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Complete photoionisation experiments1
ELSPEC 3511
Journal of Electron Spectroscopy and Related Phenomena 96 (1998) 105–115
Complete photoionisation experiments 1 Uwe Becker Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany Received 11 May 1998; accepted 15 June 1998
Abstract Experiments in atomic photoionisation are considered complete in the sense that both the amplitudes and phases of the corresponding transition matrix elements can be determined. Such experiments have recently made remarkable progress and provided a wealth of new information in this field. The researchers have employed a variety of complementary techniques such as polarised radiation and targets, spin-resolved detection of electrons and coincident detection of two electrons in cases of double photoionisation. This rapidly developing field is reviewed in a selection of show-case examples for each method. The results are compared with different calculations and critically analysed with respect to the validity of the theoretical models employed for these calculations. 䉷 1998 Elsevier Science B.V. All rights reserved. Keywords: Atomic photoionisation; Cooper–Zare model
1. Introduction Photoionisation of matter was the first in a series of experimental discoveries followed by corresponding theoretical explanations which led to the foundation of quantum physics [1, 2]. It was in fact Einstein’s interpretation of the photoelectric effect [3] that proved the concept of energy quantisation by Planck [4] to be generally valid. Despite this early success it took more than 40 years until photoelectron spectroscopy became a growing field of scientific research and, finally, an analytical tool for industrial research [5, 6]. This was because the early efforts to record line spectra in measurements of the photoelectric effect were hampered by experimental difficulties such as insufficient energy resolution. Only after the pioneering work of the Uppsala group did photoelectron spectroscopy become a powerful tool for studying 1 Dedicated to Professor H. Kleinpoppen, Stirling, for his 70th birthday.
the electronic structure of matter and its chemical composition [7]. Electron Spectroscopy for Chemical Analysis (ESCA) was a synonym for an entire field of research in atomic, molecular (gas phase) and solid state physics. This field dealt with the orbital structure of electrons in free atoms and molecules and the band structure of electronic states in solids and surfaces. The main target of these studies was to analyse the enormous variety of elements and compounds in their various forms with respect to their electronic structure and chemical composition in the ground state. The ground state properties are still the main subject of photoelectron spectroscopy; some kind of subfield, however, started to develop when some researchers concentrated more on the final state of the photoionisation process and, especially, its dynamics. Why is this of particular interest, and what can we learn from the underlying energydependent behaviour of the photoelectron? The answer to this question is closely related to the socalled complete scattering experiments to which
0368-2048/98/$ - see front matter 䉷 1998 Elsevier Science B.V. All rights reserved. PII: S0368-204 8(98)00226-6
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photoionisation as a half-scattering process belongs. The purpose of complete scattering experiments is to derive complete information on the scattering process in a quantum-mechanical sense, which means, the total number of transition matrix elements with corresponding phases necessary to describe a particular scattering process. In the case of photoionisation this is a limited number of transition matrix elements which are for intermediate energies dipole matrix elements. Therefore, complete photoionisation experiments are concerned with the determination of characteristic properties of the photoionisation process such as partial cross section, angular distribution, photoelectron spin polarisation and different types of dichroism in the angular distribution of photoelectrons. These properties may be used to derive the number of parameters necessary to describe the photoionisation process within a certain theoretical model in a quantum-mechanically complete sense. The purpose of the experimental studies in this field is to figure out which theoretical model is necessary for a complete description within certain limits of accuracy. To visualise this problem let us take the noble gases or, more generally, all closedshell atoms as an example. In the most general case of relativistic treatment of photoionisation there are three outgoing partial waves with two relative phases for each spin–orbit component of the remaining final ionic state: 1 hn ⫹ A0 ! A⫹ J ⫹ e l ^ 1; 2 j
with J ⫺ 1 ⱕ j ⱕ J ⫹ 1. It means that subshell photoionisation of an openshell atom requires ten parameters to be completely described. The situation is still worse for closed-shell atoms where the number of parameters is even higher and, in some cases, exceeds the possibilities of experimental determination. However, the situation gets better if one considers possibilities to reduce the necessary number of parameters by relating some of them to each other. The most powerful frame for such a reduction is the LS coupling scheme. It is based on the existence of a total orbital angular momentum L and a total spin S, and it enables us to reduce the number of parameters necessary to describe the subshell photoionisation of both closed-shell and open-shell atoms to three: two transition matrix
elements and one relative phase shift. This extreme reduction results from the fact that the angular momentum of the ionising photon is 1 and may be coupled to the emitted electron either to l ⫹ 1 or l ⫺ 1 because there are no longitudinal polarised photons with a vanishing projection of the angular momentum. This extremely simple but powerful model is known as the Cooper–Zare model. In the literature, it is considered as a first step approximation to a qualitative description of photoionisation without, however, the capacity to provide quantitatively acceptable results. The main reason for this assessment of the so-called three-parameter model is the fact that the simplest photoionisation parameter, the partial cross section, is mostly just poorly described by this model. Therefore, the three-parameter model does not seem to be an appropriate means to satisfactorily describe the more complex photoionisation parameters as, for example, the angular distribution and spin parameters. But there is no stringent verification for this assumption because the experimental data, particularly those of the spin polarisation, are very rare and, for technical reasons, basically limited to photon energies below 25 eV. Only very recently, new spin polarisation measurements at higher photon energies have become feasible. In addition, complementary methods such as the use of polarised targets for dichroism measurements were introduced and provided first results for open-shell atoms. The purpose of this review is to critically analyse the present data situation with respect to the validity of the different theoretical models employed to describe the photoionisation process. The review is organised in the following way. First, it describes the different methods of how to determine the various photoionisation parameters. The following sections describe selected show-case examples for each method along with the most recent experimental results. Finally, conclusions will be drawn with regard to the validity of a certain theoretical model putting special emphasis on a newly proposed model, the so-called extended threeparameter model or (3 ⫹ 1) model.
