic-like resonance and the corresponding near wing can dominate only when X"0 [102,106,114,113]. Therefore to find *º(R) it is natural to tune u in exact resonance, X"0. Apparently, the reflection method maps the longrange part of *º(R) on the nearest wings of the atomic-like profile since the dependence of the potentials on R is slow in the dissociative region (see Section 12.2.2). The extension of the reflection technique to the whole spectral region allows to map the ratio "W(0)"/"*º(R)" (232). This gives information about the space distribution of the squared wave packet (see Fig. 83). The comparison of the exact cross section with the one obtained in the reflection approximation indicates that the semiclassical approximation (232) is not perfect. The deviation with the exact cross section is stronger in the vicinity of the atomic-like resonance where the factor 1/"*º(R)" diverges and the semiclassical approximation breaks down.

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12.2.2. Red and blue wings of the atomic-like resonance The atomic-like resonance is mainly formed by a spectral transition in one of the dissociation fragments. The central part of the atomic-like resonance band is close to a Lorentzian, while the wing (the outer part) of the band has a different profile [102,106,113]. We are now prepared to describe the “near wings” (Fig. 84) more explicitly, focusing on the dissociative region where the long-range forces dominate. To reduce the molecular contribution the case with X"0 is considered (Fig. 85). The difference *º(R) of the core excited and final state potentials is easily approximated as *º(R)"u (R)#a(R /R)L, u (R)"º (R)!º (R) . (235) AD B AD A D The solution of Eq. (232), E"*º(R), is straightforward: R/R "(a/(E!*º(R))L. The final B results follow directly from Eqs. (231) and (233). Hence, the red or blue wings of the atomic-like resonance are described by Eq. (232) and in the case of a power potential (235) as p (E,u)J"*/*E">L . (236)  The spectral shape of the atomic-like resonance in the vicinity of the resonant frequency u (R) AD [102,91,106,113], is a Lorentzian exp(!(*E!X)/c) A , "*E"("*" , p (E,u)J  *E#C

(237)

Fig. 84. The conventional partition of the spectral domain in the central (Lorentzian) part, and near and far wings. *E"E!(º (R)!º (R)). A D Fig. 85. The space distributions of the wave packet (223) in the unbound core excited state (2p\pH) of HCl for different excitation energies (X"0 and 3 eV). ReW(0) and Im W(0) are depicted as solid lines while the absolute value ("W(0)") of the wave packet is shown as a dashed line.

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since in the soft X-ray region c ion correspond to these atomic-like lines: B: P, R> and C: P, R\, R\, P. To understand the main spectral features of the resonant Auger electron decay spectra for the 2p\pH core excited state, the three most important final states were accounted for [106], the bound R> and the dissociative R\ and P states. The corresponding potential surfaces (Fig. 86) were calculated by the MCSCF method [265] as described in Ref. [113]. The evaluation of the electron matrix elements of the decay transitions Q was then neglected, meaning that the RXS cross section for different final states were given for the same value of Q (the calculations in Ref. [113] refer to one of the two spin—orbit split core excited 2p\ pH states). 

Fig. 86. Potential surfaces for ground, core excited (2p\pH) and final (P,R\,R>) states of the HCl molecule. Fig. 87. The RXS cross section of HCl for bound R> and dissociative R\ and P final states. The case of monochromatic, resonant excitation (X"0).

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Fig. 87 shows the RXS profiles for different final states for X"0. One can see that the onsets of the molecular bands can be found both at the shortwave (R>) or the longwave (R\, P) sides of the atomic-like resonance. As clarified above, blue and red tails are formed if the potential surfaces of the core excited and final states converge or diverge, respectively, when R increases and approaches the dissociation region. The atomic-like resonance for the same final state can in general simultaneously have both blue and red tails [102,266]. It depends on the behaviour of the potentials º (R) and º (R) close to the equilibrium geometry. A D The spectral shape of the molecular tails strongly depends on the shape of º (R) and º (R). If the A D final (or core excited) state is bound the molecular contribution consists of a vibrational band and a smooth continuum—continuum tail (see upper panel in Fig. 87). Notice that the molecular contribution with the vibrational structure in this panel corresponds to the experimental band assigned in Ref. [106] as 5p\. The next question concerns the the role of the detuning: As is well established [102,101, 104,106,114] the molecular and atomic-like contributions depend on the excitation energy in qualitatively different manners. Indeed, the position of the atomic-like resonance does not depend on the excitation energy [102,91] while the center of gravity of the molecular band depends nonlinearly on X inside of the region of strong photoabsorption and follows a linear Raman law at the wings of the photoabsorption band [101]. Moreover, the weight of the atomic-like resonance tends to zero for large "X" faster than the molecular contribution. In this limit of sudden RXS only the molecular band contributes to the spectral shape of RXS. It is simply given by the FC factor between the ground and the final nuclear state [102,117] (see also Section 12.2.4). The commented spectral features are shown in Figs. 88, 90, 91 and 93. The quenching of the RXS cross section when u is tuned below or above the photoabsorption band is shown in Fig. 89 in comparison with the spectral shape of photoabsorption. One can see that neither the RXS nor the absorption cross sections are symmetrical functions of the detuning. It is necessary also to emphasize that both these cross sections decrease slowly, more like Lorentzians than Gaussians (see also Section 12.2.4). The switching over from Gaussian to Lorentzian behavior takes place for E(200 eV and is therefore not seen in the long wave region (Fig. 89). The RXS cross sections are depicted in Figs. 90, 91, and 93, with attention focused on the X-dependence of the RXS spectral shape. According to the reflection approximation (see Section 12.2.1) the RXS spectral profile maps the space distribution of the wave packet. This leads to the appearance of additional fine structure in the RXS profile [102,114] (Figs. 90 and 91) caused by the inhomogeneous space distribution of the core excited wave packet W(0) in the molecular region (see Fig. 85, X"3 eV, R(3 a.u.). 12.2.4. Interference between molecular and dissociative scattering channels Fig. 91 demonstrates the appearance of an additional spectral feature of the RXS profile when the narrow atomic-like resonance is embedded in a smooth molecular background. One can see that the atomic-like resonance converts into a spectral hole when u is tuned from the photoabsorption resonance [113]. This anomaly is the result of an interference between the molecular background and the atomic-like resonance channels. The possibility for such kind of interference between resonant and off-resonant contributions was mentioned earlier [102,113]. Here we give a more extended explanation; we note that the manifestation of this phenomenon is not the same as the Fano profile [58], although it reminds thereof. For the sake of transparency we assume that

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Fig. 88. The RXS cross section for the unbound R\ final state of HCl for different excitation energies u. The resonant frequency (º (R )!E ) of the vertical photoabsorption transition is equal to 202.58 eV. E is the energy of the Auger A M M electron. One can see the fast decrease of the RXS cross section when the excitation energy is tuned out of the resonant frequency of the vertical photoabsorption transition. c"0.

both the core excited and the final states are dissociative and that the incident light beam is monochromatic. This circumstance leads to p(E,u)""F", e "u#E !º (R)!E . (238) D M D It is convenient to consider here the real core excited "c2""e 2 and final " f 2""e 2 continuum A D nuclear states with the released dissociation energies e and e , respectively. Since the conA D tinuum—continuum Franck—Condon factor 1e "e 2 is singular one can conclude that D A 1e "e 21e "o2"r(e ,*e )#s(e ,*e )d(e !e ) . (239) D A A D A A A D A Here *e "e !*º and *º "º (R )!º (R). The smooth function s(e ,*e ) shows the weight A A A A A M A A A of the narrow atomic-like contribution while the smooth profile r(e ,*e ) is responsible for the D A molecular background [102,106,113]. Since the continuum wave functions are real functions,

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Fig. 89. The RXS cross section integrated over E (solid line) and the Cl 2p X-ray photoabsorption spectrum (dashed line) of HCl. The core excited and final states are 2p\pH and R\ respectively. The Gaussian profile is depicted as a dot-dashed line. The cross sections have maximum for u+202.58 eV. The half-width at half-maximum (HWHM) of the Gaussian is equal to 0.84 eV.

Fig. 90. The RXS cross section for the unbound R\ final state of HCl for different excitation energies. The RXS cross sections are normalized; the integral cross sections are the same for different excitation energies. c"0.

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Fig. 91. The RXS cross section for the unbound P final state of HCl for different excitation energies. The RXS cross sections are normalized; the integral cross sections are the same for different excitation energies. c"0. The lower panel shows the region near the spectral hole in more detail.

s(e ,*e ) and r(e ,*e ) are also real quantities. Eqs. (208) and (225), (239) infer that also the scattering A A D A amplitude is the sum of molecular and atomic-like contributions F"g(X,*E)#s(e ,X)/*E#ιC . (240) D The interference between the molecular and the atomic-like contributions emerges here naturally. One can observe a resemblance of the spectral shape of the atomic-like resonance with the well-known Fano profile [58] which describes the interference between continuum states and a discrete state embedded in this continuum. However, qualitative differences exist. In the case of interest the discrete state is missing both in the core excited and the final nuclear states. The reason for the appearance of the discrete resonance in the X-ray Raman spectrum is the cancellation of the

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Fig. 92. Spectral profile versus spectral width c of incident radiation. The final HCl state is P (see Fig. 91). The RXS cross sections are normalized. The spectral function is approximated by a Gaussian (18) with the pilot frequency 206 eV and the following values of HWHM: c"0.01 eV, 0.3 eV, 0.5 eV, 1 eV. The RXS cross section for monochromaticexcitation is depicted by a dashed line. Fig. 93. The RXS cross section for the bound R> final state of HCl for different excitation energies. The RXS cross sections are normalized; the integral cross sections are the same for different excitation energies. c"0.

kinetic energies in the resonant frequency of the decay transition, º (R)#e !º (R)!e " A A D D º (R)!º (R), due to the effective conservation of the kinetic energy under decay in the A D dissociative region. This conservation relation is reflected by the singular partJd(e !e ) in the D A continuum—continuum overlap integral 1e "e 2 (239). D A Apart from the evident asymmetry of the atomic-like profile, the interference leads to new striking spectral features: One is the total suppression of the atomic like resonance, Fig. 91. When

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the weight of the molecular pedestal exceeds the amplitude of the atomic-like part an atomic-like “hole” emerges instead of the resonance peak (Fig. 91). This “hole” disappears for large detunings where the RXS profile coincides with the photoabsorption band of the direct oPf transition (see lower panel in Fig. 91). A naive picture prompts that any narrow spectral feature is blurred when the spectral width c of the incident radiation increases. Simulations show something different, however, see Fig. 92. Indeed, one can see that the width of the spectral hole as well as of the atomic-like feature is practically independent of c [102,91,113]. Moreover, Fig. 92 demonstrates another unexpected effect, namely, the transformation of the hole into a peak when c increases. This atomic-like feature can be understood if one recalls that broadband excitation implies a summation of the RXS cross sections for different u. If the particular u with the sharp peak is included in the summation, then it might dominate for large c and there will be a peak instead of a hole. 13. Direct versus resonant X-ray Raman scattering Up to this point the present paper has reviewed the consequences of second-order perturbation theory for the interaction between matter and light for a variety different experimental situations and findings that have concerned 2nd and 3rd generation synchrotron sources. In a more general multichannel resonance scattering framework the radiative and non-radiative processes can be given a unified formulation [57,40]. However, also in the perturbation theory ansatz the radiative and non-radiative RXS processes have remarkably many common features [42], something that has been extensively utilized so far in this review. The main crossing point of these two phenomena is the one-step model with the golden rule for the scattering cross section and the Kramers—Heisenberg (KH) expression for the scattering amplitude. The obvious difference is the interaction leading to the non-radiative decay, i.e. the discrete—continuum Coulomb interaction between the core-excited state and the many continua into which it is embedded, while the radiative spectrum occurs as a result of spontaneous emission. The implication of these differences for selection rules and the resolution of fine structures has already been reviewed, see Section 9. Likewise, from the experimental point of view the important advances in synchrotron radiation sources of the 2nd and 3rd generations, has promoted measurements of both radiative and non-radiative Raman effects. This holds for atoms, molecules as well as for solids. In fact, it was in the non-radiative Raman mode that the various “Raman effects”, such as linear dispersion and resonance narrowing first were described. Another important difference between the radiative and non-radiative processes is that non-resonant scattering is more important in the case of nonradiative RXS. The “direct” transitions can thus quantitatively change the dependence of the spectral width on the detuning. 13.1. Resonant photoemission The resonant non-radiative RXS leads to the same final state as the “direct” transition — or direct photoemission. In the case when the excited electron “remains” during the transition — a spectator decay — the final state is a 2-hole 1-particle state which is identical to a “direct” photoemission satellite. In the case the excited electron participates in the process — a participator decay — the final

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state is a 1-hole state and identical to the ones reached by normal photoemission. In both cases the direct transition contributes to the total cross section and interferes with the resonant transition. From the experimental point of view the latter type, the participator decay, is the important subclass of the non-radiative Raman process. Since the final states are identical to those obtained by a normal photoemission experiment the process often goes by the name resonant photoemission (RPE). Although RPE has been studied for a long time it is evident that the current availability of narrowband polarized and tunable X-ray beams and the accompanying strong improvement of spectral resolution has promoted the studies of this process just as much as of general X-ray Raman scattering processes. Resonant photoemission has been analyzed in the framework of various theoretical approaches [42,165,44,267—271] most of them connecting to the theory of Fano resonances [58,82]. It is pertinent here to review some advances in RPE and in particular the interference with the direct photoemission process and the frequency-dependent features of RPE that now can be studied. The systematic investigation of the spectral shape versus excitation frequency u and scattering angle seems thus to be a new branch of RPE research. Very recent experiments on argon [272], on the C1s resonant photoemission in CO [273], on metallic Ni [38] (see Section 16.3.4) and on weakly interacting systems on a platinum surface [39], have shown a rather sharp dependency of the RPE profiles on u when the photon energy is tuned close to the resonant excitation of the core level. Camilloni et al. [272] (see also [274]) showed that the main reason for this spectral anomaly for the 2p\3d excitation of Ar is the interference between direct and resonant photoemission. The same  conclusion was reached by Weinelt et al. [38]. According to the recent measurements by Piancastelli et al. [273] and Carravetta et al. [75] the physical picture of the similar phenomenon in the RPE spectra of carbon monoxide is not so unambiguous. We thus focus on some recent advances of frequency-dependent molecular RPE spectra and their description using many-channel scattering (and Fano) theory, again using CO as the primary testbed. 13.1.1. Resonant vibronic photoemission It is pertinent in this review to discuss RPE theory of gas-phase molecules with resolved vibrational structure. The general theory will be reviewed for the special case of RPE by the CO molecule to illustrate its key features. In the process considered here, the photon frequency u is tuned close to the X-ray C1sPnH transition from the K-shell of carbon to the first unoccupied level (nH) of the CO molecule. This highly excited “discrete” "U 2""1s\nH2 state has an energy  A lying above the ionization thresholds of all the occupied molecular levels and is energetically well isolated (*E'5 eV) from any other core excited state. Due to the Coulomb interaction with many continuum states, this discrete state decays through several autoionization channels with emission of photoelectrons t Pt . The final electronic state of energy E can be written as H JC "W 2""t\t 2, where the index f"j,l identifies the ion electronic wave function "t\2 and the D# H JC H symmetry of the continuum wave function t of the photoelectron with energy e (for example JC t , t V , t W ). NC L C L C Only few valence shell ionization channels t\(4p\,1n\,5p\ for CO) among the many H possible autoionization channels, will be explicitly considered in the following application of the theory since the experiment usually only resolves the outermost valence ionized states. The processes corresponding to “participator Auger decay” is then considered. The number of continua interacting with the "U 2""1s\nH2 state is much larger and some of them can provide decay  A

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paths even more efficient then the participator channels (the spectator resonant Auger channels). However, their presence can be taken into account considering the total lifetime of the core excited state. Radiative decay channels are also present, but they are less effective for the decay of core excited states of light elements as CO. For photon energies u close to the discrete C1sPnH level the photoemission consists of two qualitatively different channels, namely, direct photoionization of a valence orbital t and JC a resonant process involving core excitation of the 1s electron to the molecular orbital nH with A forthcoming autoionization to the final electronic state "W 2""t\t 2. D# H JC If the incident X-ray beam is linearly polarized, the RPE cross section (13) for the j-channel summed over the final vibrational states "m 2 of the ion has the following structure: D b (241) p (u,e)" p D(u) 1# HKD (3 cos 0!1) U(u#E !e!E D,c) , HK  HK H 2 KD 4pa p D(u)" u "1W "D"W D 2", E"u#E HK   HK J# 3 J where p D(u) is the RPE cross section averaged over 0 ; 0 is the angle between the photon HK polarization vector e and the photoelectron momentum k. The electron angular distribution parameter b D depends on j and m . The label m must here be considered an “asymptotic” label D D HK because, as will be shown in the following, the total wave function of the final state cannot be simply represented as the product of an electronic and a vibronic wave function owing to the resonant contribution "U 2"m2 which depends on the vibrational wave function "m2 of the  intermediate electronic state "U 2. The total energies, E "E #e and E D"E #e D, of ground H K     HK and ionized final states, respectively, consist of the total electronic energies (E and E ) and the  H corresponding vibrational energies e and e D.  K





13.2. Fano problem for electro-vibrational transitions The solution of the Schro¨dinger equation HW "EW for E in the continuum can be obtained $# $# by the Fano approach [58,82,60,275—277,274]; F"f, m is a collective label for the electronic and D vibrational state of the ion and the continuum degeneracy. It is reasonable to use the Born—Oppenheimer (BO) approximation and to approximate the total Hamiltonian H"H # C H by the sum of the electronic and the nuclear Hamiltonians. 
The interchannel interaction between discrete and continuum molecular states » "1U "H "U 2 (242) D#  C D# mixes the electronic discrete states "U 2"m2 and the electronic continuum states "U 2"m 2 [75]  D# H » D# "UI 2#ip » "U 2 1m"m 2"m2 , "W 2""U 2"m 2# (243) $# D# D  D# D# D E!E K K D » "UI 2""U 2#  dE DY#Y "U 2 .   E!E DY#Y DY







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The profile of the cross section (241) of the RPE process in molecules may be rather complex due to the vibrational structure of both intermediate and final states and due to lifetime vibrational interference. A remarkable simplification is obtained if, as often done in experimental measurements, the cross section is integrated over the photoelectron energy, i.e. summed up over all the vibrational final states. In the case of a single electronic resonant state the so integrated cross section can be simply expressed as a sum over the intermediate vibrational states of contributions that are given by the product of an electronic factor, including both direct and resonant terms, and the FC factor between initial and intermediate vibrational state. In the case of narrowband excitation (c;C) the condition of completeness of the vibrational states and Eqs. (241) and (243) lead to the following expression for the energy integrated RPE cross section:









4pa »  HJ# Z #ip » Z u i Z # , (244) p (u)" dep (u,e)" H K HJ#  DY# DY# H 3 X #iC K K J DY where X "u!u is the detuning from the frequency u "E #e #D!E of the resonant K K K  K  photoabsorption transition to the discrete state "U 2"m2, and  i ""10"m2", Z "1W "D"UI 2, Z "1W "D"U 2 , K    D#  D#

(245)



dE C"n "» ", D" "» " . D# D#Y E!E D D This equation is valid for the angularly averaged RPE cross section or for the scattering at the magic angle 0 (70). Eq. (244) shows that the lifetime vibrational interference is absent in the cross K section integrated over the photoelectron energy and that the vibrational structure of the intermediate electronic state can be taken into account simply through the FC factor i and the K vibrational energy in the resonant denominator of the cross section [75]. A similar result was obtained in [179]. In spite of the strong mixing of the vibrational states of closed and open channels in the final state wave function (243), the BO approximation leads to an effective separation of electronic and nuclear degrees of freedom in the energy integrated partial cross section in Eq. (244). Very often the rate of direct photoionization is much smaller than the rate of the resonant channel "Z C" ;1 . m" J HJ# "Z » " J  HJ#

(246)

This ratio has been estimated from the experimental measurements of CO (see below) to be, in the average, m+0.5% for the X,A, and B final states. Because of this small parameter one can neglect » Z in Eq. (244); this corresponds to neglecting the small coupling of the continua through DY DY# DY# their interaction with the discrete state. By this approximation the partial RPE cross section (244) can be easily converted to one of the parametrized forms of the Fano profile commonly used [75]:





(X #q ) H p (u)"pM(u) i 1!o#o K H H K H H X #C K K

(247)

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with "» Z " " » (Z ) Z )" J HJ#  q" . (248) o" J HJ# HJ#  , H H » (Z ) Z ) z "» Z " J HJ# HJ#  H# J HJ#  The Fano parameters q and 1!o show the position of the Fano minimum and the depth of this H H minimum, respectively [58,275]. 13.2.1. Dependence of branching ratios on excitation frequency RPE experiments [272,273,75] have shown the dependence of the RPE spectral shape on the excitation energy. To understand the physical reason for this dependency, the RPE spectra of carbon monoxide serve as a good illustration [75]. The measurements of these spectra were carried out at the MAX I storage ring in Lund, Sweden (see Section 2). Deexcitation spectra including the three outermost final (1h) participator states X"5p\(R>), A"1n\(P), B"4p\(R>)

(249)

were recorded at various photon energies around the C1sPnH resonance. These single holes states have a significant probability of being populated via the direct photoionization channel, leaving the molecule in the same final state as would be the case for the resonant process. At 0"55° the ratio of the direct to resonant population was estimated to be on average +0.5% from comparison of spectra recorded at a photon energy corresponding to the maxima of the resonance and spectra recorded 4.5 eV below the resonance. Since !4.5 eV detuning corresponds to more than 50 times the lifetime width of the resonance (FWHM), 85(10) meV [173], the resonance population should be negligible. The C1sPnH resonance is separated by more than 5 eV from the next higher resonance, 3s Rydberg, and the lowest vibrational level, m"0, is more than 6 times more intense than the next higher vibrational level, m"1 [193]. This makes the deexcitation from the C1sPnH resonance ideal for studies of the resonant Auger process with respect to frequency detuning. Spectra recorded at 0"0° with a detuning from the C1sPnH resonance between 0 and !4.5 eV are presented in Fig. 94. The total experimental resolution is +150 meV, therefore transitions to different vibrational levels can be studied separately. From these spectra two different types of detuning effects are observed. Firstly, the vibrational progression strongly mimics the vibrational progression of the direct photoemission spectrum (!4.5 eV detuning) already at a moderate photon energy detuning. This collapse of vibrational structure upon frequency detuning [105] was described already in Section 8.3. The other striking feature of these spectra is the change of branching ratios of the participator peaks upon frequency detuning. The variation of branching ratios p(u)" p (u) (250) H H is presented in Fig. 95. In this figure the branching ratio of a particular (1h) participator state is obtained as the ratio between the intensity corresponding to the (1h) state and the total intensity of the three outermost participator states. The branching ratios of this figure show asymmetries with respect to the center of the resonance. Firstly, one observes an asymmetry of the branching ratios of the A and B states at a moderate detuning frequency, less than 1 eV. This asymmetry can be B (u)"p (u)/p(u), H H

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Fig. 94. Frequency detuned C1s\nH RPE spectra of CO leading to three main final states of CO>; the X 5p\ R>, A 1n\ P, and B 4p\ R> states [75]. Fig. 95. Comparison between experimental (symbols) and computed (lines) branching ratios vs photon energy (eV) for the channels: X(5p\) (open circles), A (1n\) (filled circles), B (4p\) (filled squares). The theoretical branching ratios have been obtained by the model neglecting the interference of direct and resonant photoemission channels (top panel) and by the model including the effect of the interference (bottom panel) [75].

explained with the higher vibrational levels of the resonance located at the positive detuning side, and can be considered to be of a “trivial” origin. However, there is also a “non-trivial” asymmetry affecting the branching ratios for a detuning of more than 1 eV. For instance, the crossing of branching ratios for the A and B states is located at +1.7 eV for negative detuning whereas on the positive side this crossing is clearly located further from the resonance, at +3.5 eV. This asymmetry cannot be explained by high vibrational levels, since the highest vibrational level unambiguously observed in a photoabsorption measurement at the C1sPnH resonance is the m"3 level, which is more than 1000 times less intense than the m"0 level. Fig. 96 shows a comparison between spectra recorded at 0° and 55° for frequency detuning between 0 and !200 meV. Also, as an inset to Fig. 96, the variation in relative intensity for energies between 0 and !700 meV is presented. Within the dipole approximation, the angular distribution of emitted electrons for completely linearly polarized light is expressed as Eq. (241). It should be noted that near the magic angle 0 (+55°) the intensity depends only on the integral K cross section and at 0° the intensity increases with an increasing b D parameter. The most obvious HK

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Fig. 96. Frequency detuned C1s\nH RPE spectra of CO. The spectra have been recorded at 0°, broken lines, and 55°, solid lines. As inset, the variation of relative intensity for 0°, open markers, and 55°, filled markers, are presented as function of photon energy detuning [75].

observation from Fig. 96 is that the relative intensities vary more rapidly at 0° than at 55°. This observation contradicts what can be expected from the strict two-step model, in which the angular anisotropy parameter b D is given by the product, b D"A ) c , where A is the so-called alignment HK ? HK parameter and c the intrinsic Auger decay parameter [278]. From ion yield studies of the ? C1sPnH resonance in CO it is known that A"!1 at resonance maximum and approaches 0 when tuning the photon energy away from the resonance maximum [279,280]. The b DHK parameter has been found to be close to 1 for all the (1h) participator states with the transitions corresponding to the A state having the lowest value of the three participator states b D+0.75 HK [281]. All the states in the C1sPnH deexcitation spectra are necessarily described by the same alignment parameter A. The results for the A band show clearly that the resonant process in a detuning experiment cannot be described by the strict two-step model in which the excitation step is assumed to be separated from the deexcitation step. One obvious difference between 0° and 55° spectra is that the direct population of (1h) final states are +50% stronger at 0° than at 55° (the angular dependence of the direct transitions are described by b D-parameters close to 2 [282], HK whereas the resonant transitions have b D-parameters +1). Thus, the angular dependence of the HK branching ratios is affected by interference or direct populations of the (1h) final states via the

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Table 4 Parameters of the Fano profile (see Eq. (247)) for the ionization channels: X(5p\), A(1n\), B(4p\). The values of o have been obtained by a fitting of experimental and theoretical branching ratios, while p (arbitrary units) and q have H H H been obtained directly from the experimental branching ratios at large and 0 detuning, see Section 13.2.1 for details Channel j

X

A

B

o H q (eV) H p H

0.27 2.58 0.102

0.27 3.27 0.194

0.21 1.35 0.215

corresponding direct channel. It can be noted that the vibrational collapse effect could be fully understood within the resonant part of the scattering formula whereas for the variations of the branching ratios it thus seems necessary to include also the direct population part. The results of the semiempirical simulations [75] are shown in Fig. 95 and the corresponding Fano parameters are collected in Table 4. 13.3. Role of interference between direct and resonant photoemission The dependency of the RPE branching ratios on the excitation energy is remarkable. It is easily understood that the branching ratio is constant if the spectral shape of the partial cross sections close to the resonance is a simple Lorentzian, i.e. if the direct photoemission is completely negligible. The deviation of p (u) from a Lorentzian profile is due to both the non-resonant H contribution of direct photoemission and to the interference of this process with the resonant photoemission. If the interference (JX q ) of direct and resonant channels is neglected, the K H approximate cross section of the photoionization channel j can be derived from Eq. (247) as






i K , p,'(u)KpM(u) 1#oq H H X #C H H K K

(251)

where the resonant contribution is described by Lorentzian profiles, one for each intermediate vibrational state, superimposed on the smooth background of the direct photoemission cross section p(u). H The p,'(u) and B,'(u) factors computed with neglect of the interference effect (by the expression H H in Eq. (251)) for the three ionization channels X (5p\), A (1n\) and B (4p\) are collected in the top panels of Fig. 95. The comparison between experimental and theoretical branching ratios in Fig. 95 shows that the no-interference model can explain the gross structure of the observed branching ratios. In this model, however, the crossing (for a single vibrational intermediate state) of two cross section curves can only occur at two points symmetric with respect to the zero detuning energy. An asymmetry of the crossing points can only be introduced by the presence of several close lying vibrational peaks of different intensity. The calculations [75] show (see top panel of Fig. 95) that the asymmetry obtained in this way is evidently much weaker than that present in the experimental data. This is a clear indication of the importance of the interference effect.

