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Physics with the new synchrotron radiation sources
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Solid State Communications,Vol. 102, No. 2-3, pp. 199-205, 1997 @ 1997 Elsevier Science lad Printed in Great Britain. All rights reserved 0038-1098/97 $17.00+.00
Pergamon
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PHYSICS WITH THE NEW SYNCHROTRON RADIATION SOURCES M. Altarelli European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France Electron or positron storage rings are the most powerful sources of X-ray photons available to experimentalists interested in spectroscopic and structural studies of condensed matter. The most recent generation of synchrotron radiation sources is characterized by an increase of several orders of magnitude in brilliance. The potential opened up by the new sources is exemplified by recent experiments on the dynamical properties of disordered systems, on high pressure physics and by experiments exploiting the spatial coherence of undulator beams. © 1997 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION In the last few decades the use of synchrotron radiation has revolutionized X-ray physics and its applications to materials science, crystallography, spectroscopy and other fields of applied and fundamental significance. This impressive record of scientific achievements made possible by synchrotron radiation has stimulated considerable effort in the design and optimization of dedicated storage rings. The source brilliance is the main figure of merit for many experimental techniques based on synchrotron radiation. Brilliance is the phase space density of photons and is expressed, for a given photon energy, as the number of photons emitted in the unit time by the unit source area in the unit solid angle in a relative bandwidth of 10 -3 and is conventionally measured in photons/s/mm2/mrad2/0.1% BW. The most recent generation of sources is characterized by two features: one is the very low emittance, generally -<10nmrad in the horizontal plane (this quantity expresses the size and the angular divergence of the particle beam circulating in the ring); the other is the extensive use of insertion devices (wigglers and especially undulators) as radiation sources. The combined effect of the low emittance, corresponding to a reduced source size and collimation and of insertion devices, with improved radiation properties, results in an increase of many orders of magnitude in brilliance of the new sources with respect to earlier ones. The brilliance of hard X-ray undulators at the 6 GeV European Synchrotron Radiation Facility (ESRF) in Grenoble exceeds 1019photons/s/mm2/mrad2/0.1% BW and this value
will soon be reached at the 7 GeV Advanced Photon Source (APS) in Argonne National Laboratory as well. Similar results for soft X-rays were obtained at lower energy machines such as the 2 GeV Advanced Light Source (AI_~) in Berkeley and Elettra in Trieste. The purpose of the present article is to describe a few experiments performed at the new sources in order to provide examples of the new possibilities they open up in condensed matter physics. 2. INSERTION DEVICES In the new generation of synchrotron sources the emission takes place not only from the bending magnets but mostly from the so-called insertion devices [1] (undulators or wigglers) installed in straight sections of the storage rings. An insertion device consists in an array of magnets producing a d.c. vertical magnetic field with a sinusoidal position dependence in the direction of electron motion (see Fig. 1). Insertion devices are classified according to the value of the parameter K = c~-3",
(1)
where ol is the maximum angle of deviation from the straight trajectory (see Fig. 1) and 3, is the relativistic parameter given by the electron energy in the ring divided by the electron rest energy (3, 2 1957 for 1 GeV electrons). It turns out that 1/3' is the characteristic angular opening of synchrotron radiation emitted by orbiting relativistic electrons. Therefore, when K >> 1 (the wiggler regime) an observer along the axis of the device receives short
199
NEW SYNCHROTRON RADIATION SOURCES
200
zI
(
=.e-
_
# # I*
Ao
,I
Fig. 1. Schematic view of the magnetic structure of an insertion device. Also shown are the maximum angle of deviation ~x of the electron trajectory and the radiation cone, of opening 3,-1, at one particular point along the trajectory. radiation pulses from a few points of the electron trajectory. The spectrum is correspondingly broad in frequency space, containing all frequencies up to some multiple of a critical energy o0, = 3c3,3/(2R), where R is the radius of curvature at the emission point and c is the speed of light. Wigglers provide therefore a spectrum analogous to that of bending magnets, although more intense and possibly with a higher critical energy. In the undulator regime (K ~ 1), on the other hand, the whole trajectory in the device is visible by the observer. The observation time encompasses the N periods of sinusoidal motion and results in a spectrum of lines, with a characteristic wavelength spread A~I~,~ l/N, centered around the fundamental wavelength X = 2-~ (1 + K2/2)
(2)
and its odd harmonics ~/n, n = 1, 3,.... In equation (2), X0 is the period of the device. Roughly speaking, one can say that, as a result of interference between the fields emitted from each point of the trajectory, the emitted power of the undulator is concentrated in wavelength in narrow intervals around the harmonics and in space in a cone of aperture 1/(3,x/~), thus enhancing the brilliance considerably. It is also important to notice that the fundamental wavelength is a function of K, which can be modified, for a given device, by varying the distance between the upper and lower sets of magnetic poles, thereby varying the peak field in the device and the angle ~x. 3. INELASTIC SCATFERING In this section recent spectacular advances in the inelastic scattering of hard and soft X-rays made possible by the high brilliance of undulator sources are described. The inelastic scattering beamline at ESRF [2] combines the high-resolution properties of high-index backscattering crystal monochromators [3] with the high photon flux in a 40/~rad cone to obtain a beam of
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-109 photons s -1 with an energy resolution of a few meV around a photon energy of 17.794 keV (when exploiting the Si (999) backreflection), focused to a spot size of 150 × 350#m 2. The photons are then scattered on the sample and analyzed in energy with the same backrefiection on a Si analyzer at a slightly different temperature (with a step of 0.015 K, corresponding to a Aho0 of 0.7 meV). X-ray scattering, in the non-resonant regime, probes the spectrum of electronic density fluctuations [4], S(k, ¢0), comprising collective excitations (phonons, plasmons) and single-particle excitations. If the energy resolution is in the meV range, the investigation of phonon modes by inelastic X-ray scattering becomes possible [5]. It is important to realize that the use of X-rays allows to access regions in (k, o0) space not accessible by neutrons, because of the very different energy-momentum relationship of the massless photon (o0 = ck) and of the neutron [o0 h2k2/(21~n)]. Denoting by o01 and kl the incoming particle frequency and momentum, the accessible (k, o0) region for X-rays is bound below the o0 = ck and the o0 -- 2o01 - k c line; for neutrons, on the other hand, the accessible region lies below the parabola o0=o01-h(k-kl)212Mn. In particular, this prevents the neutron scattering study of modes with high energy transfer at low momenta, such as acoustic modes in systems with a high speed of sound, more precisely, a speed exceeding that of the neutron (about 2200ms -1 for thermal neutrons with boo1 ~ 25 meV). In crystalline solids, this is alleviated because the momentum of an excitation is defined up to a reciprocal lattice vector and neutron scattering can become allowed in one of the higher Brillouin zones. For non-periodic structures (liquids, glasses, amorphous materials), on the other hand, the use of X-ray scattering is truly significant, as it allows to enlarge the range of accessible frequency and momenta. In a study of the dispersion of sound modes in liquid water with X-ray scattering [2], data were collected in the region of k between 4 and 14 nm-X and in a frequency range from 0 to 30 meV. These results settle a controversy, by establishing the existence of "fast sound" at higher momenta (Fig. 2), with a sound velocity of 3200 m s -l, approximately double of that measured in the hydrodynamic (long wavelength) limit and indeed very close to the speed of sound in ice. In fact the comparison of phonons in ice and water is quite instructive [6], as it reveals a strong analogy even concerning the damping of the modes (the phonon lifetimes) (Fig. 3). The S(k, o0) in two glassy systems, glycerol and LiCI:6H20 was also recently investigated [7] and demonstrated the existence of well defined propagating excitations with linear dispersion in the 2 - 8 n m -t =
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NEW SYNCHROTRON RADIATION SOURCES
30. ~ " 25.