2. Determination of photoionisation parameters Photoionisation experiments with randomly oriented targets yield two independent parameters,
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the intensity as a measure of the partial cross section s and the angular distribution of the emitted photoelectrons [8]. The angular distribution asymmetry parameter b depends on the phase between the two outgoing electron waves but also on their intensity ratio [9]. Therefore, a third parameter is required in order to perform a complete partial wave analysis in terms of the three-parameter model. This third parameter may be obtained by several methods, all of which rely on the preparation of the incident reaction partners or the analysis of the outgoing reaction partners if one considers the photoionisation process as a collision reaction hn ⫹ A ! A⫹ ⫹ e⫺ . On the side of the incoming particles, the photon is always specified by its angular momentum. The use of linearly or circularly polarised light may have certain advantages for particular measurements; however, even unpolarised radiation is sufficiently specified by the propagation vector k. In contrast to the photon, the target atom A is usually randomly orientated with respect to the magnetic subquantum number m. In order to specify this quantum number it is necessary to polarise the target atom. The change in atomic polarisation changes the photoelectron angular distribution in a certain way, yielding additional information on the photoionisation process. The deviation of the angular distribution pattern may be used to derive the necessary third parameter for a complete photoionisation experiment [10]. Different geometries regarding the atomic and light polarisation make it possible to derive different independent parameters. The polarisation of the atoms may be achieved by use of inhomogeneous magnetic fields or optical excitation/ pumping. Both methods have been successfully used to produce polarised targets. Regarding the analysis of the outgoing particles the most successful method is the determination of the spin polarisation of the outgoing electrons [11]. It allows the extraction of up to three additional parameters because the spin of the electron has three independent components. A spin-sensitive detection method is required to measure the spin polarisation of the ejected electron. The most commonly used method is still based on Mottscattering. Mott detectors are advanced instruments, and just very recently a multi-channel method for simultaneous detection of energy-resolved electron spectra, the electron time-of-flight method, has been combined with such a detector. The so-called
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Mott–TOF detector was applied successfully in spin-resolved measurements of photo- as well as Auger electrons [12]. Complementary to the spin analysis of the electron in the outgoing channel is the analysis of the polarisation of the remaining ion. It may be derived from the angular distribution of Auger electrons [13, 14] as well as fluorescence radiation [15]. In both cases one obtains the alignment of the remaining ion. If one uses circularly polarised light for primary photoionisation, the result will be an oriented final ionic state. This orientation may be determined either by the spin polarisation of subsequently emitted Auger electrons [12] or the circular polarisation of fluorescence photons [16]. In the framework of the two-step model all these processes yield the same information. The ‘alignment’ method was used by different groups as a first step towards complete photoionisation experiments [17]. Alignment and orientation refer always to the polarisation of the incoming light; they only contain information on the partial wave intensities of the outgoing photoelectron wave, the phase information is lost. To keep this information, a coincidence experiment between photo and Auger electrons has to be performed. This method yields a number of parameters equivalent to the electron spin analysis and has been successfully tested for show-case examples as the Xe 4d photoionisation [18].
3. 2p closed shell photoionisation 3.1. Argon From the experimental point of view the alignment method is the easiest approach to complete photoionisation experiments because it requires neither target polarisation nor spin-dependent measurements or coincidence experiments but simply angle-resolved electron spectroscopy. Fig. 1 shows a corresponding experimental setup with two electron time-of-flight spectrometers which allows the simultaneous angleresolved detection of all photo and Auger electron lines. In order to determine the alignment unambiguously, a L3MM Auger transition to the 1S0 final ionic state has to be chosen for the angular distribution measurement because the Auger angular distribution
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Fig. 1. Schematic setup of an angle-resolved photoemission experiment using synchrotron radiation as excitation source and time-of-flight as electron detection method.
asymmetry parameter bA is in this case directly proportional to the alignment parameter A20 which has to be a monotonous function of the photon energy because Ar 2p photoionisation exhibits no Cooper minimum. Fig. 2 shows the b-parameters for the Ar 2p photo lines and the L3MM ( 1S0) Auger lines along
with the partial wave intensities and relative phase shift between them [14]. Although these intensities and the corresponding phase shifts are still given with relatively large uncertainties, the smooth transition of the phase shift difference to the quantum defect difference at threshold is clearly demonstrated. This is
Fig. 2. Angular distribution and alignment parameters along with the derived dipole matrix elements and relative phase shifts of the outgoing es and ed partial photoelectron waves following 2p photoionization of argon. The experimental data are from fig. 10 of Ref. [14], whereas the dashed dotted lines show theoretical data obtained by the Hartree–Fock [32, 33] method. The solid curve represents RRPA calculations [34], whereas the dotted lines give semi-empirical values derived from the interpolation of the experimental data.
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Fig. 3. Angular distribution and alignment parameters along with the derived dipole matrix elements and relative phase shifts of the outgoing es and ed partial photoelectron waves following 2p photoionisation of magnesium. The experimental data are from Refs. [13] (large circles) and [35] (squares). The dashed dotted curves represent Hartree–Fock results of Ref. [36], whereas the solid b curve shows results obtained by the RRPA method [37]. Semi-empirical values derived from a critical evaluation of the experimental data in conjunction with theoretical cross sections are given by the dotted line. The quantum defect difference curve is obtained from spectroscopic measurements [21].
even more obvious for the relative intensities of the two partial waves being determined from the alignment parameter which correspond to the relative intensities of the l ⫹ 1 and l ⫺ 1 Rydberg series below threshold. From a qualitative point of view these results are quite convincing but from a quantitative point of view there still have to be made significant improvements in the future. 3.2. Magnesium The first application of the alignment method to other elements than the rare gases was performed with magnesium at a photon energy of 80 eV in the group of V. Schmidt [13]. The derived matrix elements and phase shifts were compared with theoretical data obtained using the Hartree–Fock method. Later on the same group performed a series of b and alignment measurements between 60 and 120 eV. These data were never converted to matrix elements with phase shifts because of missing partial cross sections. In order to extend the comparison with theory over a broader range of energies we used the absorption data of Henke [19] normalised to the
partial cross section at 80 eV for all cross section data above this energy. Below 80 eV the experimental data points are really scarce and we assumed a monotonous behaviour of the es partial wave toward threshold. Under this assumption we were able to derive the intensity of the ed partial wave from the alignment data. This method makes it possible to indirectly determine partial cross sections which are experimentally not easily accessible. This analysis reveals that there is a suppression of the ed wave due to a centrifugal barrier in the potential [20] giving rise to a small resonance just above threshold. Furthermore detailed experiments are necessary to corroborate these results. Fig. 3 shows the matrix elements and phases in comparison with theoretical values. The derived phase shift near threshold fits very well to the quantum defect difference obtained from spectroscopic data [21], even though the variation of this difference below threshold is somewhat peculiar.