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Considering that the symmetric Lorentzian function is the one generally adopted to describe absorption and emission line profiles in excitation and ionization of core electrons, the observed behaviour of the branching ratios with the excitation energy can be regarded as anomalous. The CO example chosen here gives a good illustration that, in spite of its small nominal strength, the direct photoionization channel plays a strong role for the ionization branching ratios even for excitation close to the first absorption resonance. The strong dependence of the branching ratios on the photon energy and their asymmetry around the resonance energy can thus not be explained without taking into account the direct photoionization channels. The contributions to the crosssection of both the direct and the resonant photoemission channels can still describe the main features of the energy dependency of the branching ratios even if the interference between them is neglected. This simplified model, however, fails to reproduce the asymmetry observed in some of the branching ratios and which cannot be quantitatively explained in terms of only the vibrational structure of the core excited state. The asymmetry can thus safely be traced to the interference between the direct and resonant ionization channels, and can in fact be considered to constitute a direct manifestation of the interference effect.

14. Doppler effects In this section we forestall some anticipated developments in high-resolution X-ray Raman scattering spectroscopy by reviewing theory for Doppler effects in RXS and by demonstrating a variety of new physical phenomena related to this effect [266]. The theory is general, covering both radiative and non-radiative X-ray scattering and any character of the states involved, but most emphasis is put on the situation where the Doppler effects are most conspicuous, namely for non-radiative scattering of molecules core excited above the dissociation threshold. As is well-known — and reviewed in the foregoing — an RXS spectrum involving dissociative states consists of two qualitatively different parts, the so-called “molecular” and “atomic-like” parts, Fig. 80. According to the theory in Refs. [103,113] the “atomic-like” contribution caused by the decay transitions in the one of the dissociation fragments has the width equal to the lifetime broadening, something confirmed by the experimental investigations of the resonant Auger spectra of HCl [91,106], that is the same example as discussed in Section 11.4.4. We review a generalization of this result and show that for large release energies following dissociation and accompanying large electron Doppler effects, the atomic-like Auger resonance can be strongly non-Lorentzian. The Doppler shift can thus exceed the lifetime broadening for a kinetic energy release in the region e&1—10 eV, which is not uncommon for dissociating molecules [283], but which is substantially larger than the thermal energy k ¹K0.03 eV. In the case of heteroatomic molecules the electron Doppler shift will be smaller for RXS of the heavy atom because the released kinetic energy is transferred mainly to the light atoms. The HCl molecule investigated in Refs. [91,106] is here a typical example, with the chlorine Auger resonance showing a Lorenzian profiles despite comparatively large release energy. In molecules with comparable atomic masses the electron Doppler shift for the dissociative resonance is sufficiently large to exceed the lifetime broadening by several times (for example, k€ &0.2!0.5 eV for a molecule like O with C"0.09 eV). Here k is   the momentum of the Auger electron.

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14.1. Shortly about Doppler effects in X-ray spectra In the soft X-ray region the thermal motion of molecules can give additional broadening. Indeed, the total initial and final energies of the photon#molecule system for the Auger process are u#E #M»/2 and (MV#p !k)/2M#E#E , respectively. Here V is the velocity of M  D a center of gravity of a molecule with mass M, u and k are the photoelectron velocity and momentum in the laboratory frame of reference, p is the incoming photon momentum, E and  M E are total internal molecular energies for the ground and final states. According to the energy D conservation law the photoelectron energy E"u/2 has a shift E"u!u !e!( p !k) ) V DM 

(252)

caused by molecular motion, where e is the kinetic energy released under dissociation of the molecule. The term p ) V is the well-known Doppler shift caused by the photon momentum p .   This shift is negligibly small in the soft X-ray region due to the small value of the photon momentum p "u/cKu/137 (for example p K0.14 a.u. for O ). This means that ordinary    Doppler broadening often can be neglected in X-ray spectroscopy. One can also anticipate that Doppler effects are difficult to identify in the hard X-ray region due to large lifetime broadening and poor spectral resolution. The shift k ) V (252) is analogous to the Doppler shift but is larger and arises from the combination with the electron momentum k [284] (for example kK6 a.u. for O ). The electron  Doppler broadening D "k»M (ln 2 depends on the temperature ¹ and on the excitation energy 2 through the thermal velocity u "(2k ¹/M and k"(2E, respectively. Here k is the Boltzmann constant. To get an idea about the Doppler broadening one can consider the X-ray resonant photoemission spectra (RPE — or resonant Auger Raman) of carbon monoxide. The comparison of different broadenings for the C1sPnH RPE spectra of the CO molecule (D K 20 meV, CK42.5 meV) [75] 2 shows that the broadening D caused by thermal motion of molecules must be taken into account 2 in the analysis of highly resolved RPE spectra. This example demonstrates the typical case when the electronic Doppler broadening D caused by thermal motion at room temperature (k ¹K0.03 2 eV) is smaller than the lifetime broadening. Recalling the large kinetic energy (1—10 eV) released under dissociation one can understand that the electron Doppler effect in dissociative states is the largest among the cases mentioned. 14.2. Phase analysis of the scattering amplitude and the Doppler effect For the sake of transparency we first review the case of resonant X-ray scattering (RXS) by a simple three-level diatomic molecule AB with the reduced mass k"m m /(m #m )   (Fig. 97) and with core excitation of atom A (X-ray scattering by homonuclear diatomic molecules is considered in Section 14.5). As discussed above, the Doppler effect in radiative RXS is usually small in comparison with the lifetime broadening C, while it is more important in the non-radiative case (Raman Auger). Due to this fact we focus the review in this section on resonant Auger scattering, but remind that the material covered easily can be extended to the radiative RXS case.

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We consider the case when an X-ray photon with energy u, wave vector p and polarization  vector e is passed during the scattering to an Auger electron of energy E"k/2 and momentum k and when the molecule is excited from the ground "o2 to the final electronic state " f 2. We take special interest in core excitation above the dissociation threshold of the potential surface º (x), G thus when specific “atomic-like” narrow resonances appear [102,106,113]. A qualitative picture of the formation of narrow atomic-like resonances with broad short- and long-wave wings [102,106] is given in Fig. 80. Contrary to the wings that follow the Raman—Stokes dispersion law, the energy position of atomic-like resonance does not depend on the excitation energy [102,91,106]. As one can see in Fig. 97 the nuclear states "e,i2 and "e, f 2 for the core excited and final electronic states are lying in the continuum having nuclear kinetic energies at infinite separation e"p/2k and e"p/2k, respectively. The continuum nuclear wave functions are here normalized to a dfunction: 1 j,e"e , j2"d(e!e ). The double differential RXS cross section for a fixed molecular   orientation and monochromatic excitation reads [42,95] p(E,u)""F",

e"u!E!u !(p !k) ) V , DM 

(253)

where u "º (R)!º (R )!u /2 ( j"i, f ), R is the ground state equilibrium interatomic HM H M M M M distance, º (x) is the interatomic potential of the jth electronic state, and u is the frequency of the H M vibrational state "o2 of the ground electronic state. For brevity the notation p(E,u) is used here instead of dp(E,u)/dE dO. The lifetime broadening of the final state C is often small in comparison D with the lifetime broadening C of the core excited states and in comparison with the spectral width of incident radiation. Therefore C is neglected in Eq. (253). D

Fig. 97. Scheme of spectral transitions.

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The “atomic-like” decay transition iPf in the radiative X-ray Raman scattering is dictated by the dipole selection rules Q "eH ) D e\ip  R . (254) DG DG In the dissociative region the spectral transition is essentially atomic-like and all electronic wave functions involved to this decay are very close to those of the atom A. We consider here the case of high-energy Auger electrons, which allows to express the wave function of this electron, tk(r), relative to the nucleus A as tk(r)"tk(r!R )eik · R .  This means that the Coulomb matrix element has a phase factor

(255)

(256) Q "Q e\ik · R , DG DG since the Auger transition in the considered case can be assumed to take place in the isolated atom A. The Coulomb matrix element Q is here calculated with tk(r!R ). DG  14.2.1. Generalized Franck—Condon factors Due to the nuclear motion, the coordinate R "RM #dR of atom A is shifted relative to the    equilibrium site RM at the distance dR . To calculate the nuclear matrix elements we need the   expression for R through the normal coordinate x"R !R for the relative motion and through   the center of gravity of the molecule R"(m R #m R )/(m #m )    R "ax#R, R "!bx#R ,  a"k/m , b"k/m . (257)  The Born—Oppenheimer approximation makes it possible to rewrite the RXS amplitude as follows



FJand(P!P!k #p) de 

1 f,e"e\ik  x?"e,i21i,e"eip  x?"o2 . X!e!ιC

(258)

The amplitude of the resonant X-ray scattering by atom B is also given by this expression if exp(!ik ) xa) and exp(ip ) xa) are replaced by exp(ik ) xb) and exp(!ip ) xb), respectively.   The momentum conservation law (P "P#p !k) describing the photon and electron recoil  effects yields immediately the Doppler shift in Eqs. (252) and (253). One here recalls that for the final description of the Doppler effect caused by the thermal motion, the cross section (5) must be convoluted with the Maxwellian distributionJexp(!M»/2k ¹). The d-function in Eq. (258) and the small photon and electron Doppler shifts (p !k) ) V will not further be taken into account  due to the smallness of the Doppler effect caused by the thermal motion of molecules. The factor an"eD Q depends on the unit vector n along the molecular axis (x"xn) through GM DG the dipole moment D . The dependence of an on the internuclear distance x enters mainly via the GM decay amplitude Q . This dependence is not so important for the here discussed dissociative or DG atomic-like resonances which are formed by the spectral transitions in one of the isolated atoms (A). So the decay factor Q (x) can be approximated by its asymptotic value Q KQ (R). DG DG DG

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The FC factors connected with the photoabsorption transition can easily be evaluated assuming the harmonic approximation for the ground-state vibrational wave function 1i,e"eip  x?"o2"G(p)ei?p  RM ,







2kb  1 *e  exp ! G(p),1i,e"o2K2k na 2 c M G

(259) if b;a , M

where *e"e!*º , *º "º (R )!º (R), c "F a , b"(2kF )\, and F "!(dº /dx) is G G G M G G G M G G G M the slope of the interatomic potential º (x) for the core excited state at the equilibrium point R . G M The phase factor exp(ιap ) R ) is very important for hard RXS by symmetrical molecules with  M identical atoms [95], since the phase multiplier then destroys the coherence between the scattering channels through the identical atoms. In the case of radiative RXS this destructive interference leads to the violation of the selection rules for the X-ray scattering tensor [95]. Here, though, we will consider only the case of soft X-rays (p R ;1) for which the phase factor exp(iap ) R ) can be  M  M replaced by unity. The dissociating atoms are moving in a constant potential º (R) with the plane wave function H "e, j2"(2k/pp)sin(p(x!x )#u ) , H H



u" H



p (p !p) dx# . H 4 VH

(260)

Here j"i, f, p "[2k(e!(º (x)!º (R)))], and x is the classical turning point where p "0. H H H H H We will assume below that "e, j2"0 in the classically inaccessible region x(x . The exact value of H the scattering phase can be found directly from the Schro¨dinger equation or from the semiclassical formula (260). The classical turning points x , x as well as the scattering phases u , u for the core G D G D excited and final state potentials depend on the energies e and e, respectively (the energy labels for these quantities are dropped here since we will need the values of x and u only for e"X). H H According to the Condon principle the value of x is close to R . G M From the continuum wave functions (260) the second generalized FC factor can be evaluated analytically as

 
 



k e\ιOVG eiPd(p!p#q)#e\iPd(p!p!q) 1 f,e"e\ik  x?"e,i2" 2(pp)



1 1 1 #ι eiP # 2sin(u ) p#p p!p!q n !e\iP

1 p!p#q

, x 5x , G D

(261)

where q"ak cos h, u"u !u #pD, D""x !x " , D G G D

(262)

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u "u #u #pD, h is the angle between the momentum k of the Auger electron and the G D molecular axis x, P is the principal value. The continuum FC factor for x (x is again given by G D Eq. (262) after permutations: I & f, p & p. It is relevant to mention that the generalized FC factors can be important also in bound—bound transitions due to a large momentum of the Auger electron. 14.3. Anomalous anisotropy of Auger electron and ion yields To succeed in the evaluation of the scattering amplitude (258) we take into account only resonance contributions in (261). In this approximation the non-resonant term (1/(p#p)) is thus neglected and one can continue the integration over p up to !R. The expression (258) for the scattering through the core excited state in atom A then becomes [266] F"f e\iOVG, u"u !u #p D if x 5x , D G M G D F"f e\iOVD, u"u !u #p D if x 'x G D M D G

(263)

with e\iP+G(p)!G(p ), eiPG(p ) M M fJan # if x 5x , G D l#kv cos h!iC l!kv cos h!iC   eiP fJanG(p )e\CDTM if x 'x , (264) M D G l#kv cos h!iC  l"E!u , and u "º (R)!º (R). The electron Doppler shift kv cos h depends on the GD GD G D  velocity v "ap/k of the atom A after emission of the Auger electron. Here  p"[2k(X!(E!u ))], v "(2X/k) is the relative atomic velocity corresponding to the GD M kinetic energy e"X. In the latter expression for f only the main contribution is retained. 14.3.1. Resonant cone of dissociation At this stage we emphasize the main contribution to the RXS cross section (253) 1 (265) (E!u #kv cos h)#C GD  which concerns electron—ion coincident spectroscopies with the experimental fixation of the molecular axis (via a direction of dissociation) and the direction of the Auger electron propagation. Experimental investigations [128,285,286] have clearly shown the anisotropy of dissociation in angle-resolved photoelectron—photoion spectroscopy. This anisotropy, hidden in the anisotropy factor an, is smoother in comparison with the strongly resonant ion yield (265) caused by the Doppler effect. The last equation shows that the electron—ion coincidence signal changes drastically if the Auger electron energy E lies in the Doppler band !kv (E!u (kv . When the  GD  lifetime broadening C is smaller than the electron Doppler shift kv the photoions or the fragment  of dissociation propagate in the narrow angular interval dhKC/kv 

(266)

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Fig. 98. Resonant cone of dissociation.

close to the cone surface (Fig. 98) hKh P u !E u !E if GD 41 . (267) cos h " GD P kv kv   The appearance of the narrow resonant cone of dissociation with “resonant” angle h is one of the P important features of the studied problem. Eq. (266) clearly demonstrates the strong correlation between the propagation directions of the Auger electron and the ion A>.





14.4. Averaging of the cross section over molecular orientations We consider the spectral shape of atomic-like resonances in the Auger spectrum of atom A in molecule AB with different atoms (the principally different case of identical atoms is treated in Section 14.5.2). In this case the cross section emanates entirely from the scattering amplitude for atom A (to be specific only the case x 'x will be considered in this section). For ordinary G D resonant Auger measurements with the gas-phase molecules, the cross section must be averaged over molecular orientations. Two qualitatively different physical reasons are responsible for the dependence of the scattering amplitude (263) on the molecular orientation. As shown in Section 14.3 the first reason is the Doppler effect which leads to a sharper resonant dependence of the cross section on the molecular orientation (265). The second reason is the orientation of the molecular axis n relative to the polarization vector e and the Auger electron momentum k. This smooth polynomial dependence is hidden in the factor an. This allows to extract the factor "anM" from the integral over the molecular orientation at the resonant angle (267), leading to an additional averaging of this factor over all nM of the cone. Using this averaging together with Eqs. (253) and (263), one obtains



C p(E,u)"p o G(p )#(G(p)!G(p ))# s cos(2u)G(p )(G(p)!G(p )) M M M M M E!u GD



.

(268)

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247

All unessential quantities are collected in p J1"anM"22p/C. The spectral shape of the cross section M is defined by the two functions






o"(1/2pkv ) arctan((E!u #kv )/C)!arctan GD  








E!u !kv GD  C

,

(269)

1 (E!u #kv )#C GD  s" ln . 2pkv (E!u !kv )#C  GD  The Doppler shift





2  cos h , kv cos h"ka (X!(E!u )) GD  k

(270)

depends on the Auger electron energy E (see Fig. 100). The energy dependence of the electron wave number k"(2E can be neglected when E is large. To understand the main spectral features of the RXS cross section one can for a while neglect the E-dependence of kv . The o function is in this case a symmetrical function relative to the resonance  E!u "0 normalized to unity: dEo"1. The integral of s is equal to zero (dEs"0) since s is GD an antisymmetrical function relative to the resonant energy E"u . The asymptots of the GD o function






E!u 1 GD H if C;kv , o"  kv 2kv   C o" if C
248

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Fig. 99. Doppler broadening caused by random orientation of molecules. o-function (269) is shown as solid line while the Doppler shifted Lorentzians (265) are depicted as dashed lines. Specific values m "m for O is used. C"0.01 eV,   E"500 eV, X"2 eV, D"0.47 a.u. Fig. 100. The dependence of the Dopler shift kv on the relative kinetic energy of Auger electron E!u for h"0°.  GD Input data are the same as for Fig. 99.

(271). According to (271) the RXS cross section tends to zero when "E!u "'kv . This means GD  that outside of the Doppler band the E-dependency of the Doppler shift is not so important (Fig. 100). 14.5. Homonuclear diatomics. Role of channel interference RXS by homonuclear diatomic molecules qualitatively differs from the scattering by heteronuclear ones [95,266]. To take into account the indistinguishability of the two atoms one needs to sum the partial scattering amplitudes F "f exp($iqR ) for both atoms   D F"F #F "f e\iO0D#f eiO0D ,     q"k cos h, R "max+x ,x , , D D G 

(273)

where f is given by Eq. (264). The phases !iqR and iqR for different scattering channels differ H D D only by the sign according to the expression for the radius vectors of the sites for the first and second atoms (257) in the center of gravity of the molecule. To avoid unnecessary complications, the RXS cross section can be analyzed taking into account only the main term in the scattering

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249

amplitude (263),(264). Contrary to intuition the scattering amplitudes f and f for atoms 1 and   2 are different ¶ X anG(p )eiP M , f &  D  l$D cos h!iC

kv D, , 2

X "1 if x 'x , D G D (274) X "e\CDTM, if x 'x . D G D Here D is the Doppler shift, v "(2X/k), the signs (!) and (#) correspond to the atoms 1 and M 2, respectively, l"E!u is the detuning of the Auger electron’s energy E from the resonant GD frequency and, and u is the phase defined in Eq. (263). The atomic (one-center) parameter an is given here for the atom 1. The distinction between f and f has a simple physical meaning. The different   signs of the Doppler shifts for atoms 1 and 2 are caused by the motion of these atoms in opposite directions (v "!v ) in the center of gravity frame. Clearly, the absolute values of the atomic   velocities are the same (v/2) (v"p/k is the relative velocity of the atoms). Another important distinction between the scattering amplitudes f and f is their relative sign [95] given by the   symmetry parameter ¶ "$1. The sign of ¶ is defined by the symmetry of the electronic   wave functions involved in the scattering process. Without loss of generality, ¶ can be chosen  equal to 1, and it is easy to understand that the sign of ¶ then depends only on the parity of the  product oj of parities o and f of ground and final electronic states ¶ "1 if of"u ,  (275) ¶ "!1 if of"g .  The atomic-like resonance is formed in the dissociative region (Fig. 97) by the decay transitions to the set of final states of the same energy. Due to this degeneracy one can not distinguish these states experimentally and, hence, the RXS cross section must be summed over the full degenerate manifold. However, as is well-known X-ray scattering by identical atoms has specific spectral and anisotropic peculiarities caused by the channel interference [95]. According to Eqs. (274) and (275) the gerade and ungerade final states give interference contributions (&f H f ) of opposite   signs. The interference terms of gerade and ungerade states can therefore suppress each other in the total cross section (for brevity we restrict the description to a couple of gerade and ungerade states below — the summation over other final state quantum numbers can easily be performed). The quantity of interest is the cross section (276) p(E,u)"p #p #p " (" f "#" f "#2Re( f H f eiI0D  F))        DES which can be measured in electron-ion coincidence spectroscopy [287,288], or in ordinary RXS spectroscopy due to the above discussed orientationally selective excitation, as recently shown by Bjo¨rneholm et al. [373]. The first two terms p " " f " correspond to the independent  D  scattering channels through the core excited state in atoms 1 and 2, respectively. The interference term p at the right-hand side describes the diffractional scattering of the Auger electron at the  A molecule. When this interference term is large, both scattering channels are strongly coherent  and vice versa.