~ I"" ..~.......
15
experiment following monochromatic excitation with photons of energy hwl, on the exciting frequency wl. This can be quantitatively explained [9-11] by describing the coupling of excitation and emission processes in a single inelastic scattering event described by a KramersHeisenherg type of formula [12]
5~ I0
.
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201
•
/(hw2) ~ Zf
q ( nrn"1 ) Fig. 2. Dispersion relation of sound modes in water, as obtained by X-ray inelastic scattering [2]. region, i.e. up to scales of distance comparable to the interparticle separation. Very interesting advances in soft X-ray resonant inelastic scattering and, more generally, in soft X-ray
emission spectroscopies have also recently been reported. The basic idea is of course not new: soft X-ray fluorescence is an obvious way to probe the valence band density of states which has the advantage, in comparison to alternative techniques such as photoemission, of being bulk-sensitive and insensitive to charging effects. The resolution is in principle limited only by the core hole lifetime broadening, which is acceptable for shallow core levels giving rise to emission in the soft X-ray range (typically 100-1500 eV). In practice, the real problem is that soft X-ray emission is a very inefficient process and its successful use requires a highly brilliant source and an efficient spectrometer [8]. This is the reason why, after pioneering work at the National Synchrotron Light Source in Brookhaven [9], more recently interesting results have been obtained at the ALS in Berkeley. A beamline based on a 5 cm period undulator, operating in the 100 eV to 1500 eV energy range delivers 1012 photons s -1 in a 0.1% bandwidth on a 100 #m spot [8]. The use of resonant fluorescence, or resonant inelastic scattering, as a probe of band structure is based on the observation of a distinct dependence of the emitted spectrum I(I~o~2), measured in a photon-in, photon-out
6050. v
40~ 3020~ 10J
ol
2 4 6 8 10 12 14 q (nm "1 )
Fig. 3. Damping parameter r(q) as a function of momentum q for sound modes in water (after [2]).
× ~(Er + h002 - Ei - l~lcol).
(3)
In equation (3), [i) is the ground state of the system, with energy Ei, [m) is one of a complete set of intermediate states to be summed over, which have a hole in a core state and an electron in a band state above the Fermi level, and I f ) is the final state, which consists of an electron and a hole in the valence band. The delta function enforces energy conservation in the scattering process. If electron-core hole interactions are neglected and appropriate Bloch functions arc inserted in equation (3), a crystal momentum conservation rule follows, k l - k 2 ----k e - k h T G, and since in the wavelength range under consideration the photon wavevector is very small compared to the size of the Brillouin zone, ke-kh +G=
0.
(4)
Furthermore, the energy denominator in equation (3) resonantly enhances intermediate states satisfying E m ~ E i + h w 1 and for a core level with a lifetime broadening comparable to the resolution of the apparatus, this correspond to an approximate energy conservation in the intermediate state. This approximate energy selectivity, together with the approximate crystal momentum conservation, equation (4), explains the dependence of the emission spectrum from the incoming energy and at the same time allows band mapping by tracking the displacement of features in the emission spectrum as the incoming energy varies. This methodology has been tested successfully on diamond [9], graphite [10] and boron nitride [11], by comparison with state-of-the-art band structure calculations. Besides the experiments performed on these well known systems, experiments have been reported in more intriguing cases, such as the C60 fullerene crystal [13], or in ferromagnetic transition metal systems in which the exciting photon beam is circularly polarized, to observe the dependence of the emission spectrum from the sense of polarization (circular dichroism in the emission spectrum) [14]. 4. HIGH-PRESSURE PHYSICS The investigation of the properties of condensed matter under pressure exceeding atmospheric pressure
202
NEW SYNCHROTRON RADIATION SOURCES
by large factors (typically 105-106 ) is of interest to various disciplines of science. Geophysicists investigate the phase diagram of materials under conditions found at the center of the earth, where the pressure exceeds 3 × 106 bar (3 Mbar or 300 GPa). Planet physicists know that even larger pressures (of the order of tens of Mbar) are encountered at the center of large planets such as Saturn or Jupiter, which are mostly composed of light elements such as H and He. The solid state physicist is more generally interested in the possibility, offered by high-pressure technology, to study how electronic properties of materials are affected when the pressure is increased. The overlap of wavefunctions from neighbouring atoms is expected intuitively to increase with pressure, thus increasing the bandwidth, so that an insulator-to-metal transition should occur at high pressure in most insulating materials. Such theoretical predictions are best carried out for simple solids such as hydrogen. The first prediction of the metallization of H at high pressures was made by Wigner and Huntington in 1935; more sophisticated modem theories predict that this insulator-to-metal transition should occur somewhere around 2.2-2.4 Mbar. Pressures of the order of Mbar can be obtained in the laboratory thanks to diamond anvil cells. In such a device, however, only samples with linear dimensions not exceeding - 1 0 #m can be introduced. This size limitation constitutes an extremely severe constraint for crystallographic and structural studies, particularly for systems consisting of low-Z atoms. In fact, neutron diffraction cannot be performed on such small samples and X-ray diffraction, which has a cross section proportional to Z 2, is 36 times more intense from the C atoms of the diamonds than from the H atoms of a hydrogen sample. This is clearly a case where the extreme collimation and the small size of high brilliance beams from the third generation sources are required. In fact measurements on H2, He, 02 and ice were performed at ESRF in energy dispersive or in monochromatic mode from samples in a diamond anvil cell. White beam wiggler radiation focused by a mirror was used in an energy dispersive diffraction experiment to measure the pressure-volume curve (the equation of state) of solid H2 and D2, up to an unprecedented pressure of 1.2 Mbar [15]. Another important structural parameter, the c / a or height-to-length ratio of the unit cell in the hexagonal structure was measured in the same range of pressures. The extension of the range of pressures over which the equation of state is experimentally accessible provides a test for the calculations based on extrapolation of the data at lower pressures, which in turn lead to predictions of the pressure for the insulator-tometal transition. The extension to still higher pressures is possible, the limit so far being the sample capability to