4. 2p open shell photoionisation: oxygen The alignment method is, from the experimental
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Fig. 4. Schematic setup of a polarised target photoemission experiment using a hexapole magnet for polarisation of the atoms, and a hemispherical electron analyser for angle-resolved detection of the photoelectrons.
point of view, a very convenient method for complete photoionisaton experiments. However, it is limited to ionisation of inner-shells or final ionic states above the ground state because it requires a radiation or radiationless transition to probe the alignment of the final ionic state. Photoionisation to the ground state of the ion is therefore a bad candidate for this method. For the photoionisation of the ground state of open-shell atoms the method of polarised target photoionisation is best. It is particularly well suited for precise phase shift measurements because the change in the angular distribution of the photoelectron induced by atomic polarisation is phase-dependent. This change is also known as dichroism in the angular distribution, if one considers the intensity difference for different polarisation directions of photoelectrons emitted from polarised atoms. Thus, two synonyms could be used: ‘polarised target’ or ‘atomic dichroism’ method. In order to perform such an experiment, an open-shell atom has to be polarised and photoionised. Since open-shell atoms occur in nature either as molecules or metals, a special source for these atoms has to be used. This can be a discharge or radio frequency
source for the molecules or an oven for the evaporation of metals. Both types of experiments have been performed recently; the first one with atomic oxygen [22], and the second one with chromium [23]. Here, we will concentrate on the oxygen experiment. Fig. 4 shows a schematic experimental setup. It basically consists of a radio frequency source to produce the free oxygen atoms, and a hexapole magnet to polarise them. The polarisation will be kept up by a small guiding field which is also used to change the polarisation direction. The such prepared atoms are photoionised by photons from a undulator beamline at the BESSY synchrotron radiation facility in Berlin. The angular distribution of the emitted photoelectron being shown in Fig. 5 is finally measured by a hemispherical electron analyzer which is mounted on a goniometer that rotates it around the photon beam. In contrast to the angular distribution of unpolarised atoms, that of polarised atoms is tilted relative to the electric vector of the light. The tilt depends on the direction of the atomic polarisation which can be chosen either parallel or anti-parallel to the photon propagation vector k. The difference between the
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Fig. 5. Photoelectron spectrum of polarised oxygen atoms along with molecular lines from undissociated O2 (shaded areas) and enlargements for three multiplet lines showing their intensities for reverse polarisation directions (linear magnetic dichroism).
two corresponding angular patterns is the so-called magnetic dichroism in the angular distribution, a quantity that can be used to derive an additional photoionisation parameter b 0 which depends on the sine of the relative phase shift between the two outgoing es and ed waves. Fig. 6 shows the measured b and b 0 parameters together with the derived partial wave intensities and their relative phase shift. Using
the Cowan Code [24], the data points are compared with calculations on the Hartree–Fock level. The agreement is surprisingly good, only the phase shifts at higher energies seem to be constantly too low. The relatively good agreement on this comparatively low level of theoretical sophistication raises the question whether a ‘three-parameter’ model is suitable to describe the photoionisation of an open-shell system
Fig. 6. Angular distribution asymmetry parameters b and b 0 [38, 22] along with the derived dipole matrix elements and relative phase shifts compared with theoretical results (dashed lines) using the Cowan code [24]. The solid line represents the quantum defect difference of the ns and nd Rydberg series which converge to the 1S threshold [39]. Semi-empirical data are shown by dotted lines.
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like oxygen where the number of possible independent partial waves within a multiplet is much larger. Our approach to the problem, both semi-empirically and theoretically, assumes that the dichroism difference between the final ionic states is properly described by geometric coupling coefficients only. To prove this assumption, b 0 parameters for the 2D and 2P multiplets were derived from the 4S values and compared with experimental results obtained for the other multiplets at a certain photon energy [25]. Again, the agreement is rather good and shows that the ‘three-parameter’ model based on the LS coupling scheme is a suitable approximation of the valence ionisation of atomic oxygen. Now, heavier elements have to be considered in order to look for a possible breakdown of this model.
5. Xenon 4d photoionisation Xenon is a showcase for inner-shell photoionisation. It served as a test case for many theoretical efforts on different levels of sophistication (see e.g. [26]). As a closed-shell atom the ‘five-parameter’ model should be valid in any case but a description with less parameters would also be sufficient if the different parameter sets would be linearly dependent. This problem has been studied for a wider range of
photon energies in valence photoionisation of xeonon [11], but the hard test of theory was made at the xenon 4d subshell photoionisation, because it experiences all kinds of interactions: from shape resonances over Cooper minima to ‘‘giant resonance’’ enhanced interchannel coupling. Would theory be able to properly describe such a complex ionisation process, and how many parameters would really be necessary to derive the dynamic properties of the 4d photoionisation within a certain range of accuracy? Two approaches were followed to tackle this problem: the photo-Auger electron coincidence method, and the spin-resolved photo and Auger electron spectroscopy. The latter is shown schematically in Fig. 7. Both methods were concentrated on a photon energy near hn 94 eV [27, 28]. Only very recently, another photon energy of hn 130 eV was studied with the coincidence method [29]. The results were compared with Relativistic Random Phase Approximation (RRPA) calculations which generally provide five parameters [30]. However, the large error bars on the experimental side were a problem, especially concerning the relativistic phase shift between two partial waves with the same orbital angular momentum l and different total angular momentum j. From a theoretical point of view, there should be no significant phase shift caused by the spin–orbit interaction only, but experimentally there was no decisive
Fig. 7. Schematic setup of a spin-resolved photoemission experiment using a time-of-flight Mott detector for the spin-resolved photoelectron detection [12] and a transmission multilayer as l/4 plate for the generation of circularly polarised VUV light [40].