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The RXS cross section (276) is here analyzed for the very common case of the gerade ground state C , p "2C *p(cos h), *p(cos h)"(w pS!w pE) , p "  S E  (lGD cos h)#C [cos(kR cos h)+l#C!(D cos h),#2CD cos h sin(kR cos h)] D D pD" , [(l!D cos h)#C][(l#D cos h)#C]

(277)

where CJ"anG(p )"(X#X). Contrary to X-ray scattering by heteronuclear molecules (267) two M E S dissociative cones and two resonant angles now exist: cos h "$2l/kv (in accordance with the P oppositely propagated atoms). These equations show that the gerade and ungerade final states take part in the scattering process with the different weights w "X/(X#X), f"g,u . (278) D D E S The physical reason of this fact was given in Section 14.3 (see also Eq. (274)). A quite unusual result is here obtained: Since w Ow the interference term does not cancel after summation over gerade E S and ungerade final states which in the dissociative region have the same energies. Fig. 101 shows the anomalously strong dependence of the RXS cross section on the angle h between directions of the Auger electron and the ion propagation. The interference term leads to striking diffractional oscillations of the cross section. A comparison of Fig. 101a and Fig. 101b demonstrates the strong dependence of the scattering anisotropy on the potential of gerade and ungerade final states through the weights w and w . As one can see from these figures the scattering E S anisotropy is very sensitive also to the energy E of the Auger electron. 14.5.1. Distinction of “left” and “right” atoms in A molecules. Large Doppler shift  The RXS profile measured by electron-ion coincidence spectroscopy has the following doublet structure in the Doppler limit D





1 1 # (l!D)#C (l#D)#C

if D
(279)

These two narrow Lorenzians correspond to opposite directions of the dissociating atoms. The large Doppler shift D destroys the coherence of the scattering channels p /p(E,u)&C/kv;1  (Fig. 102). This means that the two scattering channels now are totally independent. Moreover, one can distinguish these channels since one now can select either of them through the “left propagating” and “right propagating” atoms A [266,287]. It is possible to do so since the “left” and “right” atoms moving in opposite directions !€/2 and €/2 have different “Doppler labels” (Doppler shifts): !k · €/2 and k · €/2. One can say that the photon is scattered by the “left” (or “right”) atom when the energy of the Auger electron is equal to E"u !k · €/2 (or E"u #k · €/2). In this GD GD case only the “left” (or “right”) atom is in resonance with the Auger electron while the partial cross section for the other atom is close to zero (see Eq. (279) and Fig. 102). When the Auger electron is emitted perpendicular to the molecular axis the RXS profile collapses to a single Lorentzian since cos h"0 and the Doppler shift then is exactly equal to zero (Fig. 102). Since the Doppler shift is absent when h"90 one can not distinguish the equivalent

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251

Fig. 101. The angular dependence of the total RXS profile p(E,u) (solid line), interference contribution (dashed line) and direct term p #p (dotted line) for different relative kinetic energies l"E!u . Input data are the same as for Fig. 102.   GD The Doppler shift is equal to D"0.4 eV. Everywhere w "1, w "0 except plot (b) where w "1, w "0. These values S E E S for the weights w and w correspond to Fig. 103. S E

atoms in an A molecule. In this case both scattering channels are strongly coherent and the  interference term p takes a maximal value (see Fig. 102).  The angular dependence of the total cross section and interference contribution (Fig. 101) shows diffractional oscillations according to sin(kR cos h) and cos(kR cos h) in Eq. (277) (Fig. 101). These D D oscillatory features provide structural information and can be observed when the diffractional parameter kR exceeds n. D 14.5.2. Orientational averaging of the cross section for identical atoms We now again turn to the ordinary RXS measurements in which the flux of Auger electrons is collected from all molecules. The cross section (276) must be averaged in this case over all molecular orientations, which precisely is the procedure used in Section 14.5. Averaging the cross section (277) over h one obtains p(E,u)"2p G(p )(X#X)(o#K) , M M E S



C  dm *p(m) , K" 2p \

(280)

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Fig. 102. The dependence of the total RXS spectral profile (solid line) p(E,u) (276), (277) on the relative kinetic energy l"E!u . E"500 eV, R "2.28 a.u., X"5 eV, C"0.09 eV. The interference contribution p is depicted with the GD A  dashed line. When h"0 the “left” atom has red Doppler shift, while the “right” atom has blue Doppler shift. The equivalent atoms can be distinguished by the the Doppler labels $kv/2. One cannot distinguish these atoms when h"90 since in this case the Doppler shift is absent.

where the o-function is given by Eq. (269) with v "v/2 and m"cos h. Contrary to Eq. (277) the  averaging procedure leads to the same partial cross sections p "p "2p G(p )o. The coherence,   M M or interference, term p "2p G(p )K can be much simplified since very often the diffractional  M M parameter kR is large D kR <1 . D

(281)

For example, kR K14 for O . As it was shown in Ref. [266] the interference contribution (280) D  K"(C/2pD)(w P !w P ) , S S E E

(282)

can be approximated outside of the Doppler band as

P KP "2D D DM

(l#C!D) j (kR )#2CD n (kR ) M D M D, ((l!D)#C)((l#D)#C)

"l"'DkR <1 . D

(283)

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Here j (x)"sin x/x and n (x)"!cos x/x are the spherical Bessel functions of zero order. The M M interference term consists of two qualitatively different sub-terms when the Auger energy is inside of the Doppler band









l C exp !kR #P , "l"(D, kR <1 . P K2pkR j kR DD DD DM D D D M

(284)

The direct numerical evaluation of the integral (280) shows very good agreement with the approximate formulas (283) and (284) except for small deviations near the Doppler shifts: $D. The first term at the right-hand side of Eq. (284) is caused by the pole singularities of the integrand (280) lying in the dissociation cone: cos h"$l/D. Close to resonance (l"E!u "0) GD this term, being the main contributions to p , exceeds the second contribution (JP ) by n(kR )  DM D times. We stress that the interference term (283), (284) shows the typical oscillations caused by the diffractional scattering of the ejected Auger electron at identical atoms. However, the character of these damping oscillations is different: j (kR ), n (kR ), and j (kR l/D). M D M D M D The narrow line contribution to the interference term (284) demonstrates for large "l" a weak damping (&1/l) in comparison with P and the direct term o, whereas a quenching of this DM contribution for large C is much faster (&exp(!kR C/D) than P and o. D DM 14.5.3. Atomic-like profiles for small Doppler broadenings When the Doppler broadening is small the atomic line profile collapses to a single Lorentzian (280) p(E,u)"p (E,u)#p (E,u), p (E,u)"p w E S ES M ES

1Gj (kR ) M ES l#C

(285)

with p "2p G(p )(X#X)C/p. The second equation strongly reminds of the structure of the M M M E S radiative RXS cross section [95] if k is replaced by the change of the X-ray photon momentum under scattering and GP$. Like for radiative RXS [95] the cross section (285) demonstrates parity selection rules in the limit kR ;1. However, contrary to radiative RXS, transitions ES between ground and final states of the same parity are forbidden in resonant Auger spectra for kR ;1 due to the qualitatively different decay operators (254) and (256). ES 14.6. Super-narrowing of the atomic-like resonances At this stage we call attention to the conspicuous role of the channel interference for the narrowing of the RXS spectral profile. This narrowing occurs up to an HWHM of *lKD/kR,

R"min+R ,R , , (286) E S and does not depend on the lifetime broadening C. This lifetime free narrowing effect origins entirely from the single oscillating factor j (kR l/D) in Eq. (284). For example for O with M D  X"0.5 eV one has *lK0.03 eV (D"kv/2K0.4 eV, kRK14). Fig. 103 shows that the RXS profile consists of a narrow peak, or a hole, with the HWHM *lK0.03 eV (286) smaller than the lifetime broadening C"0.09 eV [174] and a broad Doppler pedestal (Jo) (269). Let us note that

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Fig. 103. The narrowing below lifetime broadening and below spectral width of incident radiation. D"2 a.u. The dependence of the total RXS profile p(E,u) (solid line), the interference contribution (dashed line) (276)—(268) and the direct term (Jo) (269) on the relative kinetic energy l"E!u . The interference term is scaled by a factor of 2. The GD comparison with the Lorentzian with HWHM equal to C (dot-dashed line) shows the narrowing below C. (a) x 'x , S G x (x (w !w (0). (b) x (x , x 'x (w !w '0). Input data are the same as for Fig. 102. E G S E S G E G S E Fig. 104. The relative intensity p /*p"2i exp(!i) (287) of a narrow resonance as function of the dimensionless  parameter i"CkR/D for the case w "1, w "0. Input data are the same as for Fig. 102. S E

we have here reached the thermal energy since *l&k ¹&0.03 eV. This means that rotational broadening must now be taken into account together with the recoil effect [266]. A comparison of the ratio of the maximal values of the narrowband and the direct (p"p #p )   contributions to the cross section gives (287) p /p"2(w i e\GS!w i e\GE) E E  S S which was estimated as *K/o for l"0. Here *K is the first term at the right-hand side of Eq. (284). This equation shows that the relative intensity p /p of the super-narrow contribution caused by  the channel interference depends on the dimensionless parameter i "(C/D)kR , (288) D D being the product of the diffractional parameter kR and the ratio of the lifetime broadening C and D Doppler shift D. The role of the diffractional parameter (from the point of view of observation of the narrowing effect) is two-fold. The width (287) of this resonance decreases when kR increases. However, its

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strength (287) increases only when i changes from zero up to i "1. The strength of the narrow D D resonance is quenched for i '1 (Fig. 104). So a value of i close to 1 gives optimal conditions for D D observation of this narrowing effect. 14.6.1. Qualitative picture of the narrowing effect The attention in the previous section was focussed on the formal description of the narrowing effect. As one can understand from Eqs. (282), (284), the interference term p "2p G(p )K lies  M M behind this narrowing effect. The physical reason for the super-narrowing of the atomic-like resonance can, however, also be understood as follows. The left and right atoms eject Auger electrons at different angles (cos h"$l/D, D"kv/2), leading to the suppression of the interference contribution p or coherence between scattering channels 1 and 2 if the energy of the Auger  electron does not coincide with the resonant frequency; l"E!u O0. However, both scattering GD channels are strongly coherent and so the interference term takes a maximal value for the exact resonance: l"0. In this case the Auger electron is emitted perpendicular to the molecular axis, h"p/2. But the uncertainty relation implies that this angle p/2 is known only to within *h:j/R or *h:1/kR ,

(289)

where j is the electron wavelength. The uncertainty in angle leads to an uncertainty *l of the Auger electron energy l, since cos(p/2!*h)"*l/DK*h. Combining these two results one obtains *l:D/kR which agrees with (286). 14.6.2. Role of the spectral width of incident X-ray radiation It is important to mention that the narrowing up to *l (286) does practically not depend on the spectral width of the incident radiation. The main reason for this is that the position of the super narrow atomic-like resonance (l"E!u "0) (284) does not depend on the frequency of incident GD radiation [91,102,106]. One can also conclude that the shape of the atomic-like resonance does in practice not depend on the spectral width c . A 15. Screening, relaxation and chemical shifts The differences between resonant and non-resonant X-ray emission processes are manifested in several ways as reviewed in detail in the present work. The discrete non-degenerate character of the intermediate core-excited state in the resonant process makes it polarization, symmetry and momentum selective in ways that are unmatched by the non-resonant process. At ultra-high resolution there are “Raman” related effects with resonant narrowing and linear dispersion also without counterpart in non-resonant X-ray or Auger emission. With respect to the analysis of electronic structures there is an obvious difference between resonant and non-resonant emission spectroscopy in that the presence of a core-excited electron changes the organization of the remaining electronic cloud. In the elastic resonant process this electron participates in the decay (participator decay), while in the inelastic process it remains populating the excited orbital, and the emission takes place from a level already occupied in the ground state (spectator decay) just as in non-resonant emission but then without the presence of a spectator. This spectator electron may screen the decay spectrum such that transition energies and intensities become altered. Such

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changes can be dependent on size and type of system, for instance between heterogeneous and homogeneous systems. For the latter systems, like for example conjugated polymers and metallic species, the indication is that shifts grow comparatively small due to delocalization of the core-excited electron and due to the reduced core—valence interaction. 15.1. Screening in free molecules In the heterogeneous case one has observed screening mostly for the non-radiative RXS of small molecules [289]. Recent recordings [69] also indicate screening effects for radiative RXS, in particular concerning energies; for both radiative and non-radiative emission the CO molecule has served as the best illustration. Favorable conditions for screening prevails when a low-lying spectator level, e.g. a LUMO nH level, with valence character is excited, while the population of higher excited Rydberg levels does not seem to be of much consequence for the decay spectra, and such resonant spectra often turn out quite non-resonant like. Refs. [290,69] give examples of theoretical, respectively, experimental investigations on the role of screening in molecular RXS spectra. 15.1.1. Definition of screening The screening energy, E1!0, and screening intensity, I1!0, are defined as H H E1!0"E0!E,0 H H H

(290)

I1!0"¼0!¼,0 H H H

(291)

and

where the emission from the jth level following resonant (R) and non-resonant (NR) excitation is compared. The quasi-particle or molecular orbital picture is then assumed to be valid in both cases, so that it is meaningful at all to identify common levels +j,. The screening is also dependent on the core and unoccupied levels involved (the notation of which is here suppressed); for each given unoccupied level (absorption resonance), the screening will thus be dependent on four states. 15.1.2. Rydberg level screening Water, being a small non-degenerate system, is a first obvious choice to test screening effects. As saturated it possesses a first excited orbital, 4a , that is neither pure valence- nor pure Rydberg-like,  but that is often denoted as a mixed Rydberg/valence orbital. In the more attractive potential of an open core, this level should be more compact and localized to the core site, however, the penetration to the core and therefore the screening can still be anticipated to be small. The results of the calculations, shown in Ref. [290] seem to confirm this conjecture. The addition of an electron to the LUMO 4a level reduced the intensity for the HOMO 1b level emission only by about 3%,   and the energy by 0.9 eV. Results for HOMO level emission when the photon excitation is tuned to the Rydberg 1b level, indicate that the screening effect is then of the same order of magnitude as  for 4a excitation. 

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15.1.3. Valence level screening Carbon monoxide has served as a standard example in non-resonant X-ray emission; the spectra of the two core sites illustrate the mapping the local p-density of the molecular orbitals, a fundamental aspect of the spectroscopy. The CO molecule is an obvious choice also for the resonant X-ray spectra. It represents small unsaturated species, with multiple bonds which contain (at least) one excited level which is strongly valence-like, as the 2p, or nH, level. Being valence-like this level has specific localization with respect to the two core sites, and this localization might also change upon opening of these cores. Already in the ground state it localizes towards the carbon (counterbalancing the 1n localization towards oxygen); but on opening of the carbon core it collapses radially onto that core. This leads to a strong C1s-nH dipole absorption, a factor of three stronger than O1s-nH absorption. The presence of the valence like nH electron leads to a considerable screening in energies for the HOMO (5p) emission; 3.5 eV in the carbon spectrum and 2.9 eV in the oxygen, see Table 5. One finds that the intensive carbon HOMO emission is relatively little screened [290], only 2% of the total non-resonant intensity, whereas the weak oxygen HOMO emission (only about 10% of the carbon HOMO emission) is well screened; the screening contribution is of the same order as the total non-resonant intensity. The screening is also of different sign in the two cases, which can be rationalized [290] by that much of the screening of the oxygen core takes place through the occupied orbitals rather than through the nH orbital; it actually enhances the carbon nature of the latter somewhat. Thus even a minor relaxation change of the carbon lone-pair 5p orbital with increased oxygen one-center character leads to significant changes in the emission. For the carbon case both the HOMO (5p) and LUMO (nH) levels are on the core site (being off-core-site in the oxygen case). For carbon resonant emission the nH level is thus self-screening, and the C1s-nH excitation leads directly to screening of the core without much further rearrangement of the bound electrons compared to the oxygen case. The smallness of the screening effect in the carbon case must also be assigned to special cancellation effects.

Table 5 Experimental nH-resonant (E0) and non-resonant (E,0) energies and relative intensities (I0,I,0) for the 5p and 1n X-ray emission bands of CO. The experimental energies are compared with the computed energies and intensities, which are normalized in the same way as the experimental data [69] Emission band

E0 (eV)

I0

E,0 (eV)

I,0

Carbon 5p 1n

279.1 (279.3) 277.8 (277.9)


1.25 (1.10) 1.00 (1.00)


282.0 (282.9) 278.4 (279.6)

0.86 (0.88) 1.00 (1.00)

Oxygen 5p 1n 4p

525.4 (525.3) 524.9 (523.9)
520.7 (—)

0.24 (0.20) 1.00 (1.00)
— (—)

528.3 (528.2) 525.5 (525.0) 522.7 (—)

0.13 (0.07) 1.00 (1.00) 0.24 (—)

The intensities in the nH RXS spectra are corrected for the angular dependence (168).
The values for the 1n\nH D and R\ bands are used.

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15.1.4. Core level screening For second or higher row elements it is also relevant to analyze differential screening between different core levels, either defined as main shells (K-,¸-,M-) or as subshells (¸ -, ¸ -). When the ' '' effect of screening is small, one anticipates only very small differential core level screening, especially when the core levels are localized at the same site, with almost identical valence electronic structure. Learning from the chemical shifts in hard X-ray core-to-core spectroscopy one can anticipate differential screening energies in the order of a few tenths of an eV for K emission for ? second row molecules. One might thus consider the addition of a spectator electron to act as a chemical substitution and induce a differential core level screening. The underlying mechanism of such shifts is given by the penetration of valence charge into the core which is different for different core levels and which is perturbed by chemical substitution. Calculations to test this conjecture were carried out for the H S and CH Cl molecules in Ref. [290]. The results indicated that both   K emission and ¸ emission to be screened in the order of one eV, irrespective of excitation level. @ ? The differential shifts between the main core shells were only few tenths of an eV, thus corresponding to the size of the traditional X-ray chemical shifts. The screening increases somewhat for the more shallow lying core shells. Calculations of the ¸-shell screening indicated only minor differential subshell screening. The screening intensities are found to be of a few % in most cases, and of negligible magnitude even when the screening energy is comparatively large. 15.1.5. Contributions to the screening By means of unconstrained optimization techniques using multideterminant wave functions the screening can be decomposed into different contributions, that is electrostatic, relaxation and correlation contributions [290], see Table 6. With the employed computational methods also initial versus final state relaxation, resonant versus non-resonant relaxation, and the penetration relaxation can be obtained [290]. The presence of a valence spectator electron gives a static decrease of transition energy of typically an eV, while relaxation gives the most important contribution to the total screening energy in CO [290]. The correlation contributions are significant (half an eV for nH screening in CO), but smaller than the relaxation contribution [290]. The penetration relaxation, i.e. the difference between full relaxation and only bound electron relaxation with the screening (spectator) electron frozen, seems to give a large relaxation contribution to the screening. The relaxation of the nH electron alone seems to induce strong positive contributions to the screening, thereby counteracting the relaxation of the other bound electrons (calculations of CO in Ref. [290]) The magnitude of these contributions are obviously dependent on the particular electronic structure and the localization of the levels involved, for instance the contributions for oxygen and carbon nH screening can be quite large. Concerning intensities one notes that the static interaction with the spectator electron produces an increase of about 0.5;10\ in both the carbon and oxygen spectra. However, being the property of four states and because of the non-additivity and cancellation among the various contributions it can be hard to rationalize the origin of the screening in each particular case. From studies of a few different cases involving Rydberg-, valence- and core level screening, one can conclude that the screening is quite dependent on type of molecule — saturated versus unsaturated — and on the core site, but little dependent on the particular core-shell for a given core site. A differential core level screening, relating to the X-ray chemical shift, can be noted. Although

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Table 6 Different contributions to screening energies (eV) and intensities (squared transition moments times 10\) for resonant X-ray emission of CO Calculation

C-energy


C-intensity


O-energy


O-intensity


Static SCF MCSCF Rlx Corr R-rlx NR-rlx I-rlx

F-rlx P-rlx

!1.228 !4.361 !3.546 !3.133 0.815 !13.498 7.908 !5.588 !0.002 !3.827

0.653 !0.383 !0.140 !1.036 0.243 0.518 !0.248 !0.497 0.096 !0.419

!1.277 !3.306 !2.891 !2.029 0.415 !21.393 16.809 !4.533 !0.051 !3.119

0.451 0.146 0.530 !0.305 0.384 0.363 0.234 0.308 0.189 0.086

LUMO (nH) excitation and HOMO (5p) emission levels are considered. In calculations h,i,j,k,l the core orbitals were frozen due to perfect cancellation of core relaxation energies in all cases. As in Table 5, an spd basis set and a [3!7a ,1!4b ,1!2b ] active space are used.   
Screening energies and intensities (velocity gauge) defined by eqs. (290) and (291). Using frozen ground state canonical Hartree—Fock orbitals. From SCF (open-shell restricted Hartree—Fock) energies. From MCSCF energies. Difference between SCF and static values. Difference between MCSCF and SCF values. SCF values for states involved in resonant emission, static values for states involved in nonresonant emission. Static values for states involved in resonant emission, SCF values for states involved in nonresonant emission.

Values assuming relaxed initial states (SCF), and static final states (frozen). Values assuming static initial states (frozen), and relaxed final states (SCF). Penetration relaxation defined as difference between screening obtained by full SCF and SCF with nH level frozen.

screening can produce shifts in transition energies of a few eV, its effect on the transition intensities is relatively minor. This indicates also that the large bulk of resonant spectra of organic molecules can be conducted within the frozen orbital approximation, as they indeed also have so far. From the theoretical side one finds that the use of unconstrained optimization techniques with multiconfiguration wave functions is a good tool to explore the effect of screening in resonant X-ray emission of molecules. 15.2. Screening in extended systems For homologous compounds, like conjugated polymers and metallic compounds, the screening is intimately connected with the notion of excitons [291] and will have a different meaning than for heterogeneous compounds, see also Section 16.3.1 which discusses excitons in n-electron polymers. As for small systems the screening of extended systems depends on if the core-excited electron populates a valence (conducting) level or not, and thus if it couples to the remainder of the system. If it does one can denote the level housing the core-excited electron as self-screening, i.e. the level

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populated by the excited level and the level screening the core hole (and the reminder of the electrons) is the same. In such a case one can expect significant corrections to the frozen orbital model for describing the RXS spectrum. However, actual calculations of polymers [292] and surface adsorbates [293] indicate that the frozen orbital picture is favourable for evaluation of X-ray emission intensities despite the fact that an unoccupied nH level can undergo a heavy population during the creation of the initial core hole. The spectra from the screening level itself (REXS) shows on the other hand a strong variation between relaxed and frozen orbital treatments as discussed in Section 17. This fact connects to the theory of Mahan [294], Nozieres, DeDominicis [295] (MND) for the dynamics of core-hole states in metals, see also the reviews on the soft X-ray edge problem by Kotani [296] and Ohtaka and Tanabe [297]. According to the MND theory the role of the core-hole relaxation is large for metals because of the absence of a gap between occupied and unoccupied states. The core hole creates electron—hole pairs (shake-up pairs) with largest probability for states lying close to the Fermi level, and, as a result, in the field of a core hole there is a strong reconstruction of the Bloch states close to the Fermi level. Another consequence of the absence of a gap between occupied and unoccupied states is the strong change of the matrix elements for the emission transitions from the states close to the Fermi level. However, the role of this relaxation effect is small for Bloch states lying further below the Fermi level; approximately below 0.1D, where D is the width of the valence band, it is negligible [295]. According to numerical calculations in Ref. [297] the spectral shape of the emission transitions from these states coincide with good accuracy with those obtained from the frozen orbital approximation. When the lifetime broadening of the core excited state increases the MND singularities close to the Fermi surface decrease and the frozen orbital approximation works better. One can say “that the Bloch states have no time to relax” if the lifetime of the core-excited state is short. In the case of finite band gaps the role of particle—hole (shake-up) excitations induced by the core hole will be weaker in terms of perturbation on the wave function, but will on the other hand involve a larger portion of the occupied band, also below 0.1D. In molecules, having large excitation gaps, core-hole-induced relaxation can a priori not be disregarded for any level; the outcome of the relaxation is then more dependent on the particular localization of the levels with respect to the core hole. Simulations of RXS spectra of surface adsorbates and polymers, see for instance Refs. [293,292] seem to concord with MND theory in that most of the core hole induced relaxation effects is located to the outermost, screening (nH) level, while other occupied levels seemingly are well described by the frozen orbital calculations. In this sense the surface adsorbate and polymer spectra behave more metallic-like than molecular-like. 15.3. Localization, relaxation and correlation in a two-step picture In the cases of small screening and isolated core-excitation resonances, the inelastic X-ray Raman spectra should be interpretable as the normal non-resonant X-ray emission spectra; in terms of molecular orbital theory, role of localization, relaxation and correlation contributions, etc. This also leads to tractable computational schemes for these effects, including also applications to polymers or other systems subject to a periodic boundary condition [298]. Provided that the X-ray emission can be treated within such a two-step model for electron-molecule or photon-molecule inelastic scattering, with core excitation and decay separated, the X-ray spectral rates can be

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analyzed starting out from Fermi golden rule like expressions. In relative units these take the form





I K"1W "DK "W 2"" q s V V AD A D V



(292)

where W and W are, respectively, the core ionized and valence ionized initial and final states in the A D N!1 electron system. The rate is thus expressed as a sum of terms, each of which constitutes a product of a molecular orbital (MO) factor q and a many-body factor s . The slowly varying V V energy factor preceding this expression is here, and henceforth, omitted for simplicity. In X-ray emission X stands for a double index p,q, for the many-electron dipole operator DK " 1t "dK "t 2aˆ >aˆ N O N O NO

(293)

where dK is the one-electron dipole transition operator, aˆ > and aˆ the usual creation and annihilaN O tion operators, and +t , and +t , sets of molecular orbitals that in general are mutually O N non-orthogonal. The product of molecular orbital and many-body factors resolves as q s "1t "dK "t 21W (N!1)"aˆ >aˆ "W (N!1)2 V V N O D N O A

(294)

It is well motivated only to consider final states with a double core occupancy restriction; other core occupations produce configurations with very different energy and negligible interaction [299,300]. For all practical purposes it is then motivated to limit the index p to one, corresponding to the creation of a core hole at one specific site, p"c, due to the almost perfect orthogonality (s K0) between states with holes in different core orbitals or with holes in core and valence V orbitals. For molecules with several core hole sites the result is a superposition of X-ray spectra each treated separately (different index p) according to Eq. (294). The molecular orbital factor can be obtained from separate state optimization of orbitals, or from a common set of orthogonal molecular orbitals — in practice by optimizing a single determinant wave function for the neutral ground state — thereby neglecting relaxation. With the two approximations indicated above; the truncation of the p index to one, referring to the core orbital, and the neglect of non-orthogonality (orbital relaxation) — which is the critical approximation — Eq. (294) is replaced by q s "1t "dK "t 21W (N!1)"aˆ "W (N)2 V V A O D O 

(295)

since under these assumptions the action of aˆ "W (N)2 and aˆ >aˆ "W (N!1)2 is the same. The wave O  N O A functions in Eq. (294) are completely general for both initial and final states. The only restrictions are orthogonal orbitals and the truncation of p"c. In practice there is no meaning to operate aˆ >"W (N!1)2 with a many-body expansion of "W (N!1)2 since this expansion would not N A A simulate that of any other state, let alone the ground state. So the operation has only meaning for the one-determinant case, i.e. when initial state correlation is neglected. This approximation also follows naturally from the neglect of orbital relaxation.