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withstand such pressure without breaking. In the experiment described here the H2 and D2 samples were single crystals grown inside a solid He protective cushion, to resist the initial compression shock. It is realistic to hope for higher pressure results in a near future. Solid 02 was also investigated up to 1.16 Mbar [16], by angle-dispersive powder diffraction. In this case the source was the tenth harmonic of a 4.4 cm period undulator, at a photon energy of approximately 26 keV. The incoming beam was then monochromatized at = 0.4817 A and focused by a Bragg-Fresnel lens, used as a single-bounce monochromator and focusing element. A Bragg-Fresnel lens consists of a Si or Ge platelet on which, by the use of lithographic techniques borrowed from microelectronics, a linear or circular Fresnel grating pattern is engraved. Focusing is obtained upon reflection at the Bragg angle on the platelet and is obviously a line focus from linear patterns and a point focus from circular patterns. Focal lines in the #m range and down to 0.8 #m were demonstrated [17] with a linear BFL based on a Si(1 1 1) platelet with smallest line spacing of 0.3 #m. In the case of the high-pressure set up, a focal line size of a few tens of #m was sufficient. The interest in the structural properties of solid 02 at high pressures stems in part from the observation of the onset of metallic features, such as a Dmde-like tail in the reflectivity, above 0.95 Mbar [18]. This points to the possibility of a metallic molecular state, as the transition to a monoatomic crystal does not occur until larger pressures. As a matter of fact, the data taken at ESRF [17] indicate an isostructural phase transition, with a discontinuous change in lattice constants in a monoclinic unit cell. 5. SCATYERING OF PARTLY COHERENT X-RAYS A very interesting consequence of the small source size and of the high brilliance of modem synchrotron sources is the possibility to perform experiments exploiting the spatial coherence of the X-ray beams. As it is well known from classical optics [19], the fields emitted at wavelength )~ from a source of linear size p are characterized by a transverse coherence length Lt, when viewed at a distance d from the source, given by: L t ~- )~cl/p.
(5)
This means that if the beam is intercepted by a pinhole of size comparable or smaller than L t, the pinhole constitutes a spatially coherent source, in the sense that the radiation emerging from all its points has nearly the same phase. In order to exploit this property, of course, one must have a sufficient intensity going through the pinhole of size L t, whence the interest for small p and high brilliance. Under realistic ESRF undulator conditions,
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NEW SYNCHROTRON RADIATION SOURCES
the transverse coherence length L t at 40 m from the source is of the order of 10 #m and a flux in excess of 107 photons s-a in a monochromatic band with A)C£ ~ 10-4 through the pinhole is readily achieved. A longitudinal coherence length L t can also be defined via the relationship X2
Lt ~- ,~-~,
(6)
and is more properly related to a temporal coherence, determined by the degree of monochromaticity and therefore by the length (or duration) of the wave train. A typical signature of the spatial coherence of the beam is the observation of diffraction phenomena on geometrically simple objects, such as the pinhole itself, which reproduces the Fraunhofer diffraction pattern of circular apertures. This phenomenon was observed both on a wiggler at the National Synchrotron Light Source (NSLS) in Brookhaven, U.S.A. [20] and on an undulator at ESRF [21], with a large number of clearly resolved fringes (Fig. 4). In pioneering experiments first performed in Brookhaven [20, 22] and more recently and under more favourable conditions at ESRF [23, 24], "speckle" 10 7
I
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Fig. 4. Fraunhofer diffraction patterns from circular pinhole apertures of 4 #m, 7 #m and 10 #m diameters. The intensity was recorded with a 5 #m analyser pinhole. The top (bottom) curve was displaced upwards (downwards) by a decade for clarity (from [21]).
203
scattering of X-rays was detected. Speckle scattering is a phenomenon well known from laser physics, in which intensity fluctuations as a function of scattering angles are detected in the scattering of a coherent beam by a sample, as a result of the interference between the beam components scattered (with different phase shifts) by different domains of the sample. The speckle pattern is thus uniquely related to the exact spatial arrangement of the disorder. While the direct spatial analysis of these very complicated patterns is of limited interest, the speckle pattern offers a unique opportunity to explore the time evolution (the dynamics) of the disorder, because when the spatial arrangement of the disorder changes with time, so does the scattered intensity distribution in the speckle pattern. The technique known as Intensity Fluctuation Spectroscopy, or Photon Correlation Spectroscopy or Dynamic Light Scattering, consists in observing the time fluctuations of the intensity of a single point in the speckle pattern to obtain the characteristic time scales and related information on the dynamics of disorder and of fluctuations in the sample. The detection of speckle patterns and of their time dependence is therefore a way to investigate the size of domains or grains, the dynamics of their evolution and more generally, the spatial and temporal scale of fluctuations in the sample, thus allowing the determination of static and dynamic critical exponents. While this is done more easily with visible photons from lasers, the accessible order-parameter wavelength is related to the wavelength of the probe: in going from visible light to X-rays, it can be reduced from thousands of ~, to a few ,~ so that structural and order-disorder phase transitions become accessible. In the Brooklaaven experiments speckle scattering was observed on a Cu3Au single crystal with randomly arranged antiphase domains [20] and on gold-coated films of symmetric block copolymers of polystyrene and PMMA with, #m size islands [22]. Speckle scattering at ESRF was observed on Fe3AI alloys [23] near the order-disorder transition (see Fig. 5) and in semiconducting GaAs-AIGaAs multilayer systems [24]. An X-ray intensity fluctuation experiment was also performed on the order-disorder transition of the Fe3A1 sample, by monitoring the time-dependence of the speckle pattern near the (1/2, 1/2, 1/2) superlattice reflection. The experiment demonstrated that the speckle pattern is essentially static below Tc and it starts to fluctuate in time when the sample is at equilibrium at a temperature above To where fluctuations into the ordered phase have spatial and temporal characteristic dimensions related to the reduced temperature via the static (v) and dynamic (vz) critical exponents, respectively. Intensity fluctuation spectroscopy was also performed [25] in Brookhaven on gold colloid particles dispersed in liquid glycerol and
NEW SYNCHROTRON RADIATION SOURCES
204 Si(200) Monochromator
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Fig. 5. (a) Schematics of the experimental apparatus for speckle scattering and Intensity Fluctuations Spectroscopy measurements; (b) the (1/2, 1/2, 1/2) superlattice peak of Fe3AI at T = 803 K (from [23]). allowed the determination of the dynamic density correlation function and of the diffusion coefficient. As it is apparent from equation (5), the coherence length, all things being equal, increases with the wavelength. Therefore, soft X-ray machines, such as the ALS or Elettra, which have their peak brilliance in o the region of 100A, display remarkable coherence effects. Applications range from soft X-ray microscopy to holography.
