VOLUME 78, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 3 MARCH 1997 Coherent Dynamic Structure Factor of Liquid Lithium by Inelastic X-Ray Scattering H. Sinn,1 F. Sette,2 U. Bergmann,2 Ch. Halcoussis,2 M. Krisch,2 R. Verbeni,2 and E. Burkel1 1Universität Rostock, D-18051 Rostock, Germany 2European Synchrotron Radiation Facility, F-38043 Grenoble, France (Received 5 December 1996) We present high resolution inelastic x-ray scattering measurements of the coherent dynamic structure factor S Q, v of liquid lithium at momentum transfers 0.36 # Q # 5 Å21. The determined S Q, v agrees much better with molecular dynamic simulations using the neutral pseudo-atom potential rather than the empty core potential. We observe a positive dispersion in the sound velocity confirming that in liquid lithium the longitudinal dynamics reaches a solid-like response at high frequencies. [S0031-9007(97)02529-5] PACS numbers: 61.25.Mv, 61.10.Eq, 61.20.Ne Recently, Canales et al. [1] calculated the dynamic from an undulator, was monochromized by a combination structure factor of liquid lithium by means of molecular of a cryogenically cooled heat load monochromator and a dynamic (MD) simulations using two different pair po- silicon backscattering monochromator [Si(7,7,7) with E tentials: the empty core potential derived by Ashcroft [2] 13.8 keV]. The analyzer, a two dimensional focusing and an ab initio calculation for a pair potential deduced array of silicon crystals, is positioned at a 2.5 m distance from the neutral pseudoatom method (NPA potential) from the sample in backscattering geometry. Details of the [3]. Although the shapes of these two potentials are backscattering technique and the experimental setup are quite different, most of the calculated structural and ther- discussed in Refs. [5,6]. A variation of the energy transfer modynamic properties are very similar [1]. Significant was achieved by a thermal variation of the lattice parameter differences are observed in the calculations of the coherent of the monochromator crystal. We obtained a calibration part of the dynamic structure factor. of energy shifts related to temperature differences between Inelastic neutron experiments on the dynamic structure the two crystals by measuring a phonon dispersion curve factor of liquid 7Li were performed by de Jong et al. [4]. in crystalline silicon and comparing it to results from an However, these measurements yielded no decision which inelastic neutron scattering experiment [9]. of the two pair potential approaches is the more favor- The energy resolution of the spectrometer is mainly de- able. The reason is that an essential part of the co- termined by the intrinsic reflection width of the silicon herent structure factor-the Brillouin modes-could not crystals, the alignment, and the focusing performance of be observed at small momentum transfers Q. Because the analyzer. The resolution function R v was measured of the momentum-energy relation for a classical particle at the structure factor maximum of polymethylmethacry- there exists a maximum energy transfer v at a momen- late (PMMA), yielding an energy resolution of 11 meV tum transfer Q, which is determined by the flight velocity (FWHM). of the incident neutron. In the neutron experiment men- As a sample we used pure lithium at a temperature of tioned above the Brillouin excitations were not detectable 215 ±C, well above the melting point (TM 180.5 ±C). for Q , 1.2 Å21. Moreover, the high fraction of inco- In order to avoid a contamination of the sample, it was herent scattering at small Q and the uncertainties in the contained in a UHV chamber with beryllium windows, incoherent cross section of 7Li makes it difficult to extract highly transparent for the x-ray beam. the coherent part from the experimental data. The inelastic x-ray scattering cross section with energy In contrast, these limitations do not appear in an transfer v and momentum transfer Q, following the inelastic x-ray scattering experiment with sufficient high analysis of Chihara [10], can be written in a liquid metal energy resolution. At energy transfers of about a few as meV the observed intensity originates dominantly from ds 1 coherent scattering. The energy-momentum relation of Q, v ~ s ¯hvb 2 S Q, v dV dv T fi Q 1 ry Q 2e the photon allows an almost unlimited energy transfer at any accessible momentum transfer [5,6]. 