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The electronic factors in Eq. (295) are by definition the residues of the one-particle Green’s function, res G . The Greens function G is formally defined as IJ IJ  G (u)"!i (296) dt eiSR1W (N)"¹+aˆ (t)aˆ >,"W (N)2, IJ  I J  \ where ¹ is Wick’s time ordering operator applying to aˆ >(t) in the Heisenberg picture. I Using the one-particle Green’s functions the X-ray rates are therefore given as



VWX (297) I " 1t "dK "t 21t "dK H"t 2 res DG J G A A G I S IJ AD IJ G where u is the pole corresponding to the final state ionization potential. Thus under the D assumptions given above the Green’s function analysis of the X-ray emission spectrum will be equivalent to the one for the (valence) photoelectron spectrum, only different MO factors are involved. The above equation, derived in ref. [298], can also be obtained directly from the Kramers—Heisenberg relation for the cross section for the inelastic resonance X-ray scattering. 15.3.1. Role of correlation One can distinguish five different cases in the above equation, Eq. (297), thus identifying the approximations which lead from Koopmans theorem and the quasi-particle picture to hole-mixing and breakdown effects. These approximations, which are of a quite general nature, have received much attention in the case of interpretation of photoelectron spectra [301,302] as for normal X-ray emission spectra [303]. (We remind that in the RXS case the scheme holds only in a two-step picture, with the core hole state excitation and deexcitation decoupled). 1. res G "d d for a particular q. The Hartree—Fock picture is retained which implies that the IJ IJ IO energetics of the X-ray spectrum can be analyzed by Koopmans theorem, and intensities by the MO factor q alone. This in turn can be conducted in terms of MO theory, local densities, O effective and strict selection rules. 2. res G is close to 1 for a particular q. The quasi-particle picture holds. An MO analysis is still OO possible. 3. More than one G with different q enters in the residues (often small with respect to the OO hole—particle excitations), which leads to hole-mixing effects, and electronic interference in transition cross sections. 4. More than one G with equal q enters in the residues (often large with respect to the OO hole—particle excitations). This leads to breakdown effects, that is breakdown of the molecular orbital picture that makes a 1—1 correspondence between MOs (or MO factors in Eq. (292)) and spectral bands (states). 5. No single G has a large contribution to the residues. The state is a correlation state satellite. OO The analysis can be further simplified by decomposing the molecular orbital factor q by O expanding the MOs +t, in a linear combination of atomic orbitals +u, LCAO) t " c u in the I ? I? ? spirit of the intensity model for molecular X-ray emission [304]. MO analyses based on this model have been presented for a number of species and its limitations and merits are well established by

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now. A critical examination of X-ray intensities comparing one- versus two-center contributions calculated at the self-consistent field (SCF) level of theory has been given in Ref. [305]. Further tests of the model versus full X-ray transition intensity calculations using state-specific, non-orthogonal, orbitals and considering electron correlation for both initial and final states have been given in Refs. [306—308], using configuration interaction (CI) and multi-configurational SCF (MCSCF) wave functions, respectively. The results obtained for first row species indicate that the one-center decomposition indeed is warranted. The largest errors of the original one-center model actually refer to the use of frozen orbitals, in some cases also to neglect of electron correlation in the final state. 15.3.2. Character of states From the character of the final states, the X-ray emission (XES, or two-step RIXS) and ultraviolet photoelectron spectra can be analyzed in parallel. The final states can be grouped in four energy intervals according to this character, roughly ranging as 10—20, 20—35, 35—45, and 45 eV for low-Z molecules. The transitions from core-hole states and from other satellite mechanisms are then neglected. A detailed analysis of this classification, with applications on the CO and N molecules, can be found in [303,306].  The first group of transitions, collecting the highest amplitude in XES are well characterized by Koopmans theorem with each final state corresponding to emission from a particular molecular orbital. They belong to either category 1 or 2 states described above (Koopmans- or quasi-particle states). The MO decomposition analysis should hold well in this energy region. The second energy interval contains “radiative electron rearrangement” transitions, and lead to final state wave functions dominated by particle excitations, however, the intensities of these hole-mixing states (category 3 above) are governed by the intermixing of the main, “Koopmans”, configuration. If only one such main configuration prevails, an equal branching ratio between main and satellite intensity is predicted for UPS and for the different core-sited XES spectra [306]. However, when several such main configurations are mixed in (hole mixing) there will be a complex excitation energy dependence for UPS, and a deviation from equal parent to satellite branching ratios will be the case. For XES one then has different branching ratios for different core-sited spectra (for instance between C and O spectra in carbon monoxide), with limited applicability also of the local selection rules. The hole-mixing effects become increasingly more important for larger molecules, since the near-degeneracy of the main one-hole configurations will rise. In the third energy region, 35-45 eV, considerable configuration interaction splitting of the final states prevails. In contrast to the hole-mixing region, where the final wave functions are dominated by hole—particle excitations, the wave functions are here dominated by single-hole configurations. However, there are often more than one such hole-configuration and one cannot associate a given state in this spectral region with one particular molecular orbital (breakdown state, category 4 above). Since this region has MOs with considerable 2s character it receives low XES intensity for 1st row molecules. Finally, above 45 eV most states are correlation state satellites (category 5 above) with low intensity in XES, as well as in other spectroscopies. 15.3.3. Role of localization Using a one-center decomposition X-ray transition rates can be obtained in the following way: I " dG dGH c cH res G , AD A? A@ I? J@ SD IJ G?@ IJ

(298)

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where dG "1t "dK "u 2 are the dipole matrix elements between the initial state core hole t and the A? A G ? A ath valence atomic orbital. The x,y or z components of the dipole transition operator are here explicitly introduced by index i. We consider first K transitions pertaining to a 1s hole in atom ? A and neglect terms in Eq. (298) that corresponds to cross transitions form atom A to any other atom, the so-called one-center approximation. Since t is a 1s orbital and the selection rules in A dG require that a must be a 2p orbital Eq. (298) simplifies to A? VWX c cH res DG I ""d " (299) I? J? AD QN S IJ IJ ?ZN "d " is here a common factor (the radial integral) that can be neglected for relative intensities. QN Considering now ¸ transitions from a 2p hole in the ¸ (¸ ) shell to the M-shell, then @ ''''' 2 "c cH # "2 " c cH res DG . (300) I " "d NQ IQ JQ 5 NB I? J? S IJ AD ?ZB IJ The coefficients  results from integration of the angular parts of the atomic wave functions, while  d and d now refer to radial integrals. If the radial nl wave functions are treated as equal this NQ NB can be simplified further by omitting these radial integrals. Eqs. (299) and (300) are the manyelectron analogues of the one-center intensity model for K , K .. and ¸ , ¸ .. emission. Other types ? @ @ A of transitions (M , etc.) can be treated in an analogous way, only different radial integrals with A different angular coefficients appear. Eqs. (299) and (300) hold when there is only one atomic orbital per atomic site that enters in the LCAO expansion of a particular molecular orbital, thus when using a so-called single zeta (SZ) basis set. In many calculations one uses an n-zeta (double-zeta, triple-zeta, etc.) basis set (with or without polarizing functions) in which case Eq. (299) reads as













L (301) I " d dH c c res DG , IQ? JR? S IJ AD AQ AR QR IJ ?Z where s, t are indices of the expansion of MO t over the atomic orbitals u of a particular I Q? symmetry (2p for K ) and center. In this case only the radial integrals d need to be considered. ? AQ 15.3.4. Periodic systems For periodic systems like polymers the spectral intensity distribution function I (E) can be A calculated as



L I (E)" dk d(E!E (k))"1W "DK "W (k)2" , (302) A D A D D \L where E (k) denotes the final-state bands, which may involve satellite bands due to correlation and D W (k) the corresponding periodic many-electron wave functions. In analogy with the above D procedure for evaluating the many-electron matrix element and employing the LCAO expansion of the canonical polymer orbitals [309—311] t (k)"N\ e I0&c (k)u , H H? ? &?

(303)

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where k is the quasimomentum, H the cell index and N the number of unit cells which is used to obtain the second k integration in the limit NPR, we find

 

VWX L L dk dkd(E!E (k));c (k)cH (k)res D G (k) . I (E)" dG dGH A? A@ D H? J@ # I HJ A \L HJD \L ?@ G In case of K transitions the expression can again be simplified to give ? L L VWX I (E)""d " dk dkd(E!E (k)); c (k)cH (k) res D G (k) A QN H? J? # I HJ D \L HJD \L ? for a single zeta basis set and

 

 







(304)

(305)



L VWX L L VWX I (E)" dG dGH (306) dk dkd(E!E (k)); c (k)cH (k) res D G (k) AQ AR A D HQ? JR? # I HJ \L \L HJD QR G ? for an n-zeta basis set. Corresponding equations are easily obtained for ¸ and other types of x-ray @ transitions. Within the simplifying two-step approximation energies of the emitted x-ray photons are evaluated from the simple relation u"IP !IP , where IP and IP denote the initial core A D A D and final valence state ionization potentials, respectively. IP are then determined from the poles of D the one-particle Green’s function. The correlation properties are the same as above discussed for the transition rates, since each pole corresponds to its residue entering the transition rate. The pole energies are evidently common for different spectra pertaining to different core holes, and the full spectrum is obtained from one single Green’s function calculation. Calculations within this two-step scheme has been made for systems like polyenes, polyacenes, polyacetylene and polyethylene [298,312]. 15.4. Role of chemical shifts in RXS As briefly mentioned in the introduction of this review the influence of the chemical shift became early a main topic for X-ray spectroscopy [10—13]. The measurements were conducted in the hard X-ray region involving core-to-core radiative emission and the chemical shifts observed were small, often amounting only to a fraction of an eV. They originate from the valence charge penetrating into the core region; this penetration is different for the two core levels involved in the X-ray transition and this difference changes due to the influence of the chemical environment. The shifts are thus “secondary” effects, and are an order of magnitude smaller than observed for single core levels in photoelectron spectra (XPS), the main part of which thus is canceled for transitions between two core levels. Like for XPS the X-ray shifts can be rationalized in a ground state model provided the chemical environment involve changes of groups that are strongly electropositive or electronegative, however, in other cases — e.g. for most organic systems — it becomes crucial to consider final state effects, which in general are not so easily rationalized. The basic mechanism behind core-to-core X-ray emission shifts are anyway understood and such shifts are by now well characterized for a great number of compounds. In the soft X-ray wavelength region the X-ray transitions refer to valence levels and are therefore strongly influenced by the chemical environment in terms of electronic and geometrical structures given as molecular properties or by the formation of bands for metallic elements. With modern

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spectrometers it has been possible to monitor such influences both on a molecular orbital and a vibronic level of energy resolution. However, non-resonant X-ray spectra excited by broadband photons involve excitations of all core-shifted states, and the decay spectra from these shifted states become mixed in the final spectrum unless the unusual situation previals that the initial state shifts are larger than the widths of the valence bands. Using tunable synchrotron radiation with small band passes the chemically shifted core-excitation levels can be exclusively excited and each core level spectrum can in principle be studied separately provided the shifts are in the order of an eV. As demonstrated in Ref. [144] the resonantly excited X-ray emission spectra referring to core levels shifted within the sub-eV range will be strongly dependent on interference effects, and therefore on the precise values of the chemical shifts. Even very small such shifts, far smaller than the resolution limit in the corresponding core absorption spectra, can give rise to significant changes in the spectra. With a progressive reduction of the shifts, the salient symmetry dependent features of the degenerate system are restored [144,223]. For instance in the case of the monosubstituted benzenes much of the character of the benzene resonant X-ray emission spectrum is restored in the spectra with small chemical shifts of the C1s-nH levels of the phenyl ring carbons. This restoration of symmetry selection occurs due to channel interference. This implies also that the the one- and two-step model descriptions of the resonance X-ray emission phenomenon differ strongly, and only the former is warranted for investigations of spectra referring to core-excited resonant states with small chemical shifts, while the two-step model can be appropriate for spectra with large chemical shifts depending on the particular system, see also Section 6.2. 15.4.1. Chemical shifts and channel interference The connection between chemical shifts in RXS spectra and the interference effect can be appreciated by a simple four-state model assuming C symmetry, and which connects with the T analysis of spectra of e.g. substituted benzenes [144] (see below). The model includes two core orbitals (c ,c ), one occupied n orbital ( j) and one unoccupied nH orbital (l). The dipole transition   moments between core orbitals and the unoccupied orbital and the occupied orbital are denoted as dXG and dX G, (i"1,2), respectively. In the one-step model the emission intensity then reads HA AJ p-1(u,u )Jp #p #p (307)     and K A p " ,  D #C A D D #C A A , (308) p "2K K   A A (D #C)(D #C) A A where K G"u Gu G(l)dX GdXG is defined as the transition strength of channel i. D G"u!u G (the JA HA JA A H HA A A lifetimes of the two core-excited states are assumed to be the same). According to the general two-step model the emission intensity is K A , p "  D #C A

p21(u,u )Jp #p .    To rewrite Eq. (308) one has p-1(u,u)J(p #p )(1#s )   

(309)

(310)

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and K K (D D #C) A A A A . (311) s "  K (D #C)#K (D #C) A A A A The difference between the one-step and the two-step models is thus dependent on the interference strength s , 04"s "41. If this is equal to zero, both models will give the same result. This holds   for two cases, firstly when there is only one channel available, i.e. when K  or K  is zero, and A A secondly, when D D "!C, which indicates that one can manipulate the interference pattern by A A detuning the excitation energy. When s is equal to !1, the spectral line of emission is totally  absent. Using a very narrow incoming beam it is possible to tune the excitation energy exactly resonant with one core-excited state. One can then assume that D "0, i.e. the excitation energy is A resonant with the c\nH state. Therefore, the interference strength is  s "2R /(1#R #R) , (312)  B S B where R "K /K  is the ratio between the transition strength of the near-resonant channel and B A A that of the resonant channel. R "du /C and du "E(c\l)!E(c\l), the energy difference S     between two core-excited states. For fixed transition strengths of the two channels, the maximum interference strength occurs at R "0 when two core-excited states are degenerate. When the S energy difference between the two core-excited states is unchanged, the maximum interference strength corresponds to the condition that R "(1#R ). B S By considering vibrational excitations it is quite easy to fulfill the condition that R "0 even for S two well separated core-excited states. In this case, the energy differences are no longer represented by the electronic transition energies, instead, one should consider the energy differences between vibrational levels of the core-excited states. Similar considerations should also be applied for the transition strengths of the two channels. 15.4.2. Chemical shifts in RXS of aniline The resonant X-ray emission spectra of aniline serve as an excellent illustration of the role of chemical shifts in the formation of RIXS spectra. It is then relevant to consider spectra involving the strong absorption features located at the low-energy part of the X-ray absorption spectrum which involve resonant excitations to the first nH level. This part is thus composed of C PnH G (i"1,2,3,4) transitions from the four different shifted carbon core sites. The precise energy differences — the chemical shifts — between these different core-excited states are important for the quantitative description of the RIXS process. Calculations using Hartree—Fock and static exchange methods were presented in Ref. [144] for the analysis of the absorption, non-resonant and resonant parts assuming randomly oriented aniline molecules. Due to the amino substitution the benzene symmetry D is lowered to C for aniline. The core F T orbitals for the latter molecule can be represented as C (a ), C (a ,b ), C (a ,b ) and C (a ), using           the order of the C atoms (C ) given in the insert of Fig. 105, showing the NEXAFS spectrum of G aniline. Unlike benzene, some core orbitals of aniline are non-degenerate, in particular, the C (a )   core orbital is well separated. For a given occupied orbital n, emission lines referring to the core orbitals k located at the different carbon atoms (C ) can be observed. G G The aniline NEXAFS spectrum is dominated by the first nH level, which is represented by two strong bands in the spectrum, the first containing the three C -, C - and C -nH transitions grouped   

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Fig. 105. X-ray absorption spectrum of aniline. The corresponding assignments in the spectrum are: (1) C (a )nH(b ), (2)    C (a )nH(b ), (3) C (a )nH(b ) and, (4) C (a )nH(b ).          Fig. 106. Resonant X-ray emission spectra of benzene and aniline recorded at excitation energies resonant with the lowest nH levels.

at about 285.4 eV and the second with the C -nH transition located at 286.8 eV. This could be  understood by the fact that the appearance of the NH group splits the core orbitals of the carbon  atoms, and also lifts the degeneracy of the nHe MO of benzene. The first unoccupied e orbital S S thus splits into two orbitals, b and a . Likewise is the outermost occupied e orbital, which refers   E to the strong outermost band in the non-resonant emission, and which becomes symmetry forbidden in the resonant LUMO (e ) emission spectrum of benzene [223], split into into b and S  a orbitals for aniline. The other sites follow an alternate behavior for the core-hole shifts.  The resonant X-ray emission spectra of aniline corresponding to the first two nH(b ) resonances  in the X-ray absorption spectrum are presented in Fig. 106. For comparison, the resonant X-ray emission spectrum of benzene referring to the first nH(e ) resonance is also included in this figure. S One finds the extraordinary feature that when the excitation energy is tuned resonant with the first

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nH band (referring the compound C nH levels for aniline), the two molecules show similar X-ray  emission spectra. The absence of intensity in the 280—284 eV energy region is noteworthy, since this is the result of the strict parity selection rule for RIXS applied to D benzene [223] (see also [313]). F By lowering the symmetry from D (benzene) to C (aniline), the inversion symmetry is dismissed F T and the parity selection rule does no longer hold. On this ground one expects that the intensity should be observed in the of 280—284 eV energy region for aniline. This is actually the case when the excitation energy is resonant to the second nH resonance (C (a )nH(b )). The two resonant X-ray    emission spectra shown in Fig. 106 possess the same initial and final states but different intermediate states. This indicates that multi-channel interference should be responsible for the absence of the intensity in the 280—284 eV energy region for the first, compound, nH resonance of aniline. Since the second nH resonance of aniline is dominated by only one channel, C (a )nH(b ), the    interference effect is negligible. Spectra calculated by the one- and two-step models are accordingly almost the same, see Fig. 106. However, for the spectrum referring to the first compound nH resonance of aniline, three different channels, labeled in Fig. 105, should be taken into account. Due to the small energy separations among those three intermediate core-excited states, especially between C (a )nH(b ) and C (a )nH(b ), the interference effect is expected to be large, as verified by       the simulations displayed in Fig. 107. For comparison, two simulated spectra considering a single channel, the C (a )nH(b ) channel, and two channels, the C (a )nH(b ) channels, are also shown.       These results show that when the shifts (here denoted d ) get smaller, the agreement with the  experimental spectrum gets better, and that the intensity distribution is crucially dependent on the interference effect. By fully considering this effect, not only the absence of the intensity in the energy region of 280—284 eV is explained, but the correct intensity distribution for the whole spectrum is also obtained. The spectra simulated with the two-step model using the same parameters, show poor agreement with the experimental spectrum [144], giving an example that the two-step model cannot be used for spectra where more than one dominant channel is involved in the resonant X-ray emission process, as is the case for species with small chemical shifts. 15.4.3. Site selective RXS The example given by the mono-substituted benzenes demonstrates that resonance inelastic X-ray spectroscopy of chemically shifted species is site selective. Core-excited levels with distinct, super-eV, shifts can be resonantly excited and their X-ray emission spectra analyzed separately. Core-excited levels referring to sites with small, sub-eV, chemical shifts give resonant X-ray spectra that interfere strongly. The same example also indicates that in the limit when these chemical shifts go to zero some salient symmetry selective features of the resonant X-ray emission spectrum are restored. Considering cases with small, intermediate and large shifts, one find the RIXS technique to be informative in the first and the third of these cases. For large shifts, say above 1 eV, the core level can be selectively excited, and the corresponding RIXS spectra can be analyzed in terms of electronic structure theory, local intensity rules, etc, for the particular core site. For very small shifts, below 0.5 eV, the interference effects will make RIXS spectra crucially dependent on the precise value of the shifts, and can make it possible to assign shifts although they remain unresolved in the corresponding absorption spectrum. In the intermediate region, outside the region of strong interference, but still unresolved in absorption, the RIXS spectra will overlap in a way that makes it difficult to distinguish the shifts. They will furthermore be Raman shifted with respect to each other in such cases. Precise limits of these regions will of course be set by the degree of vibrational

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Fig. 107. Calculated resonant X-ray spectra referring to C -nHb resonances using different chemical shifts.   The experimental spectrum is shown for comparison. One transition channel C -nHb , (A). Two transition   channels, C -nHb : d "0.3 eV, (B). Three transition channels, C -nHb : d "0.5 eV, (C); d "0.2 eV, (C);        d "0.05 eV, (E). 

excitations, the core-hole state lifetime, and the form of the excitation energy function. An additional source of information on chemically shifts in RIXS is provided by polarization or angular-dependent measurements, since the polarization anisotropy is very sensitive to interference effects as discussed in Section 6.2.