15.
REFERENCES
19.
1. See, e.g., P. Elleaume, Rev. Sci. lnstrum., 63, 1992, 321. 2. SeRe, F., Ruocco, G., Krisch, M., Bergmann, U., Masciovecchio, C., Mazzacurati, V., Signorelli, G. and Verbeni, R., Phys. Rev. Lett., 75, 1995, 850. 3. Graeff, W. and Materlik, G., Nucl. Instr. Meth., 195, 1982, 97. 4. See e.g. Schiilke, W. in Handbook of Synchrotron Radiation, vol. 3 (Edited by G.S. Brown and D.E. Moncton), pp. 569-57. North Holland, Amsterdam, 1991. 5. Burkel, E., Inelastic Scattering of X-rays with Very High Energy Resolution. Springer, Berlin, 1991. 6. Ruocco, G., SeRe, F., Bergmann, U., Krisch, M.,
20.
16. 17. 18.
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22. 23.
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Masciovecchio, C., Mazzacurati, V., Signorelli, G. and Verbeni, R., Nature, 379, 1996, 521; SeRe, F., Ruocco, G., Krisch, M., Masciovecchio, C., Verbeni, R. and Bergmann, U., Phys. Rev. Lett., 77, 1966, 83. Masciovecchio, C., Ruocco, G., SeRe, F., Krisch, M., Verbeni, R., Bergmann, U. and Soltwisch, M., Phys. Rev. Lett., 76, 1996, 3356. Ederer, D.L., Calcott, T.A. and Perera, R.C.C., Synchrotron Radiation News, Vol. 7, no. 4, p. 29. See, e.g. Ma, Y., Wassdahl, N., Skytt, P., Guo, J., Nordgren, J., Johnson, P.D., Rubensson, J.-E., Boske, T., Eberhardt, W. and Kevan, S.D., Phys. Rev. Lett., 69, 1992, 2598. Carlisle, J.A., Shirley, E.L., Hudson, E.A., Terminello, L.J., Calcott, T.A., Jia, J.J., Ederer, D.L., Perera, R.C.C. and Himpsel, FJ., Phys. Rev. Lett., 74, 1995, 1234. Jia, J.J., Calcott, T.A., Shirley, E,L., Carlisle, J.A., Terminello, L.J., Asfaw, A., Ederer, D.L., Himpsel, F.J. and Perera, R.C.C., Phys. Rev. Lett., 76, 1996, 4054. Ma, Y., Phys. Rev., IM9, 1994, 5799. Guo, J.-H., Glans, P., Skytt, P., Wassdahl, N., Nordgren, J., Luo, Y., Agren, H., Ma, Y., Warwick, T., Heimann, P., Rotenberg, E. and Denlinger, J.D., Phys. Rev., B52, 1995, 10681. Duda, L.C., St6hr, J., Mancini, D.C., Nilsson, A., Wassdahl, N., Nordgren, J. and Samant, M.G., Phys. Rev., B50, 1994, 16758; Hague, C.F., Matiot, J.M., Guo, G.Y., Hricovicini, K. and Krill, G., Phys. Rev., B51, 1995, 1370; Braicovich, L., Brookes, N.B., Dallera, C., Ghiringhelli, G. and Goedkoop, J.B. To be published. Loubeyre, P., LeToullec, R., Haiisermann, D., Hanfland, M., Mao, H.K., Hemley, R.J. and Finger, L.W., Nature, 383, 1996, 702. Akahama, Y., Kawamura, H., Haiisermann, D., Hanfland, M. and Shimomura, O., Phys. Rev. Lett., 74, 1995, 4690. Kuznetsov, S.M., Snigireva, I.I., Snigirev, A.A., Engstr6m, P. and Riekel, C., Appl. Phys. Lett., 65, 1994, 1. Desgreniers, S., Vohra, Y.K. and Ruoff, A.L., J. Phys. Chem., 94, 1990, 1117. See, e.g. Born, M. and Wolf, E., Principles of Optics, pp. 491-503. Pergamon, New York, 1980. Sutton, M., Mochrie, S.G.J., Greytak, T., Nagler, S.E., Bergman, L.E., Held, G.A. and Stephenson, G.B., Nature, 352, 1991, 608. Griibel, G., Als-Nielsen, J., Abernathy, D., Vignaud, G., Brauer, S., Stephenson, G.B., Mochrie, S.G.J., Sutton, M., Robinson, I.K., Fleming, R., Pindak, R., Dierker, S. and Legrand, J.F., ESRF Newsletter, no. 20, p. 14, 1994. Cat, Z.H., Lai, B., Yun, W.B., McNulty, I., Huang, K.G. and Russel, T.P., Phys. Rev. Lett., 73, 1994, 82. Brauer, S., Stephenson, G.B., Sutton, M., Briining, R., Dufresne, E., Mochrie, S.GJ., Griibel, G., Als-Nielsen, J. and Abernathy, D., Phys. Rev. Lett., 74, 1995, 2010.
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4. Robinson, I.K., Pindak, R., Fleming, R.M., Dierker, S.R., Ploog, K., Griibel, G., Abernathy, D.L. and Als-Nielsen, J., Phys. Rev., B52, 1995, 9917.
