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Molecular simulations-aided analysis of inelastic neutron scattering spectra
Nuclear Instruments and Methods in Physics Research A 354 (1995)59-65
NUCLEAR INSTNUMENTS
BMETNODS IN PHYSICS
=i!EEY”
EISEVIER
Molecular simulations-aided analysis of inelastic neutron scattering spectra F.J. Bermejo”p*, E. Encisob, A. Criado” “Institute de Estructura de la Maten’a, Consejo Superior de Investigaciones Cientijicas, Serrano 123, E-28006 Madrid, Spain “Departamento de Quimica-Fisica I, Universidad Complutense de Madrid, E-28040 Madrid, Spain ‘Institute de Ciencia de Materiales. Consejo Superior de Investigaciones Cientijicas, and Departamento de Fisica de la Materia Condensada, Universidad de Sevilla, P. 0. Box 1065. E-41080 Sevilla, Spain
Abstract The present communication illustrates, by means of three case examples, a way for extracting quantitative information from inelastic neutron spectra on samples characterized by complicated structural and/or magnetic dynamics, where the sole approach employing either experiment or computer simulation fails to provide results free of considerable ambiguities.
1. Introduction The information comprised in the S(Q,w) dynamic structure factors as derived from experiment or model calculations for simple systems such as Bravais crystals or monoatomic non-crystalline matter, suffices to specify their dynamical states, and fully quantitative comparisons between experiment and calculation are now common. However, detailed information regarding the microscopic dynamics of a vast class of materials cannot be derived directly from the measured structure factors without recourse to strong, simplifying assumptions. In particular, studies on the dynamics of non-crystalline multicomponent systems, those composed by particles of non-spherical shape, which can exhibit rather intricate dynamics as far as their internal degrees of freedom are considered, or those evidencing an interplay between magnetic and structural excitations, very often encounter substantial difficulties in specifying the microscopic origin of the observed excitations. To exploit fully the capabilities of present day neutron instrumentation when used to investigate systems belonging to the abovereferred classes, the need to recourse to various kinds of computer modelling was early recognized (to our knowledge the study of the Rahman
*Corresponding author.
constitutes the first attempt to calculate quantities directly amenable to experimental scrutiny Cl]), and for some particular applications such as disentangling the different contributions to the measured total S(Q) structure factors (or to g(r)) they now constitute a widely accepted tool (see for instance [2]). The situation is, however, rather less advanced when S(Q,o) is considered instead. Routine use of computer codes adequate for the study of time-dependent properties strongly relies upon the physical soundness of the model interparticle potentials employed for the calculation, something which in a good number of cases pre-empts any truly quantitative comparison between calculation and experiment. However, such approaches after an adequate parameterization of the model potentials, can serve, in some selected cases to separate the various contributions to the observed response, and therefore to provide a microscopic insight into the dynamics. Three case examples are discussed comprising: (a) the experimental verification of the existence of kinetic sound modes in multicomponent liquids, (b) an approximate separation of the magnetic and structural responses in magnetically disordered systems, and (c) some insight into the microscopic origin of the low-frequency excitations appearing in glassy materials. Since most of the ensuing discussion refers to work which has already appeared, the interested reader is referred to the original references for further details.
0168~9002/95/$09.500 1995 Elsevier Science B.V. All rights reserved 0168-9002(94)00934-l
SSDI
NEUTRON SCATTERING DATA ANALYSIS
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F.J. Bennejo
et al. /Nucl. Instr. and Meth. in Phys. Res. A 354 (1995) 59-65
8.0 lo'* 7.0 1c2 6.0 1C’
Q = 0.25 A-’
5.0 lo’* 4.0 1w2 3.0 lO-’ 2.0 10.’ 1.0 10’ 5.0 w
;
1
4.0 lo’*3.0 UP0.0
0.5
1.0
2.0 UT’-
Q 1.0 w-
1.5
2.0
(i’)
12
I
I
,
1
3.5 3.0 2.5 2.0 1.5 1.0 5.0 2.5 0
ito
( THz)
Fig. 1. (a) The left side shows plots of the longitudinal current correlation function J, (Q, w) of liquid (T = 200 K) methanol as calculated from MD simulations [6]. Values for the wave vectors are given in the inset. The arrows show the values of the average frequencies derived from analysis of the experimental neutron intensities.(b) Frequency positions of the main peaks in Ji (Q, o) from a polycrystalline ide sample as calculated from LD results. The solid and dash-dots lines represent hydrodynamic, orientational averaged longitudinal and transverse sound velocities. Open squares depict the frequency dependence of the most intense peak in J, (Q, w), and the lines with vertical bar and filled lozenges represent the dispersion of transverse and longitudinal sound, respectively, as seen by the dispersion of less intense peaks. (c) A comparison of the dispersion of the main peak in Ji(Q, o) (dashed line with vertical bars) with that measured experimentally in polycrystalline ice Ih [9] (open circle with error bars). Open lozenges and squares with a cross display MD data for liquid water as calculated from the position of the current-current correlations [S]. The solid and dashed straight lines passing through the origin represent longitudinal hydrodynamics sound for the solid and liquid, respectively.