16. X-ray Raman scattering by crystalline solids and polymers Many of the important concepts in the field of resonant X-ray spectroscopy can be understood in the atomic and molecular context owing to the simplicity of gas-phase systems and to the superhigh resolution of the spectra then attained in comparison with extended systems. RXS

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spectra of extended systems forming energy bands instead of single resonances are complicated by the simultaneous manifestation of different effects, such as high density of states, momentum exchange between photons, phonons and the target and interactions between the electrons and the core hole. Furthermore, one has to account for the fact that the systems transcend the soft X-ray wavelength with special consequences for the electro-magnetic interaction. Systems that are in principle infinitely extended, like long polymer chains or metals, molecular crystals, surface adsorbates and polymers, can be treated using either fully periodical approaches, where the one- or two-dimensional translational symmetry is explicitly used, or cluster or oligomer approaches, especially relevant for low-symmetry systems like surface adsorbates. The first type of approaches refer to band theory applications for systems with periodic boundary conditions and to extensions thereof to defect band theory or super-cell methods for studying spectra with localized excitations (excitons). The second type of approaches start out by treating RXS as basically a localized excitation for which the surrounding is given by a cluster model or a part of the polymer. With this approach one considers a sequence of clusters (oligomers) that models the full bulk (polymer), and studies the convergence of a property or a spectrum within this sequence. One thus attempts to find the limiting behavior by successive enlargements of the cluster models, and studies the smallest repeat unit up to a cluster of such a large size that it can be considered to represent the infinite system. An advantage thereby is that the spectra can easily be decomposed into local contributions and interpreted in terms of building blocks. The two approaches have different relative merits and limitations and often give quite complementary interpretations. The main aim of this section is to review the role of solid state effects on the formation of the RXS spectral shape of polymers and solids. We begin with the first of the two basic approaches, band theory using the periodic boundary condition. The text starts out by an investigation of the radiative RXS spectral profiles from the simplest model where the electron—core-hole interaction is neglected (Section 16.1). In the case of solids and polymers this model leads to the concept of direct transitions [195,197] with consequences like the shift of the fluorescence threshold (Section 16.1.1) and a total prohibition of RIXS channels in linear polyenes and other n-electron systems. (Sections 16.2 and 16.2.1). Because of the momentum exchange between the target and photons or between phonons (Section 16.4) or electrons and the “heavy” core hole, the RIXS transitions attain an indirect character and the RIXS channel is opened in forbidden regions (Sections 16.2.1 and 16.2.3). In spite of the small value of the X-ray momentum a strong dependence of the RIXS and REXS profiles on the exchange of photon momentum takes place in the soft X-ray region (Sections 16.2.1 and 16.2.3). The influence of the excitonic character of the unoccupied states near the Fermi level on the spectral shape of RIXS is reviewed in Section 16.3. A “tail” structure caused by the excitonic states in photoabsorption is found above the top of the occupied band. The analysis of the dispersion relation is given in Section 16.3. The analysis of some spectral peculiarities of nonradiative RXS (resonant photoemission) by solids is also reviewed. At the end of this section, Section 16.7.2, an overview of the role of electron—phonon interaction on RXS for solids and polymers is given. 16.1. Independent-electron model of RXS for solids The frozen orbital approximation (independent-electron model) in which the electron—core-hole interaction is neglected forms a suitable starting point for investigations of solids and polymers

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with respect to the understanding of many spectral features of RXS. By absorbing incoming X-ray photons with the momentum p a core electron of the nth atom is excited to the intermediate state "bkn\2 with t k , e k and k as one-electron wave function, energy and momentum, respectively, @Y @Y of the unoccupied state. This intermediate state decays by spontaneously emitting X-ray photons with the momentum p to the final electronic state "bk(ak)\2, leaving a hole in the occupied one-electron state t k and with the energy e k and momentum k. ? ? The reduced zone representation with the electron wave vectors k and k lying in the first Brillouin zone is used in the following review. In this representation the Bloch functions and the corresponding electron energies are characterized by the band indices a and b. Due to the small overlap of core electron densities of different atoms, the core shells are strongly degenerate and with good accuracy one can speak about a unique core ionization potential I (however, see below the case of scattering by polymeric systems). The spectral properties of RXS are guided by the double differential cross section p(u,u)"p (u,u)#p (u,u) ' #

(313)

which is the sum of inelastic and elastic contributions. The Bloch theorem (u k(r#R )"u k(r)) for L ? ? the Bloch function t k(r)"eιkru k(r) leads directly to the momentum conservation law in inelastic ? ? and elastic RXS k"k#q!G ,

(314)

q"G, 2p sin(0/2)"G with the corresponding RIXS and REXS cross sections [195,197,202] u p (u,u)"r Nq ' Mu G k

uk uk "D kD k " YL L ? @ Y U(u#u k k!u,c) , @ Y? (u!u k)#C ? ?6$@7$

p (u,u)"r(u/u)Nq"Fq "U(u!u,c) d(q!G) . M # M G

(315)

Here u kKe k#I, u k kKe k !e k, G is the reciprocal lattice vector, a(F and b'F ? @ Y? @Y ? ? mean states lying below and above the Fermi level, respectively, where N is the number of unit cells in the crystal. q"(2n)/v is the volume of the Brillouin zone and v is the volume of the unit M M cell in real space. The following representation for the REXS amplitude of a one-component solid is used: F"Fq eiq  RL , M L






1 1 ! (316) Fq "(e ) e)oq # uk D kDHk @ @ u!u k#iC u#u k M M k @ @ @$ neglecting the two non-resonant contributions for the inelastic scattering amplitude (7) which are small in the resonant region. The matrix elements of absorption D k and emission D k dipole ? @

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transitions are defined in the same way with help of the corresponding dipole moment operators D and D



D k" dr e\ik  ruHk(r)Du (r) , ? M ?

(317)

where u (r) is the atomic wave function of the core electron. M The change of the total (photon#electron) momentum (314) is restricted to a reciprocal lattice vector: k!k#q"G. These changes in the momenta of electrons and photons during RXS are precisely what takes to have an RXS effect. The X-ray scattering thus occurs because of the energy and momentum exchange between the light electrons and the rigid crystal lattice. The REXS process changes the photon wave vector difference q"p!p by a reciprocal lattice vector G. The well-known Bragg condition (314) for elastic or coherent RXS shows that in the soft X-ray region (p(G/2) only forward coherent scattering dominates. 16.1.1. Shift of the RIXS threshold For RXS in the soft X-ray region one can neglect the change of the photon momentum q since q;G and remove the sum over the reciprocal vectors G in Eqs. (315, 316) (however, see below). This means that in this region only “direct” transitions between unoccupied and occupied bands take place (Fig. 108) [195,197,198]; k"k .

(318)

It is important to note that this result, obtained first in [195], is valid only in the one-electron, frozen orbital, approximation (315). When the interference of the RIXS channels through core hole states on different atoms is neglected, Eq. (318) breaks down. As shown in [195] this interference (and as a consequence the momentum conservation law, Eq. (318)), leads to a strong dependence of the RIXS spectral shape on the excitation energy, which has been confirmed by recent experiments

Fig. 108. Shift of RIXS threshold.

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[199,200]. The related effect of symmetry selective excitation [92,95] in molecular systems has been observed for many molecules [37] (see Section 9). It is necessary to note that the selection rules for the RXS tensor [95] constitute the molecular analog of the momentum conservation law (see Section 9). The momentum conservation law, Eq. (318), leads to a very important consequence, namely a shift of the threshold for RIXS of alkali metals, an effect obtained in [195] in a somewhat complicated way. In this context, it is relevant to note a closely related effect in X-ray Raman scattering by polymers, namely that the RIXS intensity is equal to zero in the n-electron and frozen orbital approximations. A quite simple qualitative explanation of this effect can be given as follows. In a simple two-band system the lowest band is semi-empty and the highest one is totally unoccupied (this is the case for alkali metals). Fig. 108 shows that due to the momentum conservation law (318) the direct RIXS transitions are forbidden and hence that the RIXS cross section is equal to zero when the excitation energy is smaller than the threshold magnitude u(u , A

u "D#u , A $

(319)

where u is the threshold absorption frequency being equal to the core excitation energy at the $ Fermi level E . The definition of the energy gap D is given in Fig. 108. For example, D&2.4 eV for $ lithium [195]. Thus, the unusual effect prevails that when the absorbed X-ray radiation creates a core hole the X-ray fluorescence is absent. The reason for this suppression of the fluorescence yield is given by the interference between scattering channels through core hole states at different atoms. It is easy to understand that this “vanished” RIXS intensity is transferred to the elastic cross section, which is not equal to zero in the forbidden region (319) (see Fig. 108). The electron—photon, electron—phonon and electron—hole interactions violate the momentum conservation law or the condition (318) of direct transitions (in the sense of no momentum exchange), and allow emission in this forbidden region. The ratio of the RIXS cross sections in “forbidden” (319) and “allowed” (u'u ) regions constitutes a direct indication of the strength of A electron—photon, electron—phonon and electron—hole interactions. However, as shown recently [103,117], the effective strength of the electron—phonon interaction can be significantly reduced even by small detuning of the incident X-ray photon frequency below the absorption band. 16.1.2. Interband density of states In the “direct-transition” model (q;G, (318)) the shape of a RIXS profile is defined by the spectral features of the dipole matrix elements through the factor Q(k)"ruuNq"D kD k"n/C M ? @

(320)

and by the intersection of the u- and (u!u)-surfaces u"u k, u!u"u k k ? @ ?

(321)

through the broadened interband density of states



o(u,u!u)"

de dg o (e,g) D(u!e,C)U(u!u#g,c) . M

(322)

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The interband density of states (IDS) describing photoemission d(u!u k)d(u!u!u k k) . (323) ? @ ? ?6$@7$ has singularity lines on the (u!u)-plane. These singularities stem from the vicinities of those k-space points in which the u-plane is tangent to the (u!u)-plane (321). The properties of the IDS o (e,g) was investigated by Kane [314] (see also [315]) in connection with the theory M of photoemission. As shown below the momentum exchange between photons and electrons can be important in the soft X-ray region too. In this case the expression for the IDS (323) must be modified such that the resonant frequency u k k is replaced by u k k with k" @ Y? @ ? k#q!G and such that the right-hand side of Eq. (323) is summed over the reciprocial lattice vectors G. The singularities of interest here are connected with the small region of k space close to the point k where the surfaces (321) are tangent. Extracting the factor Q(k ) from the integral over k (315), M M that varies little close to the k , one can write the RIXS cross section as M o (u,u!u)" M k

p (u,u)" Q(k )o(u,u!u) . M ' k

(324)

M

The combination of the spectral properties of the IDS with the symmetry selection of electronic states (with help of the polarization vectors of initial and final X-ray photons) should be useful for extraction of energy-band information from experimental (u,u) or (u,u!u) plots. It is important to note that, contrary to photoemission [316], the RIXS cross section can be expressed through the IDS only if the electron—hole interaction does not significantly distort the one-electron model. 16.2. RXS by polymers Many physical phenomena associated with conjugated molecules and polymers have been studied and understood in the framework of quasi-one-dimensional n-electron theory. It provides the possibility to find analytical solutions for phenomena that are present also in 3D solids, and thereby to illustrate numerically the general theory. This approach is reviewed in the following, using the trans-polyenes with many equivalent atoms and the Hu¨ckel model for the n electrons to display the consequences of the RXS theory. The application of this model to X-ray emission spectra is justified since n and p bands of the occupied states do not in general overlap, and in particular not so for polyenes. In accordance with the molecular orbital (MO) approximation the ground-state and core excited state n orbitals are expressed through the 2pK atomic orbitals (AOs) X t " c u , tL" cLJu , tL" cL u (325) H HK K H HK K J JK K K K K of the mth carbon atom (u "2pK). Contrary to the ground state MOs t , the n orbitals tLJ and K X H H tL of the core excited state depend on the carbon n in which the core hole is created because of the J

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relaxation effect. The inelastic and elastic cross sections can then be written as  p (u,u)" p (u,u) , ' J J  p (u,u)"2ruud (e;RK )(e;RK ) " f "U(u#u !u,c) , J M QN JH JH H

(326)

p (u,u)"2r(u/u)"(e ) e)oq#(e;RK ) ) (e;RK )d uf "U(u!u,c) , QN # M where RK is the unit vector along the molecular axis. For compactness of notation the atomic dipole matrix element d is extracted from the inelastic and elastic resonant scattering QN amplitudes cLcLJ JL HL , f " eiq  RL JH u!u #iC HL L cL  JL . f" eiq  RL u!u #iC JL J L

(327)

Since the non-resonant contributionJ(u#u )\ is small in the resonant region this term is JL neglected for f (327). For disordered samples it is necessary to average the RXS cross section over all molecular orientations. In the general case, the result of this averaging is quite unwieldy due to the factor exp(iq ) R ) [214] and is given here only for the soft X-ray region qR ;1 where the LL LL factor exp(iq ) R ) in the scattering amplitudes (327) can be replaced by unity. To derive the averaged L cross section (326) one needs to know the average values only of the following polarization factors: (e;RK )(e;RK )"  (3#(e ) e)), (e;RK ) ) (e;RK )"(e ) e) .  

(328)

16.2.1. Frozen-orbital approximation To obtain a deeper understanding of the core hole—electron interaction in RXS, it is releveant to first consider the frozen orbital approximation, where this interaction is neglected. The application of the theory to the special case of linear polyenes is then reviewed, which, as already noted, reflects most aspects of the general case. The main spectral feature of the RIXS spectra of linear even polyenes (with N"2M carbons) can be understood from the case of infinite polyenes, see Fig. 109. The ground electronic state is constructed by filling each of the Mth lowest levels with two electrons (spin up and spin down). This yields a filled Fermi sea with a half-filled band, Fig. 109. In a one-electron, or frozen orbital, approximation the RIXS cross section is exactly equal to zero due to the momentum conservation law (318) (Fig. 109). In this approximation the MO coefficient for the core-excited state "1s Pl2 is the same as the corresponding ground state MO coefficients: L cLJ"cM and cL "cM . Due to the orthonormality condition HK HK JK JK cM cM "d  , GG GL G L L

(329)

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Fig. 109. Qualitative illustration of the main spectral features of RIXS and REXS spectra in polyenes.

the inelastic and elastic resonant amplitudes differ qualitatively from each other





1 d # (eiq  RL!1)cM cM , f " JL HL JH u!u #iC JH HQ L





1 1# (eiq  RL!1)cM , f" f " JL JJ u!u #iC JQ L J J

(330)

16.2.2. Coherence length of molecule—radiation interaction It is seen that the inelastic scattering is allowed in the frozen orbital approximation only due to the last term at the right-hand side of the expression for f (330). The ratio of this term to the first JH one is of the order of j/¸, so only when the molecular length ¸ is comparable with the X-ray photon wavelength j the inelastic scattering becomes significant. One can say that only parts of a long polyene with length j interact coherently with the radiation. The wavelength j can from this point of view be considered as the coherence length of interaction with the electromagnetic field. Numerical simulations of the theory reviewed above [202] has been carried out with help of Coulson’s solution of the Hu¨ckel equations for even linear polyenes with one and the same C—C bond length a and N"2M carbons eM"e #w , w "2b cos(ip/(N#1)), cM "(2/(N#1))sin(ipn/(N#1)) . G $ G G GL

(331)

The energy e has the meaning of a Fermi energy for long polyenes (N<1). The occupied and $ occupied MOs are enumerated as j"1,2,M and l"M#1,2,N, respectively. The polarization directions of initial and final photons are assumed orthogonal. The strength of momentum exchange between X-ray photons and the molecule is defined by the interference parameter f"q ) RK a"2aua sin(0/2)cos ,

(332)

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where 0 is the angle between the photon momenta, and is the angle between q and the molecular axis. Taking into account the actual values for the core ionization potential (IK290 eV) and bond length (aK2.25 a.u.), the value fK0.175 is obtained for the interference parameter for 0"60° and

"0°, which does not exceed unity in the case of polyenes. The comparison of the RIXS profiles in Fig. 110 with the ordinary non-resonant X-ray emission spectra [317] of polyenes shows the qualitative difference of these spectral shapes. Considering the long chains (N<1), the dispersion law e "e #2b cos(ka) follows directly from Coulsons solution (331), were p/a4k4p/a and with I $ q directed along the polyene axis. The effective width *e"e !e of ordinary emission is the $ M difference between the Fermi energy e and the energy of the bottom of the occupied band $ e : *e"!2b"4.8 eV. As one can see from Fig. 109 the effective width *e "e $ !e of the $ M 0'61 I >O RIXS profile for soft X-rays is *e K!2bf. 0'61 Because of the small ratio of the photon momentum change q and the Fermi momentum k "n/a, the effective width of the RIXS band is smaller by one order of magnitude than the width $ of an ordinary emission profile q *e /*e"f& , *e K0.8 eV 0'61 0'61 k $

(333)

(here f"0.175 for 0"60° was assumed, see Ref. [202]). So it can be realized that due to the electron—photon momentum exchange (Fig. 109), the X-ray fluorescence appears in the narrow region *e K0.8 eV close to the Fermi level (see dotted line in the top 0'61 panel (M"50) of Fig. 110). The RIXS profile is broadened and shifted to the shortwave region with increasing width c of the spectral function (see thick solid line in the top panel (M"50) of Fig. 110). The momentum exchange between electrons and soft X-rays allows for indirect RIXS transitions (k"k#q) and the inelastic scattering channel is opened in the forbidden region (see Figs. 110 and 111). Due to the small value of the interference parameter fK0.3 (332) X-ray fluorescence in polyenes is allowed only from the occupied states belonging to a narrow region (333) close to the Fermi level. When the frequency of X-ray photons increases, all occupied bands become active in the RIXS spectrum. 16.2.3. Effect of the electron—hole interaction Because of its simplicity, the independent-electron model provides a very attractive framework for the description of RXS. However, interelectron interaction causes this model to break down in the general case. This breakdown is particularly conspicuous for p-electron systems like polymers since the direct RIXS transitions (318) are then forbidden. In the case of highly excited states the main correlation effect is given by the electron—hole interaction considered in this section. The momentum exchange between electrons and a “heavy” core hole allows indirect transitions and opens the RIXS channels in polyenes (like the case of exchange with photon momenta). The term “heavy” core hole refers to the large effective mass k of core electrons: k "(*e (k)/*k)\&k/D , A A $ A where D and e (k) is the (small) width and energy of the core band, respectively. The change *k of A A the electron momentum under such scattering is of the order of magnitude of the Fermi momentum k . So contrary to the case of momentum exchange between electrons and photons (see Eq. (333)) $ the effective width *e of the RIXS band has now the same order of magnitude as the width of 0'61

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Fig. 110. Elastic and inelastic RXS cross sections of polyenes in the frozen orbital approximation. The RIXS and REXS cross sections (326), (330) are depicted by thick and thin solid lines, respectively. The dashed line shows the squared REXS amplitude f #f . All these quantities are reduced by 10 times. C"0.1 eV, c"1 eV. I"290 eV. f"0.175. Incident and final X-ray photons have orthogonal polarization directions. The change of the photon momentum q is directed along the molecular axis ( "0°) and 0"60°. The incident photon frequency is tuned to resonance with the first unoccupied n level. X"1.48, 0.36, 0.075 eV for the polyene with M"2, 10 and 50, respectively. X"u!(I#e ), $ X"u!(I#e ). The dotted lines in panels M"2 and M"10 show the RIXS cross sections increased by 20 and $ 5 times. The dotted line in panel M"50 shows the RIXS cross section for narrowband excitation: c"0.05 eV [202]. Fig. 111. The effect of interference scattering on the spectral shape of RIXS and REXS profiles of M"20 polyene. The notations and input data are the same as for Fig. 110. The thin dotted line in panel (c) shows the RIXS cross section increased by 8 times. The panels (a)—(d) differ by scattering angles: (a): forward scattering with 0"0°, f"0; (b): 0"51.8°, f"0.153; (c): 0"33.10°, f"0.1, (d): 0"60°, f"0.175 [202].

the occupied band *e *e /*e&*k/k &1 . (334) 0'61 $ The results of the numerical simulation (Fig. 112) based on Eqs. (326), (327) and on the Greeen’s function technique outlined in Refs. [202,312,317,291,318] confirm this estimation. A few other physical processes influence the RXS spectral shape, for example, the interference of scattering channels through core hole states at different atoms and the electron—core-hole interaction. Results of calculations with and without interference are compared in Fig. 113. This figure shows the tendency for the interference contribution to decrease close to the top of the occupied band if the photon phase factor is neglected (f"0). Comparison of Figs. 110 and 113 demonstrates

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Fig. 112. The effect of relaxation on the spectral shape of RIXS and REXS cross sections and squared REXS amplitudes (326) and (327) of polyenes. All these quantities are reduced by 10 times. C"0.1 eV, c"1 eV. M"10. Detuning energies are given relative to the core ionization potential of the mid carbon. The incident photon frequency is tuned to absorption resonance with unoccupied MOs l"11,12,2,20 (for mid carbon). 0"60°, "0°. Incident and final X-ray photons have orthogonal polarizations directions. Each panel is shifted to the right by 1 eV. The RIXS and REXS cross sections are depicted as solid and dotted lines, respectively. The squared REXS amplitude is shown by the dashed line [202].

that the electron—core-hole interaction enhances the role of the photon momentum in the formation of the RIXS profile. One can see (Fig. 113) a strong influence of the soft X-ray momentum on the formation of the spectral shape of RIXS. Contrary to the frozen orbital approximation, the role of the photon momentum now becomes important not only near the top of the unoccupied band (near the Fermi level, see Figs. 110 and 111), but for the whole emission band since RIXS now is allowed from all occupied states (Fig. 112). The reason for this is the large electron momentum *k&k change during the scattering by the “heavy” core hole due to the electron—core-hole $ interaction. This makes the RIXS transition “indirect” (see Eq. (318)) since the momenta k and k above and below the Fermi level are different, k"k#*k, see Fig. 109b. The X-ray momenta influence significantly both the RIXS and REXS spectral profiles through the phase factor exp(ιq ) R ) in the interference term (see Fig. 113). LK 16.2.4. Examples of momentum conservation and symmetry selection in RXS by polymers A recent verification of the theory presented above was given by Guo et al., who recorded the resonant and non-resonant X-ray scattering spectra of some poly-phenylene-vinylene [292] and poly-pyridine [319] compounds. The experiments were performed at beamline 7.0 at the Advanced Light Source, Lawrence Berkeley National Laboratory. As shown in Fig. 114 the poly-phenylenevinylene compounds are made up of benzene (phenyl-) rings connected by short hydrocarbon bridges. The non-resonant as well as all resonant spectra for each polymer demonstrate benzenelike features, indicating a local character of the X-ray emission in which the phenyl ring acts as a building block. Like for free benzene the outer n band in the polymer spectra shows a depletion of

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Fig. 113. The effect of interference on the spectral shape of RIXS and REXS cross sections and squared REXS amplitudes for polyenes. The notations and input data are the same as for Fig. 112. The results of relaxed calculations are shown. The thick solid and dashed lines show the RIXS cross section with and without interference effects, respectively. The REXS cross sections (accounting for interference effects) are shown by thin solid lines. The incident photon frequency is tuned to exact absorption resonance with (a) highest unoccupied n level l"20 (for mid-carbon). 0"0°, f"0, "0°; with (b) lowest unoccupied n level l"11 (for mid-carbon). 0"0°, f"0, "0°; with (c) highest unoccupied n level l"20 (for mid-carbon). 0"60°, f"0.175, "0°; with (d) lowest unoccupied n level l"11 (for mid-carbon). 0"60°, f"0.175, "0° [202].

emission on going from the non-resonant to the LUMO-resonant spectra. This transition, which is strictly symmetry forbidden for free benzene, becomes effectively forbidden in the polymer case as a result of strong interference effects. The resonant X-ray inelastic scattering spectra were recorded by tuning the incident X-ray photon beam to the first nH resonance (284.8 eV). The simulated spectra shown in Fig. 115 agree well with the experimental ones, Fig. 114. Similar to the non-resonant case, the resonant spectra of poly-phenylene-vinylene compounds demonstrate a strong similarity with the resonant spectrum of benzene. For benzene the outermost A and B bands of the non-resonant spectrum become forbidden in the resonant case owing to the partity selection rule. In particular, the HOMO 1e E level, corresponding to emission energy around 280 eV will be symmetry forbidden, which is the reason that the emission A band of highest energy in the non-resonant spectrum disappears in the resonant spectrum. The poly-phenyl-vinylene compounds, indeed show benzene-like features also in the resonant case, with the A band intensity considerably reduced. A two-step formula (without interference) shows a non-resonant type features with strong intensity for A band and a comparison with simulations using the one-step formula indicates that the restoration of the effective symmetry selection (as well as the benzene similarity) must be ascribed to channel interference; the X-ray scattering of the different close-lying core-hole states interfere in such a way that the total signal is depleted.

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Fig. 114. Experimental non-resonant and resonant X-ray scattering spectra of benzene and the PPV, PMPV and PDPV poly-phenylene-vinylene compounds excited at 310.0 and 284.8 eV photon energy, respectively. Schematic diagrams of the compounds are given in the left column [292].

As pointed out previously in this section the symmetry selection observed in molecules is analogous to the momentum conservation rule for incoming and scattered photons in the solid state. The latter approximation holds in the frozen orbital approximation and rests on the interference effect; when the interference of the RIXS channels through core-hole states on different atoms is neglected then the momentum conservation breaks down. This leads to a strong dependence of the RIXS spectral shape on the excitation energy. The momentum conservation rule implies a shift of the threshold for RIXS in n electron systems, and predicts zero RIXS intensity for occupied n bands in n electron systems, something which is nicely confirmed in by the spectra shown in Figs. 114 and 115. As noted, the electron—photon, electron—phonon and electron—corehole interactions violate the momentum conservation law and allow emission in the forbidden n region (Fig. 116). Although the poly-vinylene compounds are not pure n electron systems, one can still regard the amount of intensity in the “forbidden” A band region as an indication of the strength of these interactions. The effective strength of the electron—phonon interaction can be significantly reduced even by small detuning of the incident X-ray photon frequency below the absorption band, see Section 16.4 below. 16.3. Resonant and excitonic RIXS bands 16.3.1. Role of excitons Figs. 112 and 117 display some general features of X-ray photoabsorption to excitonic states and to continuum or quasicontinuum states. Panels with l513 in Figs. 112 and 117 show the

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Fig. 115. Calculated non-resonant and resonant X-ray scattering spectra of benzene and the PPV, PMPV and PDPV poly-phenylene-vinylene compounds [292].

high-energy sidebands above the top of the occupied band (l being the index of unoccupied levels). Fig. 117 demonstrates that the sidebands appear for excitation to all unoccupied states. The physical reason for these high-energy tails can be understood by considering the partial RIXS cross section p (u,u) (326). The density of the unoccupied states is large, or “quasicontinuous”, in the J case of solids and long polymers, and the resonant condition u"u (335) JL is therefore fulfilled for every excitation energy tuned to the unoccupied band. The core excited electron populates the resonant unoccupied level l(res) (335) corresponding to this excitation energy (Fig. 118). The partial contribution corresponding to this resonant level l(res) (335)  p (u,u)& " f "U(u!u ,c) J JH HL H

(336)

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gives a dominant contribution to the RIXS cross section (326) if the oscillatory strengths are homogeneously distributed in the X-ray photoabsorption spectrum. Because of the strict fulfillment of the resonant condition (335) for extended systems with high density of states, the resonant condition for emission u"u

(337) HL does not depend on the excitation energy, and the emission band is placed below the photoabsorption threshold. As is well known, the photoabsorption cross section of extended systems like solids and polyenes is not a homogeneous function of the frequency. The strong enhancement of photoabsorption takes places near the X-ray photoabsorption threshold since the excitonic state is close to the Fermi level. The excitonic feature is given by the sharp dependence of the scattering amplitude on the number l of the unoccupied MOs through the MO coefficient cL. The magnitude JL of this coefficient is maximal for l"l(exc) near the excitonic resonance (in the present case in the bottom of unoccupied states l(exc)KM#1). If the excitation energy u is tuned away from the excitonic resonant frequency u , the corresponding resonant contribution p (u,u) (336) is JL J suppressed as follows from the small magnitude of the MO coefficient cL . Due to this JL

Fig. 116. The effect of interference scattering on the spectral shape of RIXS and REXS cross sections and squared REXS amplitudes for polyenes. The notations and input data are the same as for Fig. 113. The thick and thin solid lines show the RIXS and REXS cross sections, respectively. The squared REXS amplitudes are given by the dashed lines. The incident X-ray photon frequency is tuned to resonance with the first unoccupied n level (for mid-carbon). "0°; (a) 0"0°, f"0; (b) Forward scattering: 0"60°, f"0.175; (c) Back scattering: 0"180°, f"0.35. Both REXS cross sections and squared REXS amplitudes in panels (a)—(c) are multiplied by factors 0.25, 0.5 and 5, respectively [202].