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Solid State Communications,Vol. 102, No. 2-3, pp. 199-205, 1997 @ 1997 Elsevier Science lad Printed in Great Britain. All rights reserved 0038-1098/97 $17.00+.00
Pergamon
PII: S0(D8-1098(96)00709-0
PHYSICS WITH THE NEW SYNCHROTRON RADIATION SOURCES M. Altarelli European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France Electron or positron storage rings are the most powerful sources of X-ray photons available to experimentalists interested in spectroscopic and structural studies of condensed matter. The most recent generation of synchrotron radiation sources is characterized by an increase of several orders of magnitude in brilliance. The potential opened up by the new sources is exemplified by recent experiments on the dynamical properties of disordered systems, on high pressure physics and by experiments exploiting the spatial coherence of undulator beams. © 1997 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION In the last few decades the use of synchrotron radiation has revolutionized X-ray physics and its applications to materials science, crystallography, spectroscopy and other fields of applied and fundamental significance. This impressive record of scientific achievements made possible by synchrotron radiation has stimulated considerable effort in the design and optimization of dedicated storage rings. The source brilliance is the main figure of merit for many experimental techniques based on synchrotron radiation. Brilliance is the phase space density of photons and is expressed, for a given photon energy, as the number of photons emitted in the unit time by the unit source area in the unit solid angle in a relative bandwidth of 10 -3 and is conventionally measured in photons/s/mm2/mrad2/0.1% BW. The most recent generation of sources is characterized by two features: one is the very low emittance, generally -<10nmrad in the horizontal plane (this quantity expresses the size and the angular divergence of the particle beam circulating in the ring); the other is the extensive use of insertion devices (wigglers and especially undulators) as radiation sources. The combined effect of the low emittance, corresponding to a reduced source size and collimation and of insertion devices, with improved radiation properties, results in an increase of many orders of magnitude in brilliance of the new sources with respect to earlier ones. The brilliance of hard X-ray undulators at the 6 GeV European Synchrotron Radiation Facility (ESRF) in Grenoble exceeds 1019photons/s/mm2/mrad2/0.1% BW and this value
will soon be reached at the 7 GeV Advanced Photon Source (APS) in Argonne National Laboratory as well. Similar results for soft X-rays were obtained at lower energy machines such as the 2 GeV Advanced Light Source (AI_~) in Berkeley and Elettra in Trieste. The purpose of the present article is to describe a few experiments performed at the new sources in order to provide examples of the new possibilities they open up in condensed matter physics. 2. INSERTION DEVICES In the new generation of synchrotron sources the emission takes place not only from the bending magnets but mostly from the so-called insertion devices [1] (undulators or wigglers) installed in straight sections of the storage rings. An insertion device consists in an array of magnets producing a d.c. vertical magnetic field with a sinusoidal position dependence in the direction of electron motion (see Fig. 1). Insertion devices are classified according to the value of the parameter K = c~-3",
(1)
where ol is the maximum angle of deviation from the straight trajectory (see Fig. 1) and 3, is the relativistic parameter given by the electron energy in the ring divided by the electron rest energy (3, 2 1957 for 1 GeV electrons). It turns out that 1/3' is the characteristic angular opening of synchrotron radiation emitted by orbiting relativistic electrons. Therefore, when K >> 1 (the wiggler regime) an observer along the axis of the device receives short
199
NEW SYNCHROTRON RADIATION SOURCES
200
zI
(
=.e-
_
# # I*
Ao
,I
Fig. 1. Schematic view of the magnetic structure of an insertion device. Also shown are the maximum angle of deviation ~x of the electron trajectory and the radiation cone, of opening 3,-1, at one particular point along the trajectory. radiation pulses from a few points of the electron trajectory. The spectrum is correspondingly broad in frequency space, containing all frequencies up to some multiple of a critical energy o0, = 3c3,3/(2R), where R is the radius of curvature at the emission point and c is the speed of light. Wigglers provide therefore a spectrum analogous to that of bending magnets, although more intense and possibly with a higher critical energy. In the undulator regime (K ~ 1), on the other hand, the whole trajectory in the device is visible by the observer. The observation time encompasses the N periods of sinusoidal motion and results in a spectrum of lines, with a characteristic wavelength spread A~I~,~ l/N, centered around the fundamental wavelength X = 2-~ (1 + K2/2)
(2)
and its odd harmonics ~/n, n = 1, 3,.... In equation (2), X0 is the period of the device. Roughly speaking, one can say that, as a result of interference between the fields emitted from each point of the trajectory, the emitted power of the undulator is concentrated in wavelength in narrow intervals around the harmonics and in space in a cone of aperture 1/(3,x/~), thus enhancing the brilliance considerably. It is also important to notice that the fundamental wavelength is a function of K, which can be modified, for a given device, by varying the distance between the upper and lower sets of magnetic poles, thereby varying the peak field in the device and the angle ~x. 3. INELASTIC SCATFERING In this section recent spectacular advances in the inelastic scattering of hard and soft X-rays made possible by the high brilliance of undulator sources are described. The inelastic scattering beamline at ESRF [2] combines the high-resolution properties of high-index backscattering crystal monochromators [3] with the high photon flux in a 40/~rad cone to obtain a beam of
Vol. 102, No. 2-3
-109 photons s -1 with an energy resolution of a few meV around a photon energy of 17.794 keV (when exploiting the Si (999) backreflection), focused to a spot size of 150 × 350#m 2. The photons are then scattered on the sample and analyzed in energy with the same backrefiection on a Si analyzer at a slightly different temperature (with a step of 0.015 K, corresponding to a Aho0 of 0.7 meV). X-ray scattering, in the non-resonant regime, probes the spectrum of electronic density fluctuations [4], S(k, ¢0), comprising collective excitations (phonons, plasmons) and single-particle excitations. If the energy resolution is in the meV range, the investigation of phonon modes by inelastic X-ray scattering becomes possible [5]. It is important to realize that the use of X-rays allows to access regions in (k, o0) space not accessible by neutrons, because of the very different energy-momentum relationship of the massless photon (o0 = ck) and of the neutron [o0 h2k2/(21~n)]. Denoting by o01 and kl the incoming particle frequency and momentum, the accessible (k, o0) region for X-rays is bound below the o0 = ck and the o0 -- 2o01 - k c line; for neutrons, on the other hand, the accessible region lies below the parabola o0=o01-h(k-kl)212Mn. In particular, this prevents the neutron scattering study of modes with high energy transfer at low momenta, such as acoustic modes in systems with a high speed of sound, more precisely, a speed exceeding that of the neutron (about 2200ms -1 for thermal neutrons with boo1 ~ 25 meV). In crystalline solids, this is alleviated because the momentum of an excitation is defined up to a reciprocal lattice vector and neutron scattering can become allowed in one of the higher Brillouin zones. For non-periodic structures (liquids, glasses, amorphous materials), on the other hand, the use of X-ray scattering is truly significant, as it allows to enlarge the range of accessible frequency and momenta. In a study of the dispersion of sound modes in liquid water with X-ray scattering [2], data were collected in the region of k between 4 and 14 nm-X and in a frequency range from 0 to 30 meV. These results settle a controversy, by establishing the existence of "fast sound" at higher momenta (Fig. 2), with a sound velocity of 3200 m s -l, approximately double of that measured in the hydrodynamic (long wavelength) limit and indeed very close to the speed of sound in ice. In fact the comparison of phonons in ice and water is quite instructive [6], as it reveals a strong analogy even concerning the damping of the modes (the phonon lifetimes) (Fig. 3). The S(k, o0) in two glassy systems, glycerol and LiCI:6H20 was also recently investigated [7] and demonstrated the existence of well defined propagating excitations with linear dispersion in the 2 - 8 n m -t =
Vol. 102, No. 2-3
NEW SYNCHROTRON RADIATION SOURCES
30. ~ " 25.