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proof for this assumption. There are two reasons for this uncertainty: first, the experimental error bars of data points obtained from highly differential measurements are usually larger than those from less sophisticated experiments and, second, the dependence of the relativistic phase shift on the experimental photoionisation parameters is quite sensitive and the solutions of the corresponding equation tend to be unstable near the true value. This makes fitting procedures a critical task and eventually gives values which deviate significantly from the correct ones. Furthermore, it causes large effective error bars. Therefore, the error bars in new experiments have to be reduced and the dependence of the phase shift on the measured data has to be studied systematically and carefully. Such studies were performed for both kinds of experiments. They gave clear evidence for a vanishing relativistic phase shift in a first approximation. Taking this into account the fits for all other parameters converge suddenly [28, 29] to values which are very close to the RRPA results with relaxation. Going even a step further, one can prove whether this remaining set of four parameters for each of the two spin–orbit components d3/2 and d5/2 may be reduced to three nonrelativistic parameters. The validity of this reduced parameter model, which is also known as the Cooper–Zare model, depends directly on the validity of LS coupling for the 4d photoionisation process. The comparison is encouraging. The simple three-parameter model reproduces the values of the five-parameter model, which in fact comprises ten parameters for 4d3/2 and 4d5/2 altogether, with a deviation of ^ 5%. An even better result is achieved if one takes into account the experimental 4d5/2/4d3/2 branching ratio for the derivation of the partial cross sections. In this case the introduction of the one additional parameter, the branching ratio r of the two fine-structure components of the final ionic state, makes it possible to derive even partial cross sections with reasonable accuracy. Formula 1 shows this modification of the original Cooper–Zare formula for the partial cross section. 9 2r > > > 1 ⫹ r = 4p 2 ahn 2R2l⫺1 ⫹ R2l⫹1 x 3 2 > > > s nll⫺1=2 ; 1⫹r
s nll⫹1=2
1
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with
r
sl⫹1=2 sl⫺1=2
Here Rl^1 is the radial dipole matrix element for the l ^ 1 partial waves and sl^1/2 is the partial cross section for the spin–orbit components l ^ 1/2 of the final ionic states. The different experimental data sets for s, b and the alignment parameter A20 available today makes it possible to derive these three parameters for a variety of photon energies between 70 and 200 eV [31]. The result is shown in Fig. 8 along with precision data points derived from the highly differential experiments. The agreement is surprising, but it also shows that scattering among the partial wave intensities is still large, which reflects the uncertainties in the alignment data [31]. Now and again, there is a need to obtain high-quality alignment data in order to compare them with the most recent results obtained with much more sophisticated methods. This is particularly urgent in the range of the Cooper minimum. It seems, however, that outside the Cooper minima the dynamics of the photoionisation of a large number of elements may be described properly by a relatively simple ‘(3 ⫹ 1)-parameter’ model. Exploration of the limits of this statement would be the aim of systematic experiments in the near future.
6. Conclusions, summary and outlook Finally, some words should be said about the theoretical plausibility of the results presented here, in particular about their interpretation in terms of partial LS coupling. Is there any simple explanation for why the continuum states are governed by LS coupling rules whereas the properties of the final ionic state are basically determined by jj-coupling? The answer is yes, this behaviour is an extension of the Rydbergstate behaviour regarding the decay of these states by spectator transitions [14]. The validity of the spectator model is based on the validity of jk-coupling, i.e. the core hole is considered to be coupled in the jjcoupling scheme whereas the excited electron couples its spin and angular momentum to this state following LS coupling rules. The j of the final ionic state couples with the angular momentum l of the photoelectron
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Fig. 8. Angular distribution asymmetry and alignment parameter b and A20 together with dipole matrix elements for the partial ep and ef wave photoelectron emission and their relative phase shift. The b-values are taken from Refs. [31] (open circles), [34] (open triangles), and [41] (open squares). The alignment parameters are from Refs. [31] (open circles) and [41] (open squares), the star points to the best A20 value compatible with the corresponding b value. The dipole matrix elements and phases derived from the higher differential measurements are taken from Refs. [27–29] (large filled squares), b and alignment values calculated in turn from these values are shown as large filled diamonds. The dasheddotted curve in part (b) represent Hartree–Fock calculations of Refs. [20, 32]. The solid curve in part (a) is a RRPA calculation from Ref. [42] whereas these curves in (c) and (d) show RRPA calculations of Ref. [30] and the dashed lines are RPA results taken from Ref. [43]. The dotted lines in all parts of the figure represent semi-empirical data obtained by critical evaluation of the experimental data.
first to an angular momentum k before the spin of the photoelectron is further coupled to the total angular momentum of the complete ionised system. The validity of this coupling scheme increases with the increasing principal quantum number n. This leads us to the conclusion that the jk-coupling scheme is also valid above threshold, which means that the photoelectron continuum behaviour is governed to a great extent by LS coupling, whereas the branching ratio between the two spin–orbit components of the final ionic state is influenced by relativistic effects. The suggested (3 ⫹ 1)-parameter model takes this separation into account, thus making it possible to parameterise the photoionisation process by a surprisingly simple model. In summary, one can say that complete photoionisation experiments or, in other words, angular momentum resolved photoelectron spectroscopy provides the ultimate insight in the dynamics of atomic photoionisation. There are now several complementary methods available to perform such
experiments with increasing accuracy. First results from highly differentiated experiments and their critical comparison with older data give strong evidence that the relatively simple modified Cooper–Zare or ‘(3 ⫹ 1)-parameter’ model is sufficient to describe the non-resonant photoionisation dynamics properly with an accuracy of better than 10% for many elements outside of Cooper minima. In the near future, it opens the possibility to generate photionisation data for a variety of elements from a limited number of experiments, covering nearly all possible properties of the photoelectron over large energy ranges.