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Fig. 117. RIXS (thick solid lines) and REXS (dotted lines) cross sections for M"10 polyene. Input data are the same as for Fig. 112. The structure below X"!0.5 eV is caused by the resonant contribution (336), while the excitonic term (338) is responsible for the sideband above X"!0.5 eV. The REXS cross sections are decreased by 10 times for l"11 and by 10 times for the higher unoccupied states [202].

suppression of the resonant term and because this excitonic term is enhanced due to a large value of cL the partial excitonic contribution JL  p (u,u)& " f "U(u!u#u ,c) (338) J JH JH H can be comparable with the resonant one (336). It is necessary to note that the excitonic contribution is also suppressed due to the large denominator u!u !ιCKeL !eL !ιC HL J J in the scattering amplitude f (327). Eq. (338) shows directly the qualitative difference between the JH resonant and the excitonic contributions. Contrary to the energy independent dispersion relation (337) for the resonant term, the RIXS scattering obeys the Raman—Stokes dispersion law u"u!u JH

(339)

for the non-resonant or tail population of the excitonic state (see Fig. 118). One can thus distinguish two dominant contributions from the sum over unoccupied states in the expression for the total RIXS cross section (326), namely, the resonant and the excitonic contributions p (u,u)Kp (u,u)#p (u,u) . ' J J

(340)

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If the excitation energy is increased above the absorption threshold it can be observed that, contrary to the resonant contribution “staying” at the same energy (337), the “excitonic” band (338) moves into the shortwave direction according to the Raman—Stokes law (339). The partial contributions p (u,u) depicted in Fig. 119 show directly the possibility for selecting resonant J contributions (l"20,19,18) and excitonic contributions (l"11,12) from the RIXS spectral profile. Since the former levels are close lying they must be considered as resonant. Figs. 112, 117 and 119 demonstrate the weak intensity of the excitonic contribution. Calculations show that the intensity of this excitonic sideband increases if the corresponding oscillatory strength of the excitonic photoabsorption transitions increases. In the general case, the intensities of resonant and excitonic contributions can be comparable. Let us note that the ratio of resonant and excitonic contributions is sensitive to the ratio between the lifetime broadening C and the effective width c of the unoccupied band, 2"b" [202]. One can show that the strength of the excitonic contribution increases when the lifetime broadening is increased. The role of excitons for interpreting the RIXS spectra of graphite has been analyzed on several occasions, see for example work by Bru¨wiler et al. [320], by Carlisle et al. [200], by van Veenendaal and Carra [321], and by Shirley [322]. Use of relaxed, excitonic, versus frozen orbital descriptions of RIXS has also been theoretically analyzed for surface adsorbates [293], see Section 17.

Fig. 118. Qualitative illustration of the origin of the resonant (left panel) and excitonic (right panel) contributions to RIXS cross sections of polyenes. The photoabsorption profile is depicted by the shaded area [202]. Fig. 119. Partial contributions (thick solid lines) (326) to the total RIXS cross section (dotted line) for core excitation of the highest unoccupied n level with l"20 (see top plot in Fig. 112). Input parameters are the same as for Fig. 112. M"10 polyene [202].

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16.3.2. Dispersion relations To show the qualitative difference between dispersion relations (337) and (339), the energy position of the highest weak excitonic- and the highest intensive resonant peaks (edges) of the RIXS cross section are depicted in Fig. 120 as functions of the detuning X. It is seen that the dispersion relation follows a Raman—Stokes law below the photoabsorption threshold. Above this threshold the dispersion splits in two qualitatively different relations. The filled circles show the constantenergy dispersion (337) caused by the above discussed resonant contribution. Only the high-energy edge of the excitonic sideband (338) follows a linear dispersion. This dispersion is shifted relative to the Raman—Stokes linear dispersion below the photoabsorption threshold by the effective width of the excitonic photoabsorption resonance, see Fig. 120. Calculations show that the dispersion relation retains a Raman—Stokes law for core excitation above the highest unoccupied n level. This is a consequence of the n-orbital approximation with a finite width of the unoccupied states. So one can expect this behaviour of the dispersion relation above an “effective” width of the photoabsorption band. It is difficult though to expect this effect in the case of polyenes due to the presence of p states and continuum states above the photoionization threshold. 16.3.3. Dispersion of resonant photoemission in solids The manifestation of the excitonic features was reviewed in this section only in terms of radiative RXS, but it is not difficult to understand that the analysis of the excitonic problem presented here is relevant for resonant photoemission as well. Indeed, the idea (and its realization) that the spectral shape of the RXS profile consists of two qualitatively different bands caused by resonant (336) and excitonic (338) contributions is general since no characteristic features of the RXS transition matrix elements were used in the analysis above. However, it is necessary to note one important difference

Fig. 120. Dispersion relations. Peak position of the highest RIXS resonance (edges of resonant and excitonic bands) versus detuning X of excitation energy, C"0.1 eV, c"0.3 eV. Relaxed calculation. M"10 polyene. 0"60°, "0° [202].

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between radiative and non-radiative RXS in this respect. In the case of resonant X-ray photoemission the role of the final photon with frequency u is played by the photoelectron with the high kinetic energy e ("e ). Due to this fact the sum over l in Eq. (326) must be omitted for ordinary J resonant photoemission, something which leads to the Raman—Stokes dispersion law. The picture is changed completely if the final state contains two photoelectrons excited above the Fermi level (usually with one electron of high energy and the second close to the Fermi level). Such type of final states leads to “satellite” bands [38]. One can see that in this case the resonant photoemission is identical to the case considered above of radiative RXS since one must sum over the quasicontinuum states l of the second low energy electron. 16.3.4. Dispersion of resonant photoemission at the 2p edges of metallic nickel The existence of a two-band structure with two qualitatively different dispersion relations has been observed in the recent resonant photoemission experiment on metallic nickel for photon energies around the ¸ threshold at 852.3 eV [38]. The experiments were performed using a Ni(100)  crystal at beam line 8.0 at ALS. This undulator beam line is equipped with a modified “Dragon” monochromator. The end station was built at Uppsala University and comprises a rotatable Scienta SES200 electron spectrometer [323]. The ground state of Ni is [3d4s] where the square brackets indicate that the valence electrons are in the metallic state. Starting with the spectra below the 2p threshold one can see two types of final states, the band like [3d4s]\ states within 2.3 eV  of the Fermi level, which dominate the spectra, and split-off atomic-like 3d satellite states at 6 eV, Fig. 121. At the 2p threshold the additional possibility exists to make excitations to 2p[3d4s]  states. These may decay by electron emission in Auger-like processes (autoionization). The Auger process in Ni leads to localized as well as delocalized final states, i.e., the same type of states as in direct photoemission. Below the 2p threshold, the whole spectrum remains at constant binding energy. Immediately  above threshold, the 6 eV feature starts to move as a function of photon energy. In the valenceband region a weak shoulder develops from the Fermi-level cutoff and disperses with photon energy. The kinetic energy of the atomiclike 3d[4s] feature is plotted in Fig. 122 versus photon energy. Below the ¸ resonance maximum it tracks the photon energy (resonant Raman behavior).  At the ¸ threshold, it transforms rapidly into a constant-kinetic-energy feature. At the ¸ edge the   same type of behaviour is seen [38]. Comparison of the experimental results (Fig. 122 with Fig. 120) shows that the theory outlined in previous sections explains the main features of the dispersion relations measured for resonant photoemission of metallic nickel [38]. 16.4. Role of electron—phonon interaction As reviewed in the foregoing, the momentum conservation plays an important role in the formation of the RXS spectral shape of solids and polymers, and results in a strong dependence of the RXS profile on the excitation energy close to the photoabsorption edge. However, the conservation of the electronic momentum is restricted to certain preconditions and might break down due to processes involving photon—, electron—hole— and and phonon interactions, the former effects treated earlier in this section. The importance of the latter effect, the electron—phonon interaction, can be anticipated by virtue of the recent — and here reviewed — discoveries concerning effects in RXS of free molecules that are induced by coupling to the nuclear vibrational motion, to

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Fig. 121. Valence electron spectra around the ¸ core-level threshold of metallic nickel (on a binding energy scale) [38].  The photon-energy increment is about 0.2 eV except for the lower set data (844.2, 848.0, 849.0, 849.8, 850.3, 850.5, 850.7 eV). The polarization e vector of the incident X-rays is in the surface plane (dashed line) or perpendicular to it (solid line). Fig. 122. The energy of the 3d[4s] spectral feature of metallic nickel as a function of photon energy [38]. Dots and crosses denote photoemission when the e vector of the incident X-rays is in the surface plane or perpendicular to it, respectively.

mention: the influence of vibronic coupling on electronic selection rules, the restoration of the selection rules by the detuning of the excitation energy, the collapse of vibrational structure, and the “control” of dissociation through detuning the frequency. These effects could be understood through the general concept of RXS duration and the time selection of the vibronic coupling responsible for the formation of the RXS profile by frequency detuning. Just as vibronic coupling for free molecules, one can thus anticipate electron—phonon coupling to be of primary importance for understanding spectral shapes of RXS of polymers and solids. The breaking and restoration of the symmetry selection rules in gas-phase systems can then be understood as breaking and restoration of the momentum conservation rule in the context of solids. Theory of electron—phonon interaction for X-ray Raman spectra of polymers and solids have recently been evaluated in two works; by Minami and Nasu [324] and by Privalov et al. [325]. From a consideration of these theories one can understand the presence of electron—phonon interaction in experimental recordings of polymers, like the aforementioned poly-phenylenevinylene [292] and poly-pyridine [319] compounds, and of crystalline solids, like cubic [326,327] and hexagonal [328] boron nitride, TiO [329,330] and CaSi [331].   In this section we give a survey of the role of electron—phonon interaction for X-ray Raman spectra of linear polyenes and solids, largely following Ref. [325]. We focus on the effect of detuning

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the frequency, on the quenching of the phonon broadening of the spectral lines, and on the selection rules and conservation of electron momentum (318). For the purpose of simplicity the above investigated electron—hole interaction is then neglected. The electron—phonon interaction is shown to open the scattering channels for all occupied states in p systems. The frequency dependence of these effects are analyzed, showing that when the duration of the scattering is shortened by a large detuning of the excitation frequency the role of electron—phonon coupling of both core excited and final states is suppressed, depleting the cross section for n systems up to zero. The detuning quenches the symmetry breaking of the core-excited electronic states and results in a restoration of the conservation of electron momentum. Specific selection rules for the zerophonon line in X-ray Raman spectra of linear polyenes are described, as is the narrowing, or collapse, of the electron-vibrational bands. When the detuning is large the spectral profile is described by a joint density of states, the singularities of which follow the Raman—Stokes dispersion law, something that allows a mapping of the band structure. The phonon broadening of these singularities is completely quenched by detuning. Non-adiabatic effects appear due to electron—phonon interaction in the intermediate core excited state and in the final state, but we focus in this subsection on the first and the major of these two effects, namely the vibronic coupling of the degenerate core-excited states. 16.4.1. Amplitudes and cross sections According to Section 10.3 vibronic coupling (VC) results in the localization of core holes and hence the violation of the symmetry of the wave function of core-excited states "i2"W (R ,r)"m,n2 , (341) L M which is a simple product of the electronic and nuclear wave functions. W (R ,r) denotes the L M electronic wave function of a core-excited state 1s Pt , with core hole in the nth atom and with L J equilibrium geometry R . The nuclear wave function "m,n2 depends on the site of the atom n due to M VC through the shift D of the position of the minimum of the core excited potential of the ath ?L vibrational mode with respect to the ground state one. This shift (caused by VC) for linear even polyenes with N"2M carbons and the bond length a reads [325]

 



2 pa pa 1 2b j sin n! sin , g" , (342) D " ?L , j "!2b ?L M uM(kuM ?L ku N N N 2 ? where b"j*º(R)/jR,"R a, *º("R ") is the non-adiabatic interaction of the core-excited atom LYL  n with adjacent atoms n"n$1, and R, is the component of R which is parallel to the molecule axis n. We denote in this section the mass of the atom as k. The strength of VC or electron—phonon interaction is defined by the dimensionless parameter g (342). M The dipole approximation for the intraatomic transitions yields for the RIXS and REXS amplitudes , 1mD, f "m,n21m,n"mM,o2 , c c eiq  RL F' "uud aJH JL HL D QN n X!w #e(mD)!e(m)#iC m H L 1mD,o"m,n21m,n"mM,o2  , iq R . F#"(eH ) e)oqdmf mo#2uud aJJ n c e  ?  QN JL m X!w #e(mM)!e(m)#iC J J L

(343)

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291

Here "mM,o2, "mD, f 2, and "m,n2 are the vibrational wave functions of the ground state, final, and core excited states, respectively; m"(m ,m ,2,N!1) is the vector formed by the vibronic   quantum numbers m of the ath normal mode; e(m)" ,\u (m #) is the vibrational energy; ? ? ? ?  is the intraatomic dipole transition moment between the 1s u "uMsin(P a/2). uM"(i/k; d QN ? ? and 2p AOs, aJH )(e ) dK ), q ) R "fn, f"2pa sin(0/2)cos , is the angle between q and n "(eH ) dK HQ JQ L molecular axis n. The notations X"u!(I#e ) and X"u!(I#e ) for the detuning frequen$ $ cies of the initial and final photons are used with I as the core ionization potential. c "cM and GL GL w are given by Eq. (331). G The multi-mode Franck—Condon (FC) factors ,\ ,\ 1mD, f "m,n2" “ 1mD, f "m ,n2, 1m, n "mM, o2" “ 1m , n"mM, o2 ? ? ? L ? ? are the products of the FC factors for each normal mode [325]. The total RXS cross section is the sum of inelastic and elastic contributions

(344)

u   . (mo)"F' "U(X!X#u ,c), p'(u,u)"2r D DM Mu o J H mD m p#(u,u)"r . (mo)"F#"U(u#e(mD)!e(mM)!u,c) .  mD mo

(345)

Here u "w !w #e(mD)!e(mM) and . (mM)"exp(!e(mM)/k ¹)/ m exp(!e(m)/k ¹) is the DM J H normalized distribution over ground state vibrational levels. As it was pointed in Section 16.2.1 the RIXS cross section for p systems is strictly equal to zero if the momentum exchange of valence electrons with other particles is neglected (318). The large value of the phonon momentum P "na/aN:k (346) ? $ opens the RIXS channel from all occupied MOs (see Eq. (334) and Fig. 109) since now *k&P &k . This makes the momentum exchange of electrons more efficient with phonons than ? $ with the soft X-ray photons (333), see Fig. 123. 16.4.2. Weak coupling. One-phonon approximation As the VC parameter g (342) decreases, the many-phonon transitions become weak. One can M show [325] that F' J1!(!1)J>H>? in the one-phonon approximation. This results in the D selection rules F' "f [1!(!1)J>H>?] . D M

(347)

16.5. Zero-phonon line in RIXS The RXS band is formed as a sum of the subbands corresponding to the electron—vibrational transitions from different occupied MOs t . Every subband can have a so-called zero phonon (ZP) H line corresponding to the decay transitions without change of vibrational quantum numbers (Fig. 124): X"w , m"mD . H

(348)

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In the framework of the reviewed model — the harmonic approximation with the same vibrational frequencies for ground, core excited, and final states — the ZP line is a single resonance. It is then assumed that the adiabatic potentials of the core excited and final states are not shifted relative to the ground state potential surface. This single line is of course broadened or split in a “real” system due to the anharmonicity and the different vibrational frequencies for different electronic states. 16.5.1. Broadband excitation The positions of the RIXS resonances do not depend on the excitation frequency, u, when the width of the spectral function is larger or comparable with the spacings between electronic levels

Fig. 123. The RIXS cross section of a N"6 polyene versus the momentum exchange of valence electrons with X-ray phonons and photons. (a) accounting only for momentum exchange between electrons and photons: g "0, f"0.175; (b) M accounting only for momentum exchange between electron and phonons: g "0.9 and f"0; (c) accounting for M momentum exchange of electron with both photons and phonons: g "0.9 and f"0.175. The cross section is decreased M by 10 times. C"0.09 eV, c"0.05 eV, I"290 eV, u "0.5 eV. The value of f"0.175 in (a) and (c) corresponds to M 0"60° and "0. The abbreviations HUMO and LUMO#1 mean the core excitation (X"w ) to the highest J unoccupied MO (l"6) and to the MO with l"5, respectively. The arrows in panel (a) mark the positions of the zero-phonon lines for the decay transitions from occupied MO with j"1,2,3. The dashed lines in the panels (b) and (c) show the excitonic bands.

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(c9b) and u is tuned to the photoabsorption band. As one can see from Fig. 124 the ZP line has blue and red vibronic sidebands. The effective width of these phonon sidebands can be estimated with help of the reflection method a "1/(ku (349) C "(2u D /a . ? ?L ? ? ? $! Apparently, it is difficult to select the ZP line from the quasicontinuum vibrational spectra of long polyenes or solids. 16.5.2. Narrowband excitation In the case of RXS being initiated by a narrowband X-ray beam, c;u , the RXS spectral profile ? is formed according to the energy conservation, or Raman—Stokes law, for the whole RXS process X"X!w #e(mM)#w !e(mD) . (350) J H The lowest vibrational levels (mM"0) of the ground state are the only populated ones if the temperature is low. According to the energy conservation law (350) (which “couples” only ground and final states) the ZP line corresponds to the emission m"0PmD"0

(351)

and possesses a long wave phonon sideband (the Stokes sideband, see Figs. 123 and 124). The effective width of this broad sideband is given by Eq. (349) since the final vibrational states are populated through the core-excited state in accordance with the corresponding FC factors. Contrary to the ZP line, the phonon sideband is formed by decay transitions that change the

Fig. 124. Qualitative picture of the formation of the zero-phonon line in the RIXS spectrum for one vibrational mode. Left panel — broadband excitation: The decay transitions without change of vibrational quantum numbers mPmD (348) form the ZP line. Right panel — narrowband excitation: The ZP line 0P0 (351) is formed according to the energy conservation law (350).

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vibrational quantum numbers, mPmD$Dm. It is interesting to note that the position of the ZP line (350) does not depend on l X"w #e(0)!e(mD)4w if X"w (352) H H J if the incident photon frequency is tuned to resonance with the lth unoccupied MO. The ZP line for higher temperatures ¹ has also a blue phonon sideband (the anti-Stokes sideband), since other vibrational levels of the ground state (mMO0) are populated, see Eq. (350). Apparently, the width of the anti-Stokes sideband is proportional to the temperature for small ¹. Comparison of Eqs. (348) and (352) allows to conclude that mainly m"0PmD"0 vibrational transitions contribute to the ZP line for narrowband excitation. 16.5.3. Selection rules for the zero-phonon line It is not hard to see that the RIXS amplitude for the ZP line is proportional to 1#(!1)H>J [325]. This results immediately in the following selection rules for the ZP line: F "0 if j#l"odd . (353) M These selection rules for occupied, t , and unoccupied, t , MOs are valid only for narrowband H J excitation. As it was pointed out above, the 0P0 vibrational decay transition gives then the main contribution to the ZP line. This is shown by Figs. 123b and c, which also confirm the selection rules (353). The contribution to the ZP line from transitions with m"mDO0 is not suppressed when the width of the incident radiation is comparable or larger than the smallest vibrational frequency u . Hence, the selection rules (353) are not valid for broadband excitation.  Let us return to the narrowband excitation and the example given by Figs. 123a—c. In this case only the ZP lines contribute to the spectrum if VC is absent (g "0), see Fig. 123a. When the M incident radiation is tuned in resonance with the even unoccupied MO (LUMO: l"4 or HUMO: l"6) the ZP line is seen only for j"2 in accordance with the selection rules (353). And vice versa, the ZP line is forbidden for j"2 and allowed for j"1,3 under core excitation to the odd unoccupied MO, l"5, (upper panel, Fig. 123b). All three ZP lines ( j"1,2,3) are allowed when the photon momentum is taken into account, see Fig. 123a (without VC) and Fig. 123c (with VC). 16.6. RIXS in the dipole approximation over molecular size To complete our discussion of selection rules, the present subsection offers a brief outline of RIXS in the dipole approximation versus the molecular size fN&2p¸/j;1 ,

(354)

where the molecular length ¸"aN is assumed to be smaller than wave length j of the X-ray photon. This approximation is valid also for long molecules when the scattering angle 0 is small. The dipole approximation breaks down for polyenes with N910 if the scattering angle is fairly large. One should note the qualitative distinction of this approximation from the dipole approximation for intraatomic transitions. Eq. (354) allows to expand the photon factor e\ιDL in the series over the small parameter fn. The zero-order term does not contribute to the amplitude because of the orthonormality of the MO coefficients (329), while the non-zero contribution to the RIXS

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amplitude appears in the first order over fn as , f c c n. (355) JL HL X!w #iC H L In fact, this term is the quadrupole contribution since the dipole approximation for intraatomic transitions was taken into account already for the atomic transition moment d . The implemenQN tation of the inversion symmetry and Eq. (331) results in the desired selection rules F' "0 if l#j"even . (356) D Fig. 125 shows clearly these selection rules (fNK0.1). The violation of the selection rules (due to fNK1) is seen in Fig. 123a. 16.7. RXS for detuned incident radiation 16.7.1. Restoration of the selection rules and collapse of vibrational structure in polymers RXS spectra (Fig. 123) are generally broadened by the vibrational structure of the intermediate and final states. The vibrational band broadening is often comparable with the spacings between electronic levels. We have shown in Section 8.3 that vibrational broadening as well as VC of core excited states can be suppressed by shortening of the RXS duration q , see also Section 4. Both A effects were illustrated in these sections for small molecules; CO and CO . The same picture is  obtained for polyenes (Figs. 126—128). We recall that the vibrational structure in elastic RXS collapses totally to a single resonance with the width defined by c [105,104,325]. In the inelastic, RIXS, case a large detuning eliminates the vibrational structure due to the core excited state, but not the vibrational structure caused by the final state. The total vibrational collapse in RIXS takes place only when the potential surfaces of ground and final states are the same [105,104,325] (Figs. 126b and c). It is not hard to see that a large detuning quenches VC and, hence, that the symmetry breaking thereby restores the selection rules also for RIXS of polymers. Indeed, according to the orthogonality condition (329) F' "0 if the photon momenta are neglected. Fig. 126c shows the D coincidence of the RIXS spectral profile (343) with the one obtained from calculations without electron—phonon interaction (except the evident Raman—Stokes shift). Due to the small photon momentum, only the highest occupied MO (HOMO), close to the Fermi level, participates in the RIXS process, see Fig. 126. As mentioned above, all occupied MOs become active in RIXS for large photon momenta, see Fig. 127 which also clearly illustrates the purification of the RIXS spectrum due to the collapse effect for each occupied MO. The collapse effect takes place here since the shifts between all adiabatic potentials are neglected. In this case the spectral profile of a certain electronic transition collapses to single resonance. The spectral shape of the RXS resonance copies the spectral function of the incident radiation with a width c for narrowband excitation. The RIXS spectral profile is broadened by the vibrational structure of the final state if the potential surfaces of the ground and final states are different (Fig. 128). 16.7.2. Restoration of selection rules and collapse effect in solids It was shown above that the electron—phonon interaction is strong with an accompanying breakdown of the momentum conservation for the electrons (318) when the excitation energy is

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Fig. 125. The RIXS cross section of an N"6 polyene in the dipole approximation over molecular size (355), f"0.0175 (fN"0.105). The incident radiation is tuned in resonance with unoccupied MO, l"5, in panel (a), and with MO l"6 in panel (b). The approximate selection rules (356) suppress the resonances in occupied band when l#j"even.

tuned to the photoabsorption band. For solids with continuum electronic spectra referring to a conduction band, the notion of a detuning is not well defined. Nevertheless, the fast RXS (61) can be reached by tuning the excitation energy below the photoabsorption threshold. For solids the translational invariance of the Bloch function for electrons of the ith band: t k(r)"eιk  ru k(r), G G u k(r#R )"u k(r), appears as a new symmetry element. The asymptotic expressions for the RIXS ? G G and REXS amplitudes in this case read F' J[1/(X#ÓC)]1mD, f "mM,o2 DHk D kd(k!k!q!G) , HY J D G F#Jdm f mo fq(X)d(q!G) .  u

(357)

The latter equation results in a collapse of the phonon structure to a single resonance for the elastic band [105,104,325]. p#(u,u)JU(u!u,c) .