F.J. Bermejo et al. /Nucl. Instr. and Meth. in Phys. Rex A 354 (1995) 59-65
2. Origin of the anomalous dispersion in two componbnt liquids The existence of propagating short-wavelength modes with velocities well above that of hydrodynamic sound exhibiting a dispersion behaviour akin to ordinary sonic waves, in fluids composed by particles with disparate mass-ratios was inferred from results of kinetic theory calculations [3] some time ago. The experimental evidence regarding such phenomena is rather controversial, and in particular, no results have yet been obtained for an archetypal system such is the alloy Li,Pb, where the existence of such anomalous dispersion was inferred from computer molecular dynamics (MD) results [4]. In the search for systems exhibiting such phenomenology, liquid water appeared as an ideal candidate since the large difference in masses of its atomic components and the easiness in carrying out experimental measurements [4]. As a matter of fact, results from MD simulations have repeatedly evidenced the presence of such an anomalous dispersion from plots of the maxima of the longitudinal current correlations Ji (Q, o) = 0’ S(Q, o)/Q’ versus wave vector [S]. However, previous experience with neutron experiments as well as with MD simulations carried out on a related liquid (fully deuterated methanol) [6], had evidenced the presence in JI (Q, o) of noticeable structure (see Fig. l(a)), something which introduces substantial uncertainties when assigning a sonic character to some of the peaks. As a matter of fact, the experimental curves giving the wave vector dependence of the excitations evidenced a dispersion steeper than that characteristic of normal sound [6], reaching values some 2.5-3 times larger than what could be expected from values of the hydrodynamic sound as well as from knowledge of the dispersion of acoustic branches along the main symmetry directions of the single crystal. The microscopic origin of such a complicated structure in JI (Q, w) was then unveiled by comparison of the functions calculated for the liquid with those for a polycrystalline solid derived from Lattice dynamics (LD) computations using the same model potential, and the complicated peak structure in J1 (Q, w) was found to have a clear correlate in the harmonic polycrystal. Following an analogous line of reasoning, and from consideration of the neutron data of Teixeira et al. [7], a LD calculation was carried out for one of the ice polymorphs [S], and some of the most representative results are shown in Fig. l(b). As can be seen, at least three different ‘dispersion’ curves can be traced following peaks of the polycrystalline ice spectra. The peaks assignable to sound propagation can be identified in the crystal spectra and show a standard dispersion behaviour approaching the correct, orientation-averaged hydrodynamic sound limits [8]. On the other hand, the steep dispersion curve which can be traced from frequencies corresponding to the maxima of Jl(Q,o) evidences its
61
non-acoustic character by the fact that the spectrum gets exhausted just past the zone boundary (i.e. near Q,/2 where Q, stands for the wave vector corresponding to the strongest Bragg reflection, Qp z 1.7 A- ‘). A clear correlate of the excitations manifest in the polycrystalline solid was also seen in the functions calculated for the melt by MD procedures, albeit as expected, most of the peaks are now somewhat displaced to lower frequencies as well as considerably broadened. As can be seen in Fig. l(c), the shape of the steep dispersion curve for the ice polycrystal is reproduced at somewhat lower frequencies in the MD calculation of the liquid. If a value for a sound velocity is inferred from the slope at low-Q of these curves, as done in MD works [S] or from analysis of experimental data [7], a speed far higher than that of hydrodynamic sound is found (notice that the ratio of such a slope to that of normal sound is approximately the same in both crystal and liquid). However, reference to the polycrystalline solid evidences that the peaks characteristic of sound propagation are the less intense ones located at lower frequencies, and upon melting, to follow the dispersion of the sound-mode peaks becomes extremely difficult since these are buried within broad bands