Acknowledgements The author would like to thank Professor V. Schmidt for providing his latest data prior to publication, and Professors S.T. Manson and N.A. Cherepkov for their valuable discussions. This work was
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Journal of Electron Spectroscopy and Related Phenomena 96 (1998) 105–115
Complete photoionisation experiments 1 Uwe Becker Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany Received 11 May 1998; accepted 15 June 1998
Abstract Experiments in atomic photoionisation are considered complete in the sense that both the amplitudes and phases of the corresponding transition matrix elements can be determined. Such experiments have recently made remarkable progress and provided a wealth of new information in this field. The researchers have employed a variety of complementary techniques such as polarised radiation and targets, spin-resolved detection of electrons and coincident detection of two electrons in cases of double photoionisation. This rapidly developing field is reviewed in a selection of show-case examples for each method. The results are compared with different calculations and critically analysed with respect to the validity of the theoretical models employed for these calculations. 䉷 1998 Elsevier Science B.V. All rights reserved. Keywords: Atomic photoionisation; Cooper–Zare model
1. Introduction Photoionisation of matter was the first in a series of experimental discoveries followed by corresponding theoretical explanations which led to the foundation of quantum physics [1, 2]. It was in fact Einstein’s interpretation of the photoelectric effect [3] that proved the concept of energy quantisation by Planck [4] to be generally valid. Despite this early success it took more than 40 years until photoelectron spectroscopy became a growing field of scientific research and, finally, an analytical tool for industrial research [5, 6]. This was because the early efforts to record line spectra in measurements of the photoelectric effect were hampered by experimental difficulties such as insufficient energy resolution. Only after the pioneering work of the Uppsala group did photoelectron spectroscopy become a powerful tool for studying 1 Dedicated to Professor H. Kleinpoppen, Stirling, for his 70th birthday.
the electronic structure of matter and its chemical composition [7]. Electron Spectroscopy for Chemical Analysis (ESCA) was a synonym for an entire field of research in atomic, molecular (gas phase) and solid state physics. This field dealt with the orbital structure of electrons in free atoms and molecules and the band structure of electronic states in solids and surfaces. The main target of these studies was to analyse the enormous variety of elements and compounds in their various forms with respect to their electronic structure and chemical composition in the ground state. The ground state properties are still the main subject of photoelectron spectroscopy; some kind of subfield, however, started to develop when some researchers concentrated more on the final state of the photoionisation process and, especially, its dynamics. Why is this of particular interest, and what can we learn from the underlying energydependent behaviour of the photoelectron? The answer to this question is closely related to the socalled complete scattering experiments to which
0368-2048/98/$ - see front matter 䉷 1998 Elsevier Science B.V. All rights reserved. PII: S0368-204 8(98)00226-6
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photoionisation as a half-scattering process belongs. The purpose of complete scattering experiments is to derive complete information on the scattering process in a quantum-mechanical sense, which means, the total number of transition matrix elements with corresponding phases necessary to describe a particular scattering process. In the case of photoionisation this is a limited number of transition matrix elements which are for intermediate energies dipole matrix elements. Therefore, complete photoionisation experiments are concerned with the determination of characteristic properties of the photoionisation process such as partial cross section, angular distribution, photoelectron spin polarisation and different types of dichroism in the angular distribution of photoelectrons. These properties may be used to derive the number of parameters necessary to describe the photoionisation process within a certain theoretical model in a quantum-mechanically complete sense. The purpose of the experimental studies in this field is to figure out which theoretical model is necessary for a complete description within certain limits of accuracy. To visualise this problem let us take the noble gases or, more generally, all closedshell atoms as an example. In the most general case of relativistic treatment of photoionisation there are three outgoing partial waves with two relative phases for each spin–orbit component of the remaining final ionic state: 1 hn ⫹ A0 ! A⫹ J ⫹ e l ^ 1; 2 j
with J ⫺ 1 ⱕ j ⱕ J ⫹ 1. It means that subshell photoionisation of an openshell atom requires ten parameters to be completely described. The situation is still worse for closed-shell atoms where the number of parameters is even higher and, in some cases, exceeds the possibilities of experimental determination. However, the situation gets better if one considers possibilities to reduce the necessary number of parameters by relating some of them to each other. The most powerful frame for such a reduction is the LS coupling scheme. It is based on the existence of a total orbital angular momentum L and a total spin S, and it enables us to reduce the number of parameters necessary to describe the subshell photoionisation of both closed-shell and open-shell atoms to three: two transition matrix
elements and one relative phase shift. This extreme reduction results from the fact that the angular momentum of the ionising photon is 1 and may be coupled to the emitted electron either to l ⫹ 1 or l ⫺ 1 because there are no longitudinal polarised photons with a vanishing projection of the angular momentum. This extremely simple but powerful model is known as the Cooper–Zare model. In the literature, it is considered as a first step approximation to a qualitative description of photoionisation without, however, the capacity to provide quantitatively acceptable results. The main reason for this assessment of the so-called three-parameter model is the fact that the simplest photoionisation parameter, the partial cross section, is mostly just poorly described by this model. Therefore, the three-parameter model does not seem to be an appropriate means to satisfactorily describe the more complex photoionisation parameters as, for example, the angular distribution and spin parameters. But there is no stringent verification for this assumption because the experimental data, particularly those of the spin polarisation, are very rare and, for technical reasons, basically limited to photon energies below 25 eV. Only very recently, new spin polarisation measurements at higher photon energies have become feasible. In addition, complementary methods such as the use of polarised targets for dichroism measurements were introduced and provided first results for open-shell atoms. The purpose of this review is to critically analyse the present data situation with respect to the validity of the different theoretical models employed to describe the photoionisation process. The review is organised in the following way. First, it describes the different methods of how to determine the various photoionisation parameters. The following sections describe selected show-case examples for each method along with the most recent experimental results. Finally, conclusions will be drawn with regard to the validity of a certain theoretical model putting special emphasis on a newly proposed model, the so-called extended threeparameter model or (3 ⫹ 1) model.