(358)

It is relevant to consider RIXS in the soft X-ray region where one can neglect the change of the photon momentum q (see, however, previous sections). According to Eq. (358) the inelastic scattering takes place with conservation of the electron momentum, k"k. The RIXS cross section

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Fig. 126. Collapse of vibrational structure in RIXS of an N"6 polyene due to the detuning of incident X-ray radiation from the absorption band. The potential surfaces of initial and final states are the same. The detuning of incident radiation from the LUMO (l"4) is defined as XM "X!w , M"3. (a) The resonant excitation of the LUMO +> (X"w ) leads to vibrational broadening of the RIXS profile (the cross section is decreased by 10 times). Panels (b) +> and (c) show the collapse effect corresponding to a detuning of incident radiation below the LUMO with XM "!5 eV and XM "!8 eV, respectively. The dashed line in panel (c) shows the RIXS cross section for resonant excitation of the LUMO and without electron—phonon interaction (g "0, XM "0). The value of the VC parameter is the same, g "0.9, M M except for the RIXS profile shown by a dashed line with g "0. All other parameters are the same as for Fig. 123. M Fig. 127. Same as for Fig. 126, but for an artificially increased X-ray photon frequency, f"2.625. Here XM "X!w is , the detuning from the HUMO. (a) Exact resonance with HUMO (l"6); the cross section is decreased by 10 times. (c) incident radiation is tuned above the HUMO, XM "8 eV. The comparison of (a) and (b) shows the “purification” of spectrum (b) due to the collapse effect.

then becomes



p'(u,u)J dk. (mM)"DHk D k1mD, f "mM,o2"U(X!X#u ,c), HY J DM f o JH m m u "u (k)#e(mD)!e(mM), u (k)"e (k)!e (k) . (359) DM JH JH J H Of particular interest here is the conservation of the electron momentum as a direct manifestation of the restoration of the translational invariance of the electronic subsystem in the core excited state. This is obviously the same effect as the symmetry purification effect.

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Fig. 128. Incomplete narrowing of the RIXS profile due to the different potential surfaces of the ground and final states. N"6 polyene, f"0.175. The dimensionless shift, gD"DD/(2a "0.3, of the final and ground state potentials is the L L L same for all vibrational modes in this calculation. All other parameters are the same as for Fig. 123. XM "X!w is the +> detuning of incident radiation from the LUMO. (a) XM "0. (b) XM "!0.5 eV. (c) XM "!2.0 eV. (d) XM "!5.0 eV.

16.7.3. The joint density of states One can see from Eq. (359) two broadening mechanisms for the RIXS profile. The first one is given by the different dispersion laws for the valence, e (k), and conduction, e (k), bands. H J The second broadening mechanism is related to the FC factors caused by the many phonon transitions (mDOmM). In this context it is relevant to examine a solid with the same potential surfaces of the ground and final states. The FC factor 1mD, f "mM,o2 then reduces to dmf mo, and one  obtains



p'(u,u)J dk"DHk D k"U(X!X#u (k),c) (360) HY J JH JH making use of Tr . "1. Inspection of this equation shows that the spectral profile is free from any phonon broadening. It can be converted to an integration over a surface of constant energy dS X!X"u (k) (361) JH in the case of narrowband excitation, U(X,c)Kd(X). As an approximation, one may suppose that ¼ (X!X),"DHk D k" does not vary strongly with the angle on a surface of constant energy, so JH HY J

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that it may be extracted outside the integral. This allows to connect the RIXS profile



dS (362) p'(u,u)J ¼ (X!X)o (X!X), o (X!X)" JH JH JH " ku (k)" JH JH with the joint (or combined) density of states o (E) for bands l and j. Such a “joint density of states” JH appears, for instance, in the theory of optical photoabsorption [332]. The essential physical result here is that the singularities, ku (k)"0, of the joint density of states follow the Raman—Stokes JH dispersion law in the sense of Eq. (361), something that allows to map the band structure. One can here recall that the RIXS spectral profile is described in the one-electron approximation by the interband density of states (322) [195,202], which, according to Eq. (362) reduces to the joint density of states if the detuning is large. Eqs. (358) and (362) show that the RXS profile is free from phonon broadening when the detuning or when the lifetime broadening C is large. This means a narrowing of the RXS spectral features caused by the singularities of the joint density of states for large X. As a special case when the bands e (k) and e (k) are parallel, the RIXS profile collapses to the single resonance; J H JU(X!X#u (0),c), with the width c [325]. JH 17. X-ray Raman scattering by surface adsorbates Most X-ray spectroscopy studies of surface adsorbates have been carried out in the absorption mode. As for other aggregates, however, the availability of 2nd and 3rd synchrotron radiation sources with high brightness has initiated a concomitant development of the more arduous X-ray emission spectroscopy, producing good-quality spectra also in this mode [333—335]. This has made it possible to exploit the many complementary features of the emission and absorption spectroscopies also for surface adsorbates and to study features like adsorbate orientation, bonding and internal reconstruction in fundamentally new ways. For instance, unlike valence band photoemission for which one obtains signals that represent the delocalized states, it has been verified that the X-ray emission signals indeed reflect the part of the electronic structure that is localized to the adsorbate [334]. The fact that XES probes the electronic structure of the occupied, rather than of the unoccupied levels as in NEXAFS, has been utilized not only to obtain information about bonding and for proposing new bonding models [336], but also for studying the geometric corruption, and the concomitant loss of bond order, of the molecule when adsorbed on the surface [337—339]. As for NEXAFS, X-ray emission spectroscopy has special symmetry, orientational and polarization selective properties when applied to surfaces. For resonantly excited spectra these properties are strongly enhanced, and qualitatively different, owing to the fact that channel interference effects then are of primary importance. In the resonant case the spectral properties are therefore closely related to the excitation frequency, to the symmetry and precise organization of the intermediate core-excited states prepared by the resonant excitation. The tuning of the incoming X-ray radiation to the discrete resonances of the adsorbate target may thus lead to scattering spectra that show very specific polarization and symmetry dependences which can provide evidence of symmetry assignments of the participating electronic levels and of orientation of the adsorbate with respect to the surface.

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The number of X-ray emission studies of surface adsorbates are still relatively few in numbers, in particular those which explicitly utilize the resonant nature of the core excitation in the analysis (in many cases resonant excitation has been used only as a means to enhance the total cross section and to remove satellite contributions). As for gases and crystalline solids one can expect the main features associated with resonant X-ray emission spectra of surface adsorbates to be understood from second-order perturbation theory between light and matter, as manifested by the Kramers—Heisenberg equation for the X-ray scattering amplitude — the consequences of which have been extensively reviewed in this work — although comparatively little is yet known about its actual consequences for resonant X-ray emission spectra of surface adsorbates with varying degrees of (dis)order; 1-, 2-, and 3-dimensional (dis)order. Some of the recent advances [214,340], in this respect are reviewed below. An evident difference for strictly ordered adsorbates with respect to 3-dimensional disordered systems as free molecules is that orientational averaging of the sample is no longer appropriate, making the orientational and polarization dependences so much sharper. Symmetry and momentum selection rules, appropriate for gases and solids, respectively, will have a different meaning, due to the quasi-continuous set of levels that often show up for surface adsorbate systems. One can anticipate that, just as for gas phase systems and solids, there is a strong connection between the channel interference and the actual observation of the symmetry selection rules. Towards the harder X-ray region, K1000 eV and beyond, one might anticipate a close connection between the channel interference and the selection rules on the one hand and the photon phase factors and the Bragg conditions on the other, making the scattering cross sections strongly anisotropic and oscillatory. Different dephasing mechanisms, such as orientational disorder, vibrational motion and vibronic coupling, may destroy the interchannel coherence and eliminate the selection rules [214]. Upon adsorption, molecules orient themselves in order to minimize the total energy. The most favorable orientation varies with the coverage and nature of the adsorbate and the substrate. The outcome of the RXS spectrum is evidently dependent on the adsorption strength between the molecule and the surface, whether the adsorption systems can be classified as physisorbed or chemisorbed. In physisorption the bonding is week, with adsorption energies in the order of 0.1 eV, whereas chemisorption energies are in the order of 1 eV. In systems with weak adsorbate—substrate interaction, the adsorbate—adsorbate interaction may also be of importance, depending on the coverage. The richness of the structural phase diagram encountered in physisorption systems arises from the subtle balance between the corrugation of the substrate potential and the adsorbate—adsorbate interaction [341] (Fig. 129). At low coverage only adsorbate—substrate interaction is important. As the coverage increases, the molecules can form a monolayer with several orientational phases [341] (Fig. 129). The present section reviews RXS for surface adsorbates in two aspects; firstly, Section 17.1 describes how the theoretical formulation using point groups for the one-step resonant X-ray emission cross section of randomly oriented molecules (Section 5.3) — applicable to any type of polarization for the absorbed and emitted photons in the soft X-ray region — carries over to samples with fixed order [141]. Some recent results for benzene and ethylene adsorbed to copper surfaces are reviewed in order to illustrate the symmetry and polarization information and the role of the channel interfence effects [293]. In the second part to follow, Section 17, radiative and nonradiative RXS following “hard” X-ray excitation of surface adsorbates is also considered, then focusing on various dephasing mechanisms for the X-ray scattering.

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Fig. 129. Four orientationally ordered phases for molecules A physisorbed on a triangular lattice [341]. (a) Two sublattice in-plane herringbone phase. (b) Four-sublattice pinwheel phase where the circles with dots indicate the molecules perpendicular to the surface. (c) Two-sublattice out-of-plane herringbone phase. (d) Ferrorotational phase where all molecules are free to rotate uniformly by a constant phase angle . A systematic out-of-plane tilt of the molecular axes is shown by arrows.

17.1. Theory for polarized RXS from adsorbed molecules The general theory for symmetry and polarization selection from randomly oriented molecules [95,141,122,342] reviewed in Section 5, forms a good reference point for the consideration of completely ordered systems, such as surface adsorbates of 1-dimensional order, see work of Triguero et al. in Ref. [293], which gives the basic material to the present subsection. A necessary ingredient in that theory was a general transformation of the transition dipole matrix elements d and d — naturally given in the molecular frame x, y, z — to the laboratory coordinates X, ½ and JA AH Z — in which the polarization vectors are expressed — through the directional cosine transformation ¹6"¹?t , see Fig. 130 and Section 5.3. Using these transformations the total X-ray scattering A A ?6 amplitude is obtained through the expressions (71), (72), (73), (74), in which the polarization, orientational and molecular information is collected in three different factors, Eq. (71). Tractable expressions for systems of fixed order — without orientational averaging — are obtained by rearranging Eq. (72) as F " ¸ ¸ F@A , JH @ A JH @A where

(363)

¸ " eHt , @  @ 

(364)

¸ " e t A  A 

(365)

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in which the summation has been reintroduced. ¸ and ¸ are the directional cosines of @ A polarization vectors e and e in the molecular frame. Therefore,





 "F "" ¸ ¸ F@A . JH @ A JH @A The total cross section is given by



(366)



 p(u,u )"r uu ¸ ¸ F@A U(u#u !u,c) . (367)  M @ A JH JH JH @A For adsorbed molecules with degenerate and quasi-degenerate core orbitals, it is a good approximation to assume u "u for all core MOs c. Eq. (73) can thus be written as HA H (C/n) 1 u u d@ dA " K@A (368) F@A" JA HA JA AH u!u #iC JH JH u!u #iC H H A and "F ""D(u!u ,C)"K ", K " ¸ ¸ K@A . JH H JH JH @ A JH @A

Fig. 130. Transformations between the laboratory and molecular coordinate frames. From Ref. [293].

(369)

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The cross section is then simply obtained as p(u,u)" p (u,u) . (370) JH JH p (u,u)"r(u/u)"K "D(u!u ,C)U(u#u !u,c) . (371) JH M JH H JH It is relevant in this context to review prototype systems in terms of few-states models. We discuss three such systems assuming narrowband excitation in a strongly absorbing n-system orthogonal to the substrate with the incident polarization e-vector along the surface normal: ¸ "d . (372) @ 8X These typical systems will also be utilized in the two examples of surface adsorbate RXS spectra, reviewed in Section 17.3 below. 17.1.1. Isolated core levels: Two-step model In the case with a single, isolated, core orbital involved, one has for the strict resonance u"u J  (373) "F ""D(u!u ,C)"IX " ¸ IA , A AH JH H JA A where I@ "u d@ , I@ "u d@ (b"z,c) JA JA JA AH HA AH  (374) p(u,u )"r "IX " ¸ I A D(u!u ,C)U(u!u ,c) . H H J M JA A IL J H A The absorption and emission processes are completely decoupled in this case, within a frozen orbital picture (no-screening approximation, see Section 15), as is the case in the traditional two-step model of RXS.









17.1.2. Core levels of different symmetry: Polarization selective excitation The situation changes qualitatively with two initial and near-degenerate core orbitals, c and c   (here assumed to be of C symmetry a and b ). Under the given circumstances only unoccupied T   orbitals with symmetries a and b will be involved in the absorption processes, and     p(u,u)" "IX " ¸ IA D(1)U(1)# "IX " ¸ IA D(2)U(2) , (375) JA A A H JA A A H J H J A A where D(i)"D(u!u G(l ),C), U(i)"U(u#u G !u,c), u G(l )"E(c\l)!E( j\l). If the screenHA G HA G JH ing effect is the same, u (l )"u (l )"u , one has HA  H HA    (376) p(u,u)" D(u!u ,C) "IX " ¸ IA U(1)# "IX " ¸ IA U(2) . JA A A H JA A A H H J J A A H Only in this case will the resonant X-ray emission have the same spectral lineshape as that of the non-resonant spectrum, and one should expect to see some difference between resonantly and non-resonantly excited X-ray emission spectra already for a system with two core levels, even





















 

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though of different symmetry. This difference is not due to the interference effect, but rather due to the polarization selective excitation which necessitates an explicit consideration of the intermediate state. Eq. (376) can be obtained from the so-called generalized two-step model [122], i.e. a model which accounts for the character of the intermediate states but which ignores channel interference. 17.1.3. Core levels of same symmetry: Channel interference The role of interference is most clearly described when the molecule has two near-degenerate core levels of the same symmetry (a , as in most chemically shifted species). With a polarization  vector along z, only unoccupied orbitals with symmetry a will then be excited. For a system with  one such unoccupied orbital l, the excitation can involve either of the core levels and, as a consequence, the total amplitude must be computed as a summation of two individual channels (i"1,2): (377) FXA"FXA (c )#FXA(c ) , JH  JH JH  K(c ) G FXA(c )" , (378) JH G u!u G#iC HA (379) K(c )"u Gu GdX GdAG G J HA JA A H with the cross section given by Eqs. (307) and (308). The cross section produced by the interference of the two channels, p (308), is dependent on the energy separation between the two core levels. If  one assumes u"u  i.e. resonance with one decay channel, then D "u !u "u   is the HA A HA HA AA energy difference between the two core-excited states, or the energy difference between the two core levels in the frozen picture. The interference term (308) can then be written as p "2K K 1/(u #C) , AA  A A i.e. the larger the energy separation, u  , the smaller is the interference effect. AA

(380)

17.2. Dephasing of X-ray Raman scattering by surface adsorbed molecules Radiative RXS by symmetrical molecules with identical orientation (1D system) shows strict selection rules (156) for radiative RXS, as demonstrated in Section 9.3. As briefly reviewed below, parity selection rules are retained also in the non-radiative (Auger Raman) process of such systems. These selection rules break down in the hard X-ray region due to orientational dephasing. The effects of the dephasing and violation of selection rules were reviewed in Section 9 for free molecules (3D disorder), and we consider here the similar problem — vibrational and librational dephasing for surface adsorbed molecules. The dephasing depends on the degree of disorder which evidently can be different for different systems. A strict theory of the orientational dephasing covering 1D, 2D, 3D (dis)order in radiative and non-radiative RXS were given in Refs. [214] and [340], respectively. To obtain a compact review of this theory, we consider here only the dephasing in 1-dimensional systems, i.e. systems with the same orientation of all molecules, as certainly is realizable by adsorption on a surface. To be specific, it is assumed that the molecules lie in the substrate plane (Fig. 131). When a diatomic homonuclear molecule is perfectly aligned the radiative RXS cross section is given by Eq. (153). The cross section for the resonant Auger scattering (Fig. 132) has the same structure [340].

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17.2.1. Librational dephasing Surface adsorbed molecules can be considered perfectly aligned only in a classical physics picture at low temperatures. In reality, the zero-point quantum librations of the molecules in their potential wells considerably broaden the orientational distribution. For example, the calculated root-mean-square (RMS) amplitude of these librations is 14° for the herringbone phase of N adsorbed on graphite [341] (Fig. 129a). Apparently, the effect of librational dephasing is  stronger than the vibrational dephasing [214,340] due to the weaker van der Waals interaction responsible for the librations. For example, the large librational dephasing for 1D ordered N molecules dominates in comparison with the very small vibrational dephasing [214,340]  (Fig. 133). The radiative and nonradiative RXS cross sections averaged over librations can be expressed in the following way [214,340]: p(u,u)"p0(1#Ps0) , M (381) p(e,u)"p,0(1!(!1)lPs,0), p ) R"ln. M The cross section for non-radiative RXS is given for the photon interference factor satisfying to the Bragg condition with l"0,1,2,2. l"0 corresponds to the soft X-ray region. We consider here the case of high-energy Auger electrons with the energy e and momentum k. Both MOs t and J t involved in the Auger decay (Fig. 132) are assumed to be n orbitals with negligible contributions H from d orbitals. Without librational dephasing the s functions, s0"cos(q ) R) and s,0"cos(k ) R), result in the following selection rules for both radiative (156) and non-radiative RXS [340] p(e,u)"0 if f"g,

(k#p) ) R "even , n

p(e,u)"0 if f"u,

(k#p) ) R "odd . n

(382)

Fig. 131. Geometry of the substrate with physisorbed molecule. Non-radiative RXS is shown; the same geometry is valid for radiative RXS if we do the following replacements: kPq, 0Ph , and uPu . O O Fig. 132. Scheme of spectral transitions for non-radiative RXS.

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The dephasing caused by the librations leads to a violation of the parity selection rules (156) and (382). The physical reason for this was discussed already in Section 9.3. The formal reason for this violation of the selection rules is the averaging of the interference terms cos(q ) R) and cos(k ) R) over librations. The average of these terms over zero-point and thermal librations is conveniently obtained using the density operator technique 1 1 s0" Re Tr(. p0(RK )eiq  R), s,0" Re Tr(. p,0(RK )eik  R) , M M p,0 p0 M M . "e\@&/Tr(e\@&), b"1/k ¹

(383)

with H as the libron Hamiltonian and the density operator normalized to unit Tr(. )"1. We consider here 1D ordering with small librations relative to the equilibrium molecular direction (Fig. 131). In the general case, the librations in the surface plane ("") and those “tilted” out of the plane (N) have different amplitudes and frequencies. Due to the smallness of librations the density operator factorizes as

. ". ) . . (384) ,  The harmonic approximation leads to the well-known expression for the density operator in configuration space [343] . (d,¹)"1/(2p1d2exp(!d/21d2), j"N,"" . H H H This immediately results in the following expression for the interference term (383): s0"cos(q ) R)e\50, s,0"cos(k ) R)e\5,0 .

(385) (386)

The libron Debye—Waller (DW) factors are strongly anisotropic ¼0"¼0 #¼0, ¼,0"¼,0#¼,0 , ,  ,  ¼0"qR1d2cos0, ¼,0" kR1d2cosh , j"N , H  H H  H O ¼0"qR1d2sin0 cosu, ¼,0" kR1d2sinh cosu , j""" . (387) H  H H  H O O The mean square of libron angles d depends on the temperature and on the amplitudes of the H zero-point librons dH (j"N,"") M ¹H if ¹/¹ H ;1 , 1d2"dHcoth
"dH M
H M 2¹





¹H 1d2"dHcoth
"dH 2¹/¹H , H M M
2¹

if ¹/¹ H <1 .


(388)

The libron temperature ¹H and amplitude of zero-point molecular librations dH depend on the
M libron frequency u,, as
uH 1 ¹ H "
, dH" , j"N,"" . (389)
M k 2kRu H
The zero-point libron temperature ¹ H /2 is about 25 K for N physisorbed on graphite, which
 serves as a good estimation for the temperature of the orientational phase transition [341].

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The investigations in Refs. [344—346,341] show that tilting and in-plane librations have the same order of magnitude d,&d& 10—20°. This means that one cannot suppress out-of-plane and M M in-plane librations simultaneously. It is therefore appropriate to stress that the librational dephasing in non-radiative RXS is never completely absent (¼,0O0), and that the librations will violate the parity selection rules given by the ideal 1D system for all directions of the Auger electron detection, Fig. 133. Such kind of librational dephasing is the result of the large magnitude of the interference parameter, kR; for example kRK11 for N .  Librational blurring of the interference patterns takes place also for radiative RXS in the hard X-ray region, Fig. 134. One can suppress the in-plane librations by choosing u"90° and consider the typical case of large kR. From (386) it is obvious that the parity selection rules then are fulfilled K59.1° in Fig. 133) and low temperatures approximately (Fig. 133b) for large Bragg angles (0   since the DW factor then is small. Another important distinction between an ideal 1D system (Fig. 133a) and 1D ordered molecules with the librational degrees of freedom is that the position of the K59.1°, Fig. minimum of p (e,u) (Figs. 133b and c) does not coincide with the Bragg angle (0   S 133a). The librational shift of this minimum position is caused by the DW factor (387). Let us recall that q changes from 0 to 2k when the scattering angle increases from 0° (forward scattering) to 180° (backward scattering). It is therefore appropriate to stress that the librational dephasing in radiative RXS is absent for small angle scattering. This possibility to “operate” the vibrational and librational dephasings is an important feature of radiative RXS. Let us remind that this opportunity is absent for the non-radiative counterpart since the discussed dephasing is then defined by the momentum of the Auger electron [340]. 17.2.2. Role of the zero-point librations In contrast to the classical view the “thermal” quenching of coherence does not disappear when ¹P0 since 1d2PdH (388) (Fig. 133b). The dephasing at zero temperature is caused by the H M zero-point librations with amplitudes dH. As one can see from Eq. (387) and Fig. 133 that the M librational dephasing is never completely absent in non-radiative RXS. Fig. 133 shows that zero-point librations violate the parity selection rules (p (e,u)O0) even for zero temperature, and ES that the thermal librations lead to a “melting” of the interference pattern when ¹'¹ . It is here
relevant to repeat that the librational dephasing in radiative RXS can be neglected in the soft X-ray region or in the case of small angle scattering. 17.3. Sample applications: Ethylene and benzene on copper surfaces Ethylene and benzene on copper-110 surfaces [293] represent well the fundamental aspects of RXS of surface adsorbates, namely symmetry and polarization selectivity and the role of interference. They do also illustrate the simplifing theories presented in Sections 17.1.3 and 17.1.2, respectively. As shown by Triguero et al. in Ref. [293] the spectra are well be simulated using cluster models containing up to 100 surface atoms, within the framework of density functional and/or effective core potential electronic structure theories [347,348], see Fig. 135. The cluster model approach was found more demanding for resonant excitation than in the nonresonant case with respect to convergence with cluster size, owing to the requirement of accurate descriptions of the unoccupied levels.