1 zS0y Q, v Previous experiments on liquid lithium were performed 1 Z 2 z Sinc at the synchrotron laboratory HASYLAB in Hamburg i Q, v , (1) [7,8]. The experiments reported here were carried out where sT is the Thomson cross section, fi Q and ry Q during the commissioning phase of the inelastic x-ray are the form factors for the ions and for the valence scattering beamline (ID16) at the European Synchrotron electrons, Z is the number of electrons, and z is the Facility (ESRF) in Grenoble. The x-ray radiation, coming number of the valence electrons. 0031-9007 97 78(9) 1715(4)$10.00 © 1997 The American Physical Society 1715 VOLUME 78, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 3 MARCH 1997 S Q, v is equal to the coherent dynamic structure collective behavior of this excitations, is evident already factor appearing in the cross section for neutron scattering. in the raw data. The exponential factor in Eq. (1) (detailed balance factor) The first steps of the data reduction were the normaliza- is a first order quantum correction that allows one to tion to a monitor signal and the subtraction of the empty consider S Q, v as a symmetrical function in v [11]. cell measurements. In order to get our results comparable The incoherent dynamic structure factor for the valence to the available results from neutron scattering, we ana- electrons, S0y Q, v , arises from excitations of the electron lyzed the coherent structure factor S Q, v in terms of the gas. For simple metals, S0y Q, v can be replaced, as model of the extended hydrodynamic modes [13]: a first approximation, by the dynamic structure factor A z A z of a jellium Sjell 0 0 s s 1 v 1 vs tan f y Q, v [10]. In this approximation, S Q, v 1 no significant incoherent scattering for energy transfers p v2 1 z20 p v 1 vs 2 1 z2s below a few eV is expected. Possible deviations from As zs 2 v 2 vs tan f this behavior are discussed in Ref. [12]. 1 . (2) p v 2 vs 2 1 z2s Incoherent scattering consisting of electron-hole exci- tations of the core electrons is described by Sinc Equation (2) describes a central Lorentzian (Rayleigh i Q, v . At high Q the incoherent scattering terms zS0 line) and two asymmetrical Lorentzians (Brillouin lines) y Q, v 1 Z 2 z Sinc at 6v i Q, v turn into the Compton cross section. s, where f determines the asymmetry. The pa- The measured inelastic x-ray scattering spectra I Q, v rameters Ai and zi define the areas and the half-widths of are shown in Fig. 1 for selected values of momentum the three lines. The measured intensity can be described transfers. The solid lines are the fits discussed in the by a convolution of the dynamic structure factor and the following, and the full curve shown at the bottom of resolution function the figure is the resolution function. The spectrum Z is characterized by a central peak component due to 1 I Q, v ~ e ¯hv0b 2 S Q, v0 R v 2 v0 dv0. (3) quasi-elastic scattering, and the two side components are 2 due to inelastic x-ray scattering (energy loss and gain) Using all six parameters in Eq. (2) as adjustable, the x2 from collective atom excitations, generally referred to values of the fitting functions were in the expected range as Brillouin lines. At low Q values, the dispersion of of statistical noise of the data. these lines, and therefore the persistence of a propagating The results for the deconvoluted fitting curves at three different Q values are shown in Fig. 2 as vertical bars. The half lengths of the bars correspond to the 1s un- certainty of the fitting curves due to the statistical error in the raw data. An intensity normalization of the x-ray data in Fig. 2 was achieved by setting the second fre- quency moment of S Q, v to the theoretical value Q2 Z vl v2S Q, v dv , (4) Mb 2vl where M is the particle mass and vl is a suitable integration limit [14]. The full lines and the dashed lines are molecular dy- namic calculations performed by Canales et al. using the NPA potential and empty-core potential, respectively. Clearly, the NPA simulation fits much better the ex- perimental data at Q 0.72 Å21 and Q 1.25 Å21. At higher Q values the dependence of dynamic