~ I"" ..~.......
15
experiment following monochromatic excitation with photons of energy hwl, on the exciting frequency wl. This can be quantitatively explained [9-11] by describing the coupling of excitation and emission processes in a single inelastic scattering event described by a KramersHeisenherg type of formula [12]
5~ I0
.
,
•
,
•
,
-
,
•
,
•
100 ,
•
~. ~flp'Alm)(mlp'A[i) 2 ~ , Em _Ei _ h w 1 _ it,m/2
m / s ,
0 2 4 6 8 10 12 14
201
•
/(hw2) ~ Zf
q ( nrn"1 ) Fig. 2. Dispersion relation of sound modes in water, as obtained by X-ray inelastic scattering [2]. region, i.e. up to scales of distance comparable to the interparticle separation. Very interesting advances in soft X-ray resonant inelastic scattering and, more generally, in soft X-ray
emission spectroscopies have also recently been reported. The basic idea is of course not new: soft X-ray fluorescence is an obvious way to probe the valence band density of states which has the advantage, in comparison to alternative techniques such as photoemission, of being bulk-sensitive and insensitive to charging effects. The resolution is in principle limited only by the core hole lifetime broadening, which is acceptable for shallow core levels giving rise to emission in the soft X-ray range (typically 100-1500 eV). In practice, the real problem is that soft X-ray emission is a very inefficient process and its successful use requires a highly brilliant source and an efficient spectrometer [8]. This is the reason why, after pioneering work at the National Synchrotron Light Source in Brookhaven [9], more recently interesting results have been obtained at the ALS in Berkeley. A beamline based on a 5 cm period undulator, operating in the 100 eV to 1500 eV energy range delivers 1012 photons s -1 in a 0.1% bandwidth on a 100 #m spot [8]. The use of resonant fluorescence, or resonant inelastic scattering, as a probe of band structure is based on the observation of a distinct dependence of the emitted spectrum I(I~o~2), measured in a photon-in, photon-out
6050. v
40~ 3020~ 10J
ol
2 4 6 8 10 12 14 q (nm "1 )
Fig. 3. Damping parameter r(q) as a function of momentum q for sound modes in water (after [2]).
× ~(Er + h002 - Ei - l~lcol).
(3)
In equation (3), [i) is the ground state of the system, with energy Ei, [m) is one of a complete set of intermediate states to be summed over, which have a hole in a core state and an electron in a band state above the Fermi level, and I f ) is the final state, which consists of an electron and a hole in the valence band. The delta function enforces energy conservation in the scattering process. If electron-core hole interactions are neglected and appropriate Bloch functions arc inserted in equation (3), a crystal momentum conservation rule follows, k l - k 2 ----k e - k h T G, and since in the wavelength range under consideration the photon wavevector is very small compared to the size of the Brillouin zone, ke-kh +G=
0.
(4)
Furthermore, the energy denominator in equation (3) resonantly enhances intermediate states satisfying E m ~ E i + h w 1 and for a core level with a lifetime broadening comparable to the resolution of the apparatus, this correspond to an approximate energy conservation in the intermediate state. This approximate energy selectivity, together with the approximate crystal momentum conservation, equation (4), explains the dependence of the emission spectrum from the incoming energy and at the same time allows band mapping by tracking the displacement of features in the emission spectrum as the incoming energy varies. This methodology has been tested successfully on diamond [9], graphite [10] and boron nitride [11], by comparison with state-of-the-art band structure calculations. Besides the experiments performed on these well known systems, experiments have been reported in more intriguing cases, such as the C60 fullerene crystal [13], or in ferromagnetic transition metal systems in which the exciting photon beam is circularly polarized, to observe the dependence of the emission spectrum from the sense of polarization (circular dichroism in the emission spectrum) [14]. 4. HIGH-PRESSURE PHYSICS The investigation of the properties of condensed matter under pressure exceeding atmospheric pressure
202
NEW SYNCHROTRON RADIATION SOURCES
by large factors (typically 105-106 ) is of interest to various disciplines of science. Geophysicists investigate the phase diagram of materials under conditions found at the center of the earth, where the pressure exceeds 3 × 106 bar (3 Mbar or 300 GPa). Planet physicists know that even larger pressures (of the order of tens of Mbar) are encountered at the center of large planets such as Saturn or Jupiter, which are mostly composed of light elements such as H and He. The solid state physicist is more generally interested in the possibility, offered by high-pressure technology, to study how electronic properties of materials are affected when the pressure is increased. The overlap of wavefunctions from neighbouring atoms is expected intuitively to increase with pressure, thus increasing the bandwidth, so that an insulator-to-metal transition should occur at high pressure in most insulating materials. Such theoretical predictions are best carried out for simple solids such as hydrogen. The first prediction of the metallization of H at high pressures was made by Wigner and Huntington in 1935; more sophisticated modem theories predict that this insulator-to-metal transition should occur somewhere around 2.2-2.4 Mbar. Pressures of the order of Mbar can be obtained in the laboratory thanks to diamond anvil cells. In such a device, however, only samples with linear dimensions not exceeding - 1 0 #m can be introduced. This size limitation constitutes an extremely severe constraint for crystallographic and structural studies, particularly for systems consisting of low-Z atoms. In fact, neutron diffraction cannot be performed on such small samples and X-ray diffraction, which has a cross section proportional to Z 2, is 36 times more intense from the C atoms of the diamonds than from the H atoms of a hydrogen sample. This is clearly a case where the extreme collimation and the small size of high brilliance beams from the third generation sources are required. In fact measurements on H2, He, 02 and ice were performed at ESRF in energy dispersive or in monochromatic mode from samples in a diamond anvil cell. White beam wiggler radiation focused by a mirror was used in an energy dispersive diffraction experiment to measure the pressure-volume curve (the equation of state) of solid H2 and D2, up to an unprecedented pressure of 1.2 Mbar [15]. Another important structural parameter, the c / a or height-to-length ratio of the unit cell in the hexagonal structure was measured in the same range of pressures. The extension of the range of pressures over which the equation of state is experimentally accessible provides a test for the calculations based on extrapolation of the data at lower pressures, which in turn lead to predictions of the pressure for the insulator-tometal transition. The extension to still higher pressures is possible, the limit so far being the sample capability to
Vol. 102, No. 2 - 3