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Fig. 133. Role of the librational dephasing on the angular dependence of the non-radiative RXS cross sections p (e,u) and E p (e,u) for gerade and ungerade final states, respectively. 1D system. Input data for K-spectrum of the N molecule: S  e"380 eV, ¹ K47.6 K, d "14°, KRK11 a.u. u"90°. (a) No librations, d,"0. (b,c) d,"14°. The angular
M M M dependences of the RAS cross sections are close to the case (b) up to ¹K40 K. The Bragg angles are equal 0 "0°,16.6°,34.9°,59.1°.   Fig. 134. The role of librations on the angular dependence of the radiative RXS cross sections (381), (386) of a 1D ordered system. h is the angle between q and the normal to the molecular axes n. ¹ K5.8 K for the physisorbed Cl molecules. O
 u "90°, the scattering angle is equal to 90°. The dashed curves depict p (u,u) and the solid curves depict p (u,u) for O E S gerade and ungerade final states. It is assumed that the polarization vectors e and e are orthogonal to the plane of vectors q and n, so that polarization factors remain the same with the change of h . pR"2.8 for K-spectrum of Cl . O 

The experimental and simulated spectra [293], see Figs. 136—139, show a subtle but significant dependence on the exciting photon energy, which in the particular case of chemisorbed ethylene is due to polarization selective excitation and not to interference of different scattering channels. The ethylene on copper RXS spectrum shows strong polarization (or directional) selectivity in that the X,½,Z component spectra appear quite differently. There is a strong entanglement between the symmetry of the emitting levels and the polarization (or direction) of the corresponding outgoing photon. The polarization selectivity is found responsible for the differences between the resonant and non-resonant spectra. Due to the fixed orientation of the ethylene adsorbate, only certain orbitals can be observed along given directions, see Table 7. In the case of benzene on copper the shift among the core levels induced by surface adsorption leads to interference effects in the decay. Such interference is shown to alter the spectra substantially and restore some of the symmetry selectivity of the free species. A close similarity is obtained with the symmetry selective RXS process in chemically shifted spectra, as illustrated by the comparison between aniline and benzene spectra, and also with symmetry and momentum

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selective polymer spectra, as illustrated by the phenyl ring containing poly-phenylene-vinylene spectra described in Section 16.2.4. The restructuring of the adsorbate upon bonding can lead to significant and characteristic changes in the X-ray spectra [337,338]. This is well illustrated by surface adsorbed benzene for which the main effect is a breaking of the planarity and a separation of the p and n parts of the emission spectrum. Geometry optimization of C H on copper clusters actually leads to two   optimum adsorbate conformations, one with the carbon ring bent in an inverted boat-like quinoid form with the hydrogen atoms flipping upwards, and one in which the carbon ring planarity is maintained but with the hydrogens bending [339]. RXS spectra of the latter are shown in Fig. 138. Such changes have been anticipated using other techniques but not accepted as a statistically significant fact [349]. The differences in the predicted X-ray spectra referring to the different

Fig. 135. Cluster convergency of C(1s) resonant and non-resonant X-ray emission spectra of chemisorbed ethylene on Cu(110). Calculations are performed at the DFT(BP86) level with the program DeMon [347]. From Ref. [293].

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Table 7 The relationship between polarization vector of emitted light and the symmetry allowed emission orbitals for nonresonant and resonant X-ray emission spectra of physically and chemically adsorbed ethylene Polarization vector

Symmetry allowed emission orbitals Non-resonant

X ½ Z

a ,b   a ,b   b ,a  

Resonant Physical

Chemical

a  a  b 

a ,b   a ,b   b ,a  

Fig. 136. Comparison with experiment for two different excitation energies. Theory (left) and experiment (right). C H /Cu , FWHM"2.0 eV. From Ref. [293].    Fig. 137. Analysis of the emission components of calculated RXS spectra of C H /Cu . From Ref. [293].   

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Fig. 138. C(1s) XES of chemisorbed benzene on Cu(110). Comparison of the one- and two-step models. From Ref. [293]. Fig. 139. Comparison of theoretical spectra (left) with experiment (right) for two different exciting photon energies. C H /Cu . From Ref. [293].   

molecular states of the benzene molecule involved in the interaction with the substrate, and the p—n symmetry-breaking manifested in the orientational dependence of these spectra, should be significant enough to make it possible to test structural changes upon adsorption by actual X-ray measurements. A further issue to be studied in this context is the role of vibronic and phonon coupling for surface adsorbates, effects which are known to lead to symmetry breaking of electronic selection rules of RXS spectra of gases (e.g. free benzene [223] and free ethylene [244]), and for solids [325], see Sections 10 and 16.4. Even closely related molecules, like hydrocarbons, behave differently in this respect. The symmetry breaking is strongly frequency dependent and quenched by detuning the frequency from resonance, see Section 10.3. It is still an open question whether the presence of the surface quenches or enhances such couplings, and if the dynamic features evident for the free species are observable also for adsorbates. The ethylene- and benzene-copper systems also illustrate the fact that new states appear on the high-energy side in the emission spectra of the adsorbates not present in the free molecular spectra.

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Some dependence of the appearance of these states on the excitation energy is observed for both ethylene and benzene. This was understood in terms of chemical bonding as large mixings of n and nH with the metal orbitals upon adsorption, and interpreted in terms of the spin-uncoupling mechanism proposed in Ref. [339] or the traditional Dewar—Chatt—Duncanson model [350,351]. For other systems, like surface adsorbed N and CO the organization of the emission spectra (here  nonresonant) have led to the proposal of new bonding models [336] in some contrast to the traditional ones (Blyholder model [352]). An interesting and useful result from simulations of the resonant [293] and non-resonant [353,354] spectra is the validity of the frozen orbital approximation. Account of relaxation, either though separate self-consistent field or transition potential procedures [353], does thus not necessarily improve upon frozen orbital calculations, and it seems that the surface adsorbate species behave more metallic-like than molecular-like in this respect. An underlying explanation using the theory of Mahan [294], Nozieres, DeDominicis [295] (MND) for the dynamics of the core—hole state screening in metallic systems has been discussed in this connection [293], and was briefly commented in Section 15.2. The support from simulations of the ground state model has an important ramification for the use of RXS to analyze the bonding and other ground state properties of surface adsorbates.

18. X-ray absorption spectra measured in the Raman mode As briefly mentioned in the introduction of this review the earliest development of X-ray spectroscopy in terms of element characterization, chemical shifts, multi-electron transitions, etc., took place for X-ray measurements in the emission mode. The modern developments in the soft X-ray region using synchrotron radiation sources concerned at first mostly absorption measurements, and near-edge X-ray absorption spectroscopy (NEXAFS) is by now an established and well-reviewed field within X-ray physics [184,188]. Ordinarily, the X-ray absorption cross sections are defined by direct counting of the absorbed photons (transmission spectrum). An alternative and popular model to obtain X-ray absorption cross sections is based on the assumption of proportionality between these cross sections and the resonant X-ray scattering cross section [188]. This assumption is based on a two-step model, in which the absorption cross section is obtained by detecting the emission of X-ray photons or Auger electrons associated with the secondary process of core hole annihilation. However, certain limitations of such a two-step model for radiative and nonradiative RXS have also been realized, and a unified picture of X-ray fluorescence or Auger electron emission as a process of inelastic scattering of X-ray photons has sometimes been called for (see Section 3). As reviewed here the use of radiative or non-radiative RXS gives a qualitatively different information on X-ray absorption, and opens new possibilities for studies of X-ray absorption both near and far beyond the edge (NEXAFS and EXAFS). 18.1. High-resolution NEXAFS measured in the Raman mode It is relevant to start the description of this region in X-ray Raman spectroscopy from a result obtained by Tulkki and As berg [355] and by Ha¨maa¨la¨inen et al. [356]. As one can see from Eq. (16),

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the spectral width of the RIXS resonance is restricted only by the width (c) of the spectral function of incident radiation (provided the final state lifetime is small: C ;c). This leads directly to the D conclusion that the resolution of the X-ray absorption spectra measured in the RXS mode for some fixed value of emission frequency (u"u ) M pPVQ(u),p(u ,u) M

(390)

may be obtained narrower than the lifetime width of the core excited state C [355]. G 18.1.1. Experimental evidence of NEXAFS with resonance narrowing Ha¨ma¨la¨inen et al. [356], see also [49], demonstrated super-high-resolution NEXAFS by monitoring the energy of the fluorescence photons near the ¸ edge of dysprosium with a ''' high-resolution spectrometer. The experiment was carried out on the double focusing X25 wiggler beam line [357] at NSLS at Brookhaven National Laboratory. A two-crystal Si (220) monochromator gave an incident beam resolution of 0.7 eV at 6.5 keV and a photon flux of the order of 10 photons/s with a spot size about 0.5 mm at the sample. A near-backscattering geometry (83—87°) provided the necessary energy resolution which was limited by the source size to about 0.3 eV. A detailed description of the instrument is given in Ref. [358]. Fig. 140 shows the measured ¸ -fluorescence spectrum from a Dy O sample when the   ? incident energy is well above the absorption edge. The measured line width of about 9 eV is very large compared with the analyzer resolution of 0.3 eV. Only the fluorescence radiation within a narrow band (Fig. 140) of the fluorescence peak was monitored while the incident frequency was scanned through the ¸ absorption edge. Fig. 141 shows the resulting absorp''' tion spectrum from dysprosium nitrate (solid line). The improvement of the resolution is dramatic when compared with the transmission spectrum from the same sample with the same incident beam resolution (dashed line) where the resolution is limited by the ¸ shell ''' lifetime broadening of 4.2 eV. The high-resolution spectrum reveals a well separated true absorption edge followed by the white line corresponding to strong dipole allowed transitions to empty 5d states. The first edge is due to the quadrupole transitions to partially filled 4f states. A detailed scan in this region is shown as an inset in Fig. 141. All this fine structure is smeared out in the ordinary X-ray absorption spectrum (dashed line) due to the large ¸ -lifetime ''' broadening. 18.2. Polarization features of NEXAFS measured in the radiative Raman mode The RIXS cross section is proportional to the ordinary photoabsorption cross section p(u) only in the two-step model. However, p(u) and the absorption cross section pPVQ(u) (390) measured by the RXS method can differ significantly, something that can be ascribed three physical reasons. The first one refers to the interchannel interference effect in the RXS process (see Section 6). The second reason refers to the energy conservation law (9) reflected by the spectral function of incident radiation (65). This leads to a spectral profile with the width equal to the photon bandpass c independent of the lifetime broadening C (see previous section). In this section we briefly review G the third source of deviation of pPVQ(u) from p(u), namely the one that refers to the strong

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Fig. 140. Experimental ¸  fluorescence spectrum from Dy O [356]. The dashed line represents the analyzer resolution. ?  

Fig. 141. High-resolution fluorescence absorption from dysprosium nitrate. The dashed line represents the conventional transmission spectrum (ordinary photoabsorption). An inset shows the absorption spectrum measured in the RIXS mode around the absorption edge with a better statistical accuracy. The structure is related to the quadrupole transitions to the partly unoccupied 4f states [356].

dependence of the RIXS spectral shape on the polarization directions of incident and emitted X-ray photons. As seen from Eq. (66) the spectral profile (390) depends on the angle u between the DG dipole moments of absorption and emission transitions.

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One can illustrate the findings described above by computing features of the frequency and polarization dependences of the RIXS absorption cross section (390) for the H S molecule,  Fig. 142. This molecule is illustrative because of the organization of its lowest core-excited states, with two close lying MOs, 3b and 6a , with disputed order [139,189,248] (see also Section 7.3).   One can see that the polarization dependence is different for different combinations of symmetries of occupied (2b , 5a , 2b ) and vacant (3b , 6a , 7a , 4p) MOs. The angle u characterizes       DG this polarization dependence: u "0° or 90° when the symmetries of the corresponding DG unoccupied and occupied MOs are the same or different, respectively. Figs. 142a and b demonstrate this directly, for example the two transitions SKP3b and SKP6a have opposite   h-dependence. The special case of the H S molecule considered here and in Section 7.5 shows that the state- and  the polarization dependence of an absorption spectrum measured in the RXS mode can be used as a tool for resolving core excited states and for determining symmetries of unoccupied MOs of molecules in the gas phase.

18.3. EXAFS measured in the Raman mode X-ray absorption cross sections can show oscillatory features extending far beyond the ionization edge. The physical mechanism behind these oscillations, commonly denoted as EXAFS (extended X-ray absorption fine structure), is given by the interference between the outgoing photoelectron wave from the absorbing atom and the backscattered waves from the surrounding atoms [359,157,188]. A remarkable increase in the attention on EXAFS took place after the publication of the paper of Sayers et al. [360], showing that EXAFS-based techniques actually can yield structural information of complex molecular or condensed systems even in cases when other traditional structure methods, such as X-ray diffraction, are not applicable. EXAFS analysis of disordered systems can give information on distances R  between the ?? absorbing atom n and the neighboring high-Z atoms n . However, the angle between the polariza tion vector e of the absorbed X-ray photons and the internuclear axis direction R  LL can be measured only when the EXAFS technique is applied to well-ordered systems like crystals, surfaces and adsorbates on single crystal surfaces [359,157,188]. In the latter case, measurements of surface extended X-ray absorption fine structures (SEXAFS) [188,361,362] make it possible to determine the internuclear distances R  and the chemisorption sites. At the present time LL several modifications of the EXAFS technique are utilized [359,157,188,363]; for example, ordinary EXAFS spectra measured in the direct transmission mode; EXAFS obtained by a partial electron yield detection mode; EXAFS obtained by the partial fluorescence yield (FY) mode, and; EXAFS obtained by the electron energy-loss method [359,157,188]. In some cases it is possible to detect EXAFS only in the fluorescence yield mode because of the smaller penetration depth of electrons with respect to photons. The corresponding measurements became possible by the advent of high-intensity X-ray sources derived from synchrotron radiation. As was the case for NEXAFS described above, the utility of EXAFS detected in the FY mode has been based on the assumption of proportionality between the X-ray fluorescence yield and the X-ray absorption cross section p6(u) for samples of small thickness [359,188]. Assuming this model, the absorption cross section is obtained by detecting the emission of X-ray photons associated with the

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Fig. 142. The RIXS SK -absorption spectra (390) of H S as a function of relative incident X-ray photon frequency and @  the angle h between the polarization vectors e and e [372]. The following order of unoccupied MOs is assumed: 3b , 6a ,   7a and 4p. The spectra (a—c) correspond to the tuning of emission frequency u to the resonance: (a) 2b PSK, (b)   5a PSK, (c) 2b PSK. The ordinary X-ray absorption spectrum is represented by a dotted line in panel (a). c"0.1 eV,   C"0.5 eV [372].

secondary process of core hole annihilation. However, just as was the case for NEXAFS, the limitations of such a two-step model for the X-ray fluorescence process has been unraveled and a unified picture of X-ray fluorescence as a process of inelastic X-ray scattering is also desirable for high energies beyond the edge. As reviewed here, the use of inelastic X-ray scattering can give

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qualitatively new information on the geometrical structure of condensed or molecular systems [364,365]. 18.3.1. Principles of EXAFS measured in the Raman mode It is worthwhile here to first review the essence of ordinary EXAFS. The oscillatory (EXAFS) part of the X-ray K-absorption cross section p(u) is given by [359,188] eiI0LL>BL p(u)!p (u) M "!3Im f (n)(e ) RK ) , s" LL kR  L p (u) LL M L

(391)

where f (n) is the backscattering amplitude of the photoelectron with momentum k"(2(u!I) L by atom n , dL is the p-wave phase shift by the absorbing atom n and p (u) is the smooth   M (nonoscillating) part of the X-ray absorption cross section. One can see that s is proportional to (e ) R ) for well-ordered samples, like crystals or metallic surfaces [359,188]. This fact makes it LL possible to determine, for example, the chemisorption site of an adsorbed molecule at a surface [188,361,362], but it is still difficult to define the orientation of adsorbed molecules relative to the surface by this method (however, see Ref. [188]). Moreover, traditional EXAFS does not give directional information, like angles between internuclear axes, for non-crystalline systems such as amorphous semiconductors, liquids or free molecules, because in this case (e ) RK ) must be LL replaced by .  This situation changes when the X-ray absorption spectrum is measured in the RIXS mode. To make this possible the incident photon frequency u must exceed the X-ray photoionization threshold I. A simple model with a fluorescence spectrum with one resonance jP1s nonL overlapping with other emission lines is a suitable first consideration. One can tune the emission spectrometer to this resonance (uKu ) with the resonant frequency u of an emission transition HL HL from the occupied MO t to the core level of the n:th atom. The measured fluorescent signal will be H proportional to (e ) D )p(u) (392) HL and as a result it will depend on the mutual orientation of the polarization vector e of the emitted X-ray photon and the dipole moment D of the emission transition jP1s . Taking into account HL L that the direction of the dipole moment D is connected directly with the space orientation of the HL molecular axis, and that the oscillatory part s of the RIXS cross section (392) is proportional to, (e ) D )(e ) R ) one can understand that the detection of EXAFS in the RXS mode can give HL LL directional information as well as information on internuclear distances. At first sight one cannot receive information about “molecular structure” or bond angles of disordered systems using the RXS method due to the averaging of the scattering cross section over molecular orientations (i.e. orientation of the nearest surroundings of the absorbing atom). However, this is not the case; a closer inspection reveals that the oscillatory part of the RXS cross section averaged over molecular orientations (63) must depend on the e, e, DM and R  vectors only through the two HL LL scalar products (e ) e) and (DM ) R ) of external and internal vectors, respectively. As a result the HL LL RIXS cross section cannot be given as a simple product of emission and absorption probabilities. Moreover, this circumstance causes a strong dependence of the EXAFS spectral shape on polarization directions e and e and on the frequency of the emitted X-ray photon.

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One can show that the oscillatory part of the X-ray “absorption cross section” measured in the RIXS mode reads [364] eiI0LL>BL p(u,u)!p (u,u) M "!9Im f (n)f (h) , (393) s" LL kR  L p (u,u) LL M L where p6(u,u) is the smooth (non-oscillating) part of the RIXS cross section. The polarization M function f (h) is given by Eq. (66) with the bond angle LL u "uH ,arccos(DK ) RK ) (394) DG LLY HL LLY between the directions of the transition dipole moment DK of emission and the direction RK  to HL LL atom n , where DK "D/D. The two s-functions (391) and (393) coincide only for scattering at the  magic angle (70). The EXAFS formulas for these s-functions including a multiple-scattering correction can be found in Ref. [359,364]. The effect of lattice vibrations is taken into account by introducing into the right-hand side of Eqs. (391) and (393) the EXAFS Debye—Waller factor exp(!2pk) [366,360,359,188] (Eqs. (391) and (393) are written in the one-scattering approximaL tion). Let us here point out the strong polarization dependence of the fluorescence EXAFS profile (393) caused by the polarization function f (h) (66) [364]. LL Expression (394) can be easily generalized to non-resonant scattering of light in which uOu HL or to the case of strongly overlapping X-ray emission lines. In the region of resonant fluorescence for arbitrary frequencies u of emitted photons one needs to use for f (h) in Eq. (66) only a more LL general definition of the angle u ,u , namely DG LL cosuHD D(u!u ,C ) LL HL HL G . (395) cos u " H LL D D(u!u ,C ) H HL HL G The measurement of these angles would completely solve the problem of the definition of bond angles. Unfortunately, the bond angles uH (394) can probably not be measured directly, and, in LL accordance with Eqs. (393) and (66), only the angles u  (395) are measurable quantities. The X-ray LL emission spectrum has often one or several resonances which do not overlap or only slightly overlap one another. In this case the angles uH and u  coincide if the emission photon frequency LL LL u is tuned into exact resonance with one of these non-overlapping resonances jPn (u"u ). HL 18.3.2. Oxygen EXAFS-RIXS spectra in amorphous silicon oxide The possibilities for structure determination with EXAFS measured in the RIXS mode can be demonstrated by sample calculations on amorphous silicon oxide (a-SiO ). As shown by contem porary investigations [367—369] it is more relevant to consider an amorphous structure for this compound on a short length scale. But even on this scale the notion of an elementary amorphous unit is not well defined, and it is possible to speak only about a probability to find this elementary unit. Galeener [368] was forced to consider regions of increased order in the form of highly regular rings of bonds, connected into the otherwise more disordered network at sites with unknown character. The nearest neighbors for each oxygen atom lying in the ring plane are two silicon atoms. Since the EXAFS method allows to select signals from different coordinate spheres [359,188], one can

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analyze the EXAFS signals from the first coordinate sphere of the Si—O—Si fragment. According to ab initio calculations [364] the X-ray emission spectrum of the chosen model system has several OK-resonances which do not overlap or only slightly overlap with other emission resonances. As seen directly from Eq. (394) uH "0 when the dipole moment D of the emission transition jPn LLY HL lies perpendicular to the ring (the corresponding value of the s-function (393) is denoted s ). The , angle u is expressed directly through the Si—O—Si bond angle H (u "H/2) if D is the bisector LLY LLY HL of the Si—O—Si angle lying in the ring plane (the s-function (393) is in this case denoted s ). As , a result, the following connection between the Si—O—Si bond angle HK190.5° [369] and the experimentally measured values for the s-function is obtained cos H/2"(s /s !1)N(h) , (396) , , where the function N(h)"(2!cos h)/(3cos h!1) depends on the polarization vectors e and e through the angle h between them. This formula shows directly how bond angles can be defined by EXAFS measured in the RIXS mode. One can see from Eq. (395) that formulas (394) and (396) give only a rough connection between the bond angle uH "H/2 and the experimentally measured LLY s function or angle u when the frequency of X-ray emission is tuned into a region of strongly LLY overlapping resonances. The ab initio calculations of the oxygen X-ray emission spectrum of the 6-fold ring consisting of 6 silicon and 12 oxygen atoms performed in Ref. [364] illustrated the connection between the bond angles uH and the measurable angles u more specifically. LLY LLY 18.3.3. EXAFS measured in the nonradiative Raman mode EXAFS can be detected in several ways as mentioned above, either directly in the transmission mode, or indirectly by measuring the products of absorption, namely, the fluorescent radiation or the non-radiative Auger — or secondary — electrons. The cross section for the latter process is considerably larger than that of radiative RIXS for low-Z atoms. An analogy with radiative RIXS can be obtained with the following Auger process: An incoming X-ray photon excites a core electron into the continuum state tk, and is followed by an electron decay from MO t to core shell H (n). The energy of this transition is transferred through a Coulomb interelectron interaction



(kj"nj)" tH k (r )t (r )(1/r )tH(r )t (r ) dr dr  H   L  H   

(397)

to the second electron from this MO. For simplicity, we thus consider an Auger transition that empties two electrons from the same orbital, for which the exchange term is lacking. tk denotes here the continuum orbital housing the expelled electron. The amplitude of this process (eH ) Dk )(kj"nj) L (398) u!uk #iC L G is strongly reminiscent of the RIXS amplitude (7) with the cross section given by Eq. (13). D is D# proportional to (398) after integration over k, and will depend on the mutual orientations of the vectors e, k, R and on the space orientation of the t MO. The orientation of this MO in the case LLY H of a K-spectrum is defined only by the direction of the transition dipole moment D , and it is easy HL to understand that the cross section (13) for the non-radiative RXS for disordered systems only

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Fig. 143. Dependence of the experimentally measurable bond angle (395) on the energy of an X-ray emission photon (solid line) and the HWHM C for the first coordinate sphere amorphous SiO . Positions and relative intensities of the G  ordinary oxygen X-ray emission spectrum of the planar 6-fold ring in amorphous SiO are displyed for comparison by  vertical bars. The energy of an X-ray emission photon is given relative to the OK ionization threshold. The values of cos uH for bond angles (394) are 0, 0.008 and 0.992 for the OK-emission transitions from MOs of B , A and LLY   B symmetries, respectively [364]. 

depends on the scalar products (e ) k) and (D ) R ). As a result, EXAFS detected in the nonHL LLY radiative Auger mode will also be described by the s-function (393), but with a different polarization function f (m), depending now on the angles m"arccos(eˆ ) kK ) and u . The spectral dependLLY LLY ence of the experimentally measurable bond angle u on the energy e of the Auger electron LLY reminds of the one shown in Fig. 143 and makes it possible to determine bond angles just as when EXAFS is detected in the radiative RIXS mode. The here discussed and earlier predicted [364] strong polarization dependence of the EXAFS profile of disordered systems measured in the radiative Raman mode (393),(66) was recently studied experimentally for some catalytic systems [370]. Apparently, such a polarization dependence takes place also in the nonradiative EXAFS (see Eq. (398) and discussion below this equation) [364].

Acknowledgements We acknowledge many useful discussions with Yi Luo concerning various aspects of the theory of resonant X-ray Raman scattering. We also thank Vincenzo Carravetta, Amary Cesar, Yang Li, Christoph Liegener, Lars Pettersson, Timofei Privalov, Pawe" Sa"ek, Luciano Triguero and Olav Vahtras for a nice collaboration on theory and computations of RXS, some results of which are included in this review. Without having the stimulus of the contacts with experimentalists this review would not have been written. We cordially thank Olle Bjo¨rneholm, Paul Bru¨hwiler, Peter Glans, Kerstin Gunnelin, Jing-Hua Guo, Nohubiro Kosugi, Martin Magnusson, Nils Ma rtensson, Mattias Neeb, Anders Nilsson, Joseph Nordgren, Jan-Erik Rubensson, Per Skytt, Stefan Sundin and Svante Svensson. This work was supported by the Swedish National Research council (NFR) and by scholarships from the STINT foundation and the Royal Swedish Academy of Science (KVA).

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