2. Determination of photoionisation parameters Photoionisation experiments with randomly oriented targets yield two independent parameters,
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the intensity as a measure of the partial cross section s and the angular distribution of the emitted photoelectrons [8]. The angular distribution asymmetry parameter b depends on the phase between the two outgoing electron waves but also on their intensity ratio [9]. Therefore, a third parameter is required in order to perform a complete partial wave analysis in terms of the three-parameter model. This third parameter may be obtained by several methods, all of which rely on the preparation of the incident reaction partners or the analysis of the outgoing reaction partners if one considers the photoionisation process as a collision reaction hn ⫹ A ! A⫹ ⫹ e⫺ . On the side of the incoming particles, the photon is always specified by its angular momentum. The use of linearly or circularly polarised light may have certain advantages for particular measurements; however, even unpolarised radiation is sufficiently specified by the propagation vector k. In contrast to the photon, the target atom A is usually randomly orientated with respect to the magnetic subquantum number m. In order to specify this quantum number it is necessary to polarise the target atom. The change in atomic polarisation changes the photoelectron angular distribution in a certain way, yielding additional information on the photoionisation process. The deviation of the angular distribution pattern may be used to derive the necessary third parameter for a complete photoionisation experiment [10]. Different geometries regarding the atomic and light polarisation make it possible to derive different independent parameters. The polarisation of the atoms may be achieved by use of inhomogeneous magnetic fields or optical excitation/ pumping. Both methods have been successfully used to produce polarised targets. Regarding the analysis of the outgoing particles the most successful method is the determination of the spin polarisation of the outgoing electrons [11]. It allows the extraction of up to three additional parameters because the spin of the electron has three independent components. A spin-sensitive detection method is required to measure the spin polarisation of the ejected electron. The most commonly used method is still based on Mottscattering. Mott detectors are advanced instruments, and just very recently a multi-channel method for simultaneous detection of energy-resolved electron spectra, the electron time-of-flight method, has been combined with such a detector. The so-called
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Mott–TOF detector was applied successfully in spin-resolved measurements of photo- as well as Auger electrons [12]. Complementary to the spin analysis of the electron in the outgoing channel is the analysis of the polarisation of the remaining ion. It may be derived from the angular distribution of Auger electrons [13, 14] as well as fluorescence radiation [15]. In both cases one obtains the alignment of the remaining ion. If one uses circularly polarised light for primary photoionisation, the result will be an oriented final ionic state. This orientation may be determined either by the spin polarisation of subsequently emitted Auger electrons [12] or the circular polarisation of fluorescence photons [16]. In the framework of the two-step model all these processes yield the same information. The ‘alignment’ method was used by different groups as a first step towards complete photoionisation experiments [17]. Alignment and orientation refer always to the polarisation of the incoming light; they only contain information on the partial wave intensities of the outgoing photoelectron wave, the phase information is lost. To keep this information, a coincidence experiment between photo and Auger electrons has to be performed. This method yields a number of parameters equivalent to the electron spin analysis and has been successfully tested for show-case examples as the Xe 4d photoionisation [18].
3. 2p closed shell photoionisation 3.1. Argon From the experimental point of view the alignment method is the easiest approach to complete photoionisation experiments because it requires neither target polarisation nor spin-dependent measurements or coincidence experiments but simply angle-resolved electron spectroscopy. Fig. 1 shows a corresponding experimental setup with two electron time-of-flight spectrometers which allows the simultaneous angleresolved detection of all photo and Auger electron lines. In order to determine the alignment unambiguously, a L3MM Auger transition to the 1S0 final ionic state has to be chosen for the angular distribution measurement because the Auger angular distribution
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Fig. 1. Schematic setup of an angle-resolved photoemission experiment using synchrotron radiation as excitation source and time-of-flight as electron detection method.
asymmetry parameter bA is in this case directly proportional to the alignment parameter A20 which has to be a monotonous function of the photon energy because Ar 2p photoionisation exhibits no Cooper minimum. Fig. 2 shows the b-parameters for the Ar 2p photo lines and the L3MM ( 1S0) Auger lines along
with the partial wave intensities and relative phase shift between them [14]. Although these intensities and the corresponding phase shifts are still given with relatively large uncertainties, the smooth transition of the phase shift difference to the quantum defect difference at threshold is clearly demonstrated. This is
Fig. 2. Angular distribution and alignment parameters along with the derived dipole matrix elements and relative phase shifts of the outgoing es and ed partial photoelectron waves following 2p photoionization of argon. The experimental data are from fig. 10 of Ref. [14], whereas the dashed dotted lines show theoretical data obtained by the Hartree–Fock [32, 33] method. The solid curve represents RRPA calculations [34], whereas the dotted lines give semi-empirical values derived from the interpolation of the experimental data.
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Fig. 3. Angular distribution and alignment parameters along with the derived dipole matrix elements and relative phase shifts of the outgoing es and ed partial photoelectron waves following 2p photoionisation of magnesium. The experimental data are from Refs. [13] (large circles) and [35] (squares). The dashed dotted curves represent Hartree–Fock results of Ref. [36], whereas the solid b curve shows results obtained by the RRPA method [37]. Semi-empirical values derived from a critical evaluation of the experimental data in conjunction with theoretical cross sections are given by the dotted line. The quantum defect difference curve is obtained from spectroscopic measurements [21].
even more obvious for the relative intensities of the two partial waves being determined from the alignment parameter which correspond to the relative intensities of the l ⫹ 1 and l ⫺ 1 Rydberg series below threshold. From a qualitative point of view these results are quite convincing but from a quantitative point of view there still have to be made significant improvements in the future. 3.2. Magnesium The first application of the alignment method to other elements than the rare gases was performed with magnesium at a photon energy of 80 eV in the group of V. Schmidt [13]. The derived matrix elements and phase shifts were compared with theoretical data obtained using the Hartree–Fock method. Later on the same group performed a series of b and alignment measurements between 60 and 120 eV. These data were never converted to matrix elements with phase shifts because of missing partial cross sections. In order to extend the comparison with theory over a broader range of energies we used the absorption data of Henke [19] normalised to the
partial cross section at 80 eV for all cross section data above this energy. Below 80 eV the experimental data points are really scarce and we assumed a monotonous behaviour of the es partial wave toward threshold. Under this assumption we were able to derive the intensity of the ed partial wave from the alignment data. This method makes it possible to indirectly determine partial cross sections which are experimentally not easily accessible. This analysis reveals that there is a suppression of the ed wave due to a centrifugal barrier in the potential [20] giving rise to a small resonance just above threshold. Furthermore detailed experiments are necessary to corroborate these results. Fig. 3 shows the matrix elements and phases in comparison with theoretical values. The derived phase shift near threshold fits very well to the quantum defect difference obtained from spectroscopic data [21], even though the variation of this difference below threshold is somewhat peculiar.