withstand such pressure without breaking. In the experiment described here the H2 and D2 samples were single crystals grown inside a solid He protective cushion, to resist the initial compression shock. It is realistic to hope for higher pressure results in a near future. Solid 02 was also investigated up to 1.16 Mbar [16], by angle-dispersive powder diffraction. In this case the source was the tenth harmonic of a 4.4 cm period undulator, at a photon energy of approximately 26 keV. The incoming beam was then monochromatized at = 0.4817 A and focused by a Bragg-Fresnel lens, used as a single-bounce monochromator and focusing element. A Bragg-Fresnel lens consists of a Si or Ge platelet on which, by the use of lithographic techniques borrowed from microelectronics, a linear or circular Fresnel grating pattern is engraved. Focusing is obtained upon reflection at the Bragg angle on the platelet and is obviously a line focus from linear patterns and a point focus from circular patterns. Focal lines in the #m range and down to 0.8 #m were demonstrated [17] with a linear BFL based on a Si(1 1 1) platelet with smallest line spacing of 0.3 #m. In the case of the high-pressure set up, a focal line size of a few tens of #m was sufficient. The interest in the structural properties of solid 02 at high pressures stems in part from the observation of the onset of metallic features, such as a Dmde-like tail in the reflectivity, above 0.95 Mbar [18]. This points to the possibility of a metallic molecular state, as the transition to a monoatomic crystal does not occur until larger pressures. As a matter of fact, the data taken at ESRF [17] indicate an isostructural phase transition, with a discontinuous change in lattice constants in a monoclinic unit cell. 5. SCATYERING OF PARTLY COHERENT X-RAYS A very interesting consequence of the small source size and of the high brilliance of modem synchrotron sources is the possibility to perform experiments exploiting the spatial coherence of the X-ray beams. As it is well known from classical optics [19], the fields emitted at wavelength )~ from a source of linear size p are characterized by a transverse coherence length Lt, when viewed at a distance d from the source, given by: L t ~- )~cl/p.
(5)
This means that if the beam is intercepted by a pinhole of size comparable or smaller than L t, the pinhole constitutes a spatially coherent source, in the sense that the radiation emerging from all its points has nearly the same phase. In order to exploit this property, of course, one must have a sufficient intensity going through the pinhole of size L t, whence the interest for small p and high brilliance. Under realistic ESRF undulator conditions,
Vol. 102, No. 2-3
NEW SYNCHROTRON RADIATION SOURCES
the transverse coherence length L t at 40 m from the source is of the order of 10 #m and a flux in excess of 107 photons s-a in a monochromatic band with A)C£ ~ 10-4 through the pinhole is readily achieved. A longitudinal coherence length L t can also be defined via the relationship X2
Lt ~- ,~-~,
(6)
and is more properly related to a temporal coherence, determined by the degree of monochromaticity and therefore by the length (or duration) of the wave train. A typical signature of the spatial coherence of the beam is the observation of diffraction phenomena on geometrically simple objects, such as the pinhole itself, which reproduces the Fraunhofer diffraction pattern of circular apertures. This phenomenon was observed both on a wiggler at the National Synchrotron Light Source (NSLS) in Brookhaven, U.S.A. [20] and on an undulator at ESRF [21], with a large number of clearly resolved fringes (Fig. 4). In pioneering experiments first performed in Brookhaven [20, 22] and more recently and under more favourable conditions at ESRF [23, 24], "speckle" 10 7
I
o 10
/~rn pinhole /~m pinhole 3.5 /~m pinhole 7
106
•
105 tO
l° 4
03
~ i0 3 c-
P, ~o 2 (D
101
i " i~
i l~'~
X
10 °
!
,. ,
I
I
I
I
-2OO - 1 0 0 0 100 200 Horizontal Deflection (#rads)
Fig. 4. Fraunhofer diffraction patterns from circular pinhole apertures of 4 #m, 7 #m and 10 #m diameters. The intensity was recorded with a 5 #m analyser pinhole. The top (bottom) curve was displaced upwards (downwards) by a decade for clarity (from [21]).
203
scattering of X-rays was detected. Speckle scattering is a phenomenon well known from laser physics, in which intensity fluctuations as a function of scattering angles are detected in the scattering of a coherent beam by a sample, as a result of the interference between the beam components scattered (with different phase shifts) by different domains of the sample. The speckle pattern is thus uniquely related to the exact spatial arrangement of the disorder. While the direct spatial analysis of these very complicated patterns is of limited interest, the speckle pattern offers a unique opportunity to explore the time evolution (the dynamics) of the disorder, because when the spatial arrangement of the disorder changes with time, so does the scattered intensity distribution in the speckle pattern. The technique known as Intensity Fluctuation Spectroscopy, or Photon Correlation Spectroscopy or Dynamic Light Scattering, consists in observing the time fluctuations of the intensity of a single point in the speckle pattern to obtain the characteristic time scales and related information on the dynamics of disorder and of fluctuations in the sample. The detection of speckle patterns and of their time dependence is therefore a way to investigate the size of domains or grains, the dynamics of their evolution and more generally, the spatial and temporal scale of fluctuations in the sample, thus allowing the determination of static and dynamic critical exponents. While this is done more easily with visible photons from lasers, the accessible order-parameter wavelength is related to the wavelength of the probe: in going from visible light to X-rays, it can be reduced from thousands of ~, to a few ,~ so that structural and order-disorder phase transitions become accessible. In the Brooklaaven experiments speckle scattering was observed on a Cu3Au single crystal with randomly arranged antiphase domains [20] and on gold-coated films of symmetric block copolymers of polystyrene and PMMA with, #m size islands [22]. Speckle scattering at ESRF was observed on Fe3AI alloys [23] near the order-disorder transition (see Fig. 5) and in semiconducting GaAs-AIGaAs multilayer systems [24]. An X-ray intensity fluctuation experiment was also performed on the order-disorder transition of the Fe3A1 sample, by monitoring the time-dependence of the speckle pattern near the (1/2, 1/2, 1/2) superlattice reflection. The experiment demonstrated that the speckle pattern is essentially static below Tc and it starts to fluctuate in time when the sample is at equilibrium at a temperature above To where fluctuations into the ordered phase have spatial and temporal characteristic dimensions related to the reduced temperature via the static (v) and dynamic (vz) critical exponents, respectively. Intensity fluctuation spectroscopy was also performed [25] in Brookhaven on gold colloid particles dispersed in liquid glycerol and
NEW SYNCHROTRON RADIATION SOURCES
204 Si(200) Monochromator
I
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|
undulalor source
SiC Mirror . . .sensitwe . posdmn _J d e ; c t ~ . . . ~
(a) ~
~
2.0
I
7.