struc- ture factor on differences in the pair potential is much weaker. The MD curves and the measured spectra at Q 3.53 Å21 are indistinguishable within the experi- mental error. The dispersion vs Q of the Brillouin lines in compari- son with results from MD simulation and inelastic neutron scattering is shown in Fig. 3. The results from the NPA FIG. 1. Experimental data for liquid lithium from inelastic x-ray scattering. The full lines denote fits according to the simulation (solid line) are in good agreement with the model of the extended hydrodynamic modes, the full curve at dispersion from inelastic x-ray scattering, with respect to the bottom is the resolution function of the spectrometer. the maximum observed frequency. This implies that the 1716 VOLUME 78, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 3 MARCH 1997 FIG. 3. Dispersion relation of the Brillouin mode. Open circles: this work; open triangles: inelastic neutron scattering [4]; solid line: MD simulations [1] with the NPA potential; dashed line: empty core potential; dot-dashed line: macroscopic sound velocity; dotted line: viscoelastic sound velocity [17]. In conclusion, the dispersion and the shapes of the Bril- louin lines in liquid lithium could be obtained from high resolution inelastic x-ray scattering. The high statistical accuracy of the data and the large accessible Q-v range yields a picture of the propagating modes that is much more precise than from previous neutron scattering ex- periments. From the comparison with two MD simula- tions a preference for the NPA pair potential rather than FIG. 2. Deconvoluted fitting curves for Q 0.72, 1.25, and for the empty core potential can be concluded. 3.53 Å21. The vertical bars represent the error interval due We wish to thank the ESRF for the technical support to the statistical uncertainty of the data. The full line is a and the good collaboration. This work was partially sup- result from molecular dynamics [1] with the NPA potential; ported by the European network (ERBCHRXCT930118) the broken line corresponds to simulations with the empty core potential. and the Bundesministerium für Forschung und Technolo- gie (05-650 WEA1). One of us (H. S.) would like to NPA method of calculating the pair potential leads to a thank M. Canales for providing the data from molecular dynamics, and the participants of the EEC science pro- better description of the repulsive part of the potential gram (SC1-CT91-0754), especially P. Verkerk, for fruit- and, consequently, gives a more realistic description of ful discussions. the generalized Einstein frequency as described, e.g., in Ref. [15]. The dispersion of the data from the MD simulation with the empty-core potential (dashed line) lies significantly above the x-ray data in the small Q range. [1] M. Canales, L. E. Gonzales, and J. A. Padro, Phys. Rev. E The discrepancy between x-ray data and the neutron 50, 3656 (1994). data for Q . 1.2 Å21, which suggest a lower lying [2] N. W. Ashcroft, Phys. Lett. 23, 48 (1966). dispersion, is not fully understood now. However, for a [3] L. E. Gonzales, D. J. Gonzales, M. Silbert, and J. A. further investigation, the error bars of the neutron data Alonso, J. Phys. Condens. Matter 5, 4283 (1993). have to be diminished. [4] P. H. K. de Jong, P. Verkerk, and L. A. de Graaf, J. Non- For small Q values, the dispersion observed by inelastic Cryst. Solids 156­158, 48 (1993). x-ray scattering is steeper than the slope corresponding [5] E. Burkel, Inelastic Scattering of X-Rays with Very High to the macroscopic sound velocity in the liquid (dashed- Energy Resolution (Springer, Berlin, 1991). dotted line in Fig. 3). This so-called positive dispersion [6] F. Sette, G. Ruocco, M. Krisch, U. Bergmann, C. Mascio- was observed before, e.g., on liquid cesium by inelastic vecchio, V. Mazzacurati, G. Signorelli, and R. Verbeni, Phys. Rev. Lett. 75, 850 (1995). neutron scattering [16] and to a much larger extent [7] E. Burkel and H. Sinn, J. Phys. Condens. Matter 6, A225­ in water [6]. For a monoatomic liquid this effect is A228 (1994). associated with the onset of a viscoelastic shear relaxation [8] E. Burkel and H. Sinn, Int. J. Thermophys. 16, 1135­1142 (dotted line) [17]. 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