4. 2p open shell photoionisation: oxygen The alignment method is, from the experimental
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Fig. 4. Schematic setup of a polarised target photoemission experiment using a hexapole magnet for polarisation of the atoms, and a hemispherical electron analyser for angle-resolved detection of the photoelectrons.
point of view, a very convenient method for complete photoionisaton experiments. However, it is limited to ionisation of inner-shells or final ionic states above the ground state because it requires a radiation or radiationless transition to probe the alignment of the final ionic state. Photoionisation to the ground state of the ion is therefore a bad candidate for this method. For the photoionisation of the ground state of open-shell atoms the method of polarised target photoionisation is best. It is particularly well suited for precise phase shift measurements because the change in the angular distribution of the photoelectron induced by atomic polarisation is phase-dependent. This change is also known as dichroism in the angular distribution, if one considers the intensity difference for different polarisation directions of photoelectrons emitted from polarised atoms. Thus, two synonyms could be used: ‘polarised target’ or ‘atomic dichroism’ method. In order to perform such an experiment, an open-shell atom has to be polarised and photoionised. Since open-shell atoms occur in nature either as molecules or metals, a special source for these atoms has to be used. This can be a discharge or radio frequency
source for the molecules or an oven for the evaporation of metals. Both types of experiments have been performed recently; the first one with atomic oxygen [22], and the second one with chromium [23]. Here, we will concentrate on the oxygen experiment. Fig. 4 shows a schematic experimental setup. It basically consists of a radio frequency source to produce the free oxygen atoms, and a hexapole magnet to polarise them. The polarisation will be kept up by a small guiding field which is also used to change the polarisation direction. The such prepared atoms are photoionised by photons from a undulator beamline at the BESSY synchrotron radiation facility in Berlin. The angular distribution of the emitted photoelectron being shown in Fig. 5 is finally measured by a hemispherical electron analyzer which is mounted on a goniometer that rotates it around the photon beam. In contrast to the angular distribution of unpolarised atoms, that of polarised atoms is tilted relative to the electric vector of the light. The tilt depends on the direction of the atomic polarisation which can be chosen either parallel or anti-parallel to the photon propagation vector k. The difference between the
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Fig. 5. Photoelectron spectrum of polarised oxygen atoms along with molecular lines from undissociated O2 (shaded areas) and enlargements for three multiplet lines showing their intensities for reverse polarisation directions (linear magnetic dichroism).
two corresponding angular patterns is the so-called magnetic dichroism in the angular distribution, a quantity that can be used to derive an additional photoionisation parameter b 0 which depends on the sine of the relative phase shift between the two outgoing es and ed waves. Fig. 6 shows the measured b and b 0 parameters together with the derived partial wave intensities and their relative phase shift. Using
the Cowan Code [24], the data points are compared with calculations on the Hartree–Fock level. The agreement is surprisingly good, only the phase shifts at higher energies seem to be constantly too low. The relatively good agreement on this comparatively low level of theoretical sophistication raises the question whether a ‘three-parameter’ model is suitable to describe the photoionisation of an open-shell system
Fig. 6. Angular distribution asymmetry parameters b and b 0 [38, 22] along with the derived dipole matrix elements and relative phase shifts compared with theoretical results (dashed lines) using the Cowan code [24]. The solid line represents the quantum defect difference of the ns and nd Rydberg series which converge to the 1S threshold [39]. Semi-empirical data are shown by dotted lines.
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like oxygen where the number of possible independent partial waves within a multiplet is much larger. Our approach to the problem, both semi-empirically and theoretically, assumes that the dichroism difference between the final ionic states is properly described by geometric coupling coefficients only. To prove this assumption, b 0 parameters for the 2D and 2P multiplets were derived from the 4S values and compared with experimental results obtained for the other multiplets at a certain photon energy [25]. Again, the agreement is rather good and shows that the ‘three-parameter’ model based on the LS coupling scheme is a suitable approximation of the valence ionisation of atomic oxygen. Now, heavier elements have to be considered in order to look for a possible breakdown of this model.
5. Xenon 4d photoionisation Xenon is a showcase for inner-shell photoionisation. It served as a test case for many theoretical efforts on different levels of sophistication (see e.g. [26]). As a closed-shell atom the ‘five-parameter’ model should be valid in any case but a description with less parameters would also be sufficient if the different parameter sets would be linearly dependent. This problem has been studied for a wider range of
photon energies in valence photoionisation of xeonon [11], but the hard test of theory was made at the xenon 4d subshell photoionisation, because it experiences all kinds of interactions: from shape resonances over Cooper minima to ‘‘giant resonance’’ enhanced interchannel coupling. Would theory be able to properly describe such a complex ionisation process, and how many parameters would really be necessary to derive the dynamic properties of the 4d photoionisation within a certain range of accuracy? Two approaches were followed to tackle this problem: the photo-Auger electron coincidence method, and the spin-resolved photo and Auger electron spectroscopy. The latter is shown schematically in Fig. 7. Both methods were concentrated on a photon energy near hn 94 eV [27, 28]. Only very recently, another photon energy of hn 130 eV was studied with the coincidence method [29]. The results were compared with Relativistic Random Phase Approximation (RRPA) calculations which generally provide five parameters [30]. However, the large error bars on the experimental side were a problem, especially concerning the relativistic phase shift between two partial waves with the same orbital angular momentum l and different total angular momentum j. From a theoretical point of view, there should be no significant phase shift caused by the spin–orbit interaction only, but experimentally there was no decisive
Fig. 7. Schematic setup of a spin-resolved photoemission experiment using a time-of-flight Mott detector for the spin-resolved photoelectron detection [12] and a transmission multilayer as l/4 plate for the generation of circularly polarised VUV light [40].