44 micron . . . diameter . pinno,e spenura
Fs3AI crystal in vacuum fumacs I
1.0
/
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~ /1 ]I
I
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9.
10.
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1.2
7
11.
~o.a 12. 13.
o
~ 0.4
-0.0
(b)
I -I.O
i -0.5
0.0
ze-20 B
I 0.5
(mrad)
I
1,0
14.
Fig. 5. (a) Schematics of the experimental apparatus for speckle scattering and Intensity Fluctuations Spectroscopy measurements; (b) the (1/2, 1/2, 1/2) superlattice peak of Fe3AI at T = 803 K (from [23]). allowed the determination of the dynamic density correlation function and of the diffusion coefficient. As it is apparent from equation (5), the coherence length, all things being equal, increases with the wavelength. Therefore, soft X-ray machines, such as the ALS or Elettra, which have their peak brilliance in o the region of 100A, display remarkable coherence effects. Applications range from soft X-ray microscopy to holography.
15.
REFERENCES
19.
1. See, e.g., P. Elleaume, Rev. Sci. lnstrum., 63, 1992, 321. 2. SeRe, F., Ruocco, G., Krisch, M., Bergmann, U., Masciovecchio, C., Mazzacurati, V., Signorelli, G. and Verbeni, R., Phys. Rev. Lett., 75, 1995, 850. 3. Graeff, W. and Materlik, G., Nucl. Instr. Meth., 195, 1982, 97. 4. See e.g. Schiilke, W. in Handbook of Synchrotron Radiation, vol. 3 (Edited by G.S. Brown and D.E. Moncton), pp. 569-57. North Holland, Amsterdam, 1991. 5. Burkel, E., Inelastic Scattering of X-rays with Very High Energy Resolution. Springer, Berlin, 1991. 6. Ruocco, G., SeRe, F., Bergmann, U., Krisch, M.,
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16. 17. 18.
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22. 23.
Vol. 102, No. 2-3
Masciovecchio, C., Mazzacurati, V., Signorelli, G. and Verbeni, R., Nature, 379, 1996, 521; SeRe, F., Ruocco, G., Krisch, M., Masciovecchio, C., Verbeni, R. and Bergmann, U., Phys. Rev. Lett., 77, 1966, 83. Masciovecchio, C., Ruocco, G., SeRe, F., Krisch, M., Verbeni, R., Bergmann, U. and Soltwisch, M., Phys. Rev. Lett., 76, 1996, 3356. Ederer, D.L., Calcott, T.A. and Perera, R.C.C., Synchrotron Radiation News, Vol. 7, no. 4, p. 29. See, e.g. Ma, Y., Wassdahl, N., Skytt, P., Guo, J., Nordgren, J., Johnson, P.D., Rubensson, J.-E., Boske, T., Eberhardt, W. and Kevan, S.D., Phys. Rev. Lett., 69, 1992, 2598. Carlisle, J.A., Shirley, E.L., Hudson, E.A., Terminello, L.J., Calcott, T.A., Jia, J.J., Ederer, D.L., Perera, R.C.C. and Himpsel, FJ., Phys. Rev. Lett., 74, 1995, 1234. Jia, J.J., Calcott, T.A., Shirley, E,L., Carlisle, J.A., Terminello, L.J., Asfaw, A., Ederer, D.L., Himpsel, F.J. and Perera, R.C.C., Phys. Rev. Lett., 76, 1996, 4054. Ma, Y., Phys. Rev., IM9, 1994, 5799. Guo, J.-H., Glans, P., Skytt, P., Wassdahl, N., Nordgren, J., Luo, Y., Agren, H., Ma, Y., Warwick, T., Heimann, P., Rotenberg, E. and Denlinger, J.D., Phys. Rev., B52, 1995, 10681. Duda, L.C., St6hr, J., Mancini, D.C., Nilsson, A., Wassdahl, N., Nordgren, J. and Samant, M.G., Phys. Rev., B50, 1994, 16758; Hague, C.F., Matiot, J.M., Guo, G.Y., Hricovicini, K. and Krill, G., Phys. Rev., B51, 1995, 1370; Braicovich, L., Brookes, N.B., Dallera, C., Ghiringhelli, G. and Goedkoop, J.B. To be published. Loubeyre, P., LeToullec, R., Haiisermann, D., Hanfland, M., Mao, H.K., Hemley, R.J. and Finger, L.W., Nature, 383, 1996, 702. Akahama, Y., Kawamura, H., Haiisermann, D., Hanfland, M. and Shimomura, O., Phys. Rev. Lett., 74, 1995, 4690. Kuznetsov, S.M., Snigireva, I.I., Snigirev, A.A., Engstr6m, P. and Riekel, C., Appl. Phys. Lett., 65, 1994, 1. Desgreniers, S., Vohra, Y.K. and Ruoff, A.L., J. Phys. Chem., 94, 1990, 1117. See, e.g. Born, M. and Wolf, E., Principles of Optics, pp. 491-503. Pergamon, New York, 1980. Sutton, M., Mochrie, S.G.J., Greytak, T., Nagler, S.E., Bergman, L.E., Held, G.A. and Stephenson, G.B., Nature, 352, 1991, 608. Griibel, G., Als-Nielsen, J., Abernathy, D., Vignaud, G., Brauer, S., Stephenson, G.B., Mochrie, S.G.J., Sutton, M., Robinson, I.K., Fleming, R., Pindak, R., Dierker, S. and Legrand, J.F., ESRF Newsletter, no. 20, p. 14, 1994. Cat, Z.H., Lai, B., Yun, W.B., McNulty, I., Huang, K.G. and Russel, T.P., Phys. Rev. Lett., 73, 1994, 82. Brauer, S., Stephenson, G.B., Sutton, M., Briining, R., Dufresne, E., Mochrie, S.GJ., Griibel, G., Als-Nielsen, J. and Abernathy, D., Phys. Rev. Lett., 74, 1995, 2010.
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4. Robinson, I.K., Pindak, R., Fleming, R.M., Dierker, S.R., Ploog, K., Griibel, G., Abernathy, D.L. and Als-Nielsen, J., Phys. Rev., B52, 1995, 9917.
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25. Dierker, S.B., Pindak, R., Fleming, R.M., Robinson, I.K. and Berman, L., Phys. Rev. Lett., 75~ 1995, 449.