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proof for this assumption. There are two reasons for this uncertainty: first, the experimental error bars of data points obtained from highly differential measurements are usually larger than those from less sophisticated experiments and, second, the dependence of the relativistic phase shift on the experimental photoionisation parameters is quite sensitive and the solutions of the corresponding equation tend to be unstable near the true value. This makes fitting procedures a critical task and eventually gives values which deviate significantly from the correct ones. Furthermore, it causes large effective error bars. Therefore, the error bars in new experiments have to be reduced and the dependence of the phase shift on the measured data has to be studied systematically and carefully. Such studies were performed for both kinds of experiments. They gave clear evidence for a vanishing relativistic phase shift in a first approximation. Taking this into account the fits for all other parameters converge suddenly [28, 29] to values which are very close to the RRPA results with relaxation. Going even a step further, one can prove whether this remaining set of four parameters for each of the two spin–orbit components d3/2 and d5/2 may be reduced to three nonrelativistic parameters. The validity of this reduced parameter model, which is also known as the Cooper–Zare model, depends directly on the validity of LS coupling for the 4d photoionisation process. The comparison is encouraging. The simple three-parameter model reproduces the values of the five-parameter model, which in fact comprises ten parameters for 4d3/2 and 4d5/2 altogether, with a deviation of ^ 5%. An even better result is achieved if one takes into account the experimental 4d5/2/4d3/2 branching ratio for the derivation of the partial cross sections. In this case the introduction of the one additional parameter, the branching ratio r of the two fine-structure components of the final ionic state, makes it possible to derive even partial cross sections with reasonable accuracy. Formula 1 shows this modification of the original Cooper–Zare formula for the partial cross section. 9 2r > > > 1 ⫹ r = 4p 2 ahn 2R2l⫺1 ⫹ R2l⫹1 x 3 2 > > > s nll⫺1=2 ; 1⫹r
s nll⫹1=2
1
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with
r
sl⫹1=2 sl⫺1=2
Here Rl^1 is the radial dipole matrix element for the l ^ 1 partial waves and sl^1/2 is the partial cross section for the spin–orbit components l ^ 1/2 of the final ionic states. The different experimental data sets for s, b and the alignment parameter A20 available today makes it possible to derive these three parameters for a variety of photon energies between 70 and 200 eV [31]. The result is shown in Fig. 8 along with precision data points derived from the highly differential experiments. The agreement is surprising, but it also shows that scattering among the partial wave intensities is still large, which reflects the uncertainties in the alignment data [31]. Now and again, there is a need to obtain high-quality alignment data in order to compare them with the most recent results obtained with much more sophisticated methods. This is particularly urgent in the range of the Cooper minimum. It seems, however, that outside the Cooper minima the dynamics of the photoionisation of a large number of elements may be described properly by a relatively simple ‘(3 ⫹ 1)-parameter’ model. Exploration of the limits of this statement would be the aim of systematic experiments in the near future.
6. Conclusions, summary and outlook Finally, some words should be said about the theoretical plausibility of the results presented here, in particular about their interpretation in terms of partial LS coupling. Is there any simple explanation for why the continuum states are governed by LS coupling rules whereas the properties of the final ionic state are basically determined by jj-coupling? The answer is yes, this behaviour is an extension of the Rydbergstate behaviour regarding the decay of these states by spectator transitions [14]. The validity of the spectator model is based on the validity of jk-coupling, i.e. the core hole is considered to be coupled in the jjcoupling scheme whereas the excited electron couples its spin and angular momentum to this state following LS coupling rules. The j of the final ionic state couples with the angular momentum l of the photoelectron
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Fig. 8. Angular distribution asymmetry and alignment parameter b and A20 together with dipole matrix elements for the partial ep and ef wave photoelectron emission and their relative phase shift. The b-values are taken from Refs. [31] (open circles), [34] (open triangles), and [41] (open squares). The alignment parameters are from Refs. [31] (open circles) and [41] (open squares), the star points to the best A20 value compatible with the corresponding b value. The dipole matrix elements and phases derived from the higher differential measurements are taken from Refs. [27–29] (large filled squares), b and alignment values calculated in turn from these values are shown as large filled diamonds. The dasheddotted curve in part (b) represent Hartree–Fock calculations of Refs. [20, 32]. The solid curve in part (a) is a RRPA calculation from Ref. [42] whereas these curves in (c) and (d) show RRPA calculations of Ref. [30] and the dashed lines are RPA results taken from Ref. [43]. The dotted lines in all parts of the figure represent semi-empirical data obtained by critical evaluation of the experimental data.
first to an angular momentum k before the spin of the photoelectron is further coupled to the total angular momentum of the complete ionised system. The validity of this coupling scheme increases with the increasing principal quantum number n. This leads us to the conclusion that the jk-coupling scheme is also valid above threshold, which means that the photoelectron continuum behaviour is governed to a great extent by LS coupling, whereas the branching ratio between the two spin–orbit components of the final ionic state is influenced by relativistic effects. The suggested (3 ⫹ 1)-parameter model takes this separation into account, thus making it possible to parameterise the photoionisation process by a surprisingly simple model. In summary, one can say that complete photoionisation experiments or, in other words, angular momentum resolved photoelectron spectroscopy provides the ultimate insight in the dynamics of atomic photoionisation. There are now several complementary methods available to perform such
experiments with increasing accuracy. First results from highly differentiated experiments and their critical comparison with older data give strong evidence that the relatively simple modified Cooper–Zare or ‘(3 ⫹ 1)-parameter’ model is sufficient to describe the non-resonant photoionisation dynamics properly with an accuracy of better than 10% for many elements outside of Cooper minima. In the near future, it opens the possibility to generate photionisation data for a variety of elements from a limited number of experiments, covering nearly all possible properties of the photoelectron over large energy ranges.
Acknowledgements The author would like to thank Professor V. Schmidt for providing his latest data prior to publication, and Professors S.T. Manson and N.A. Cherepkov for their valuable discussions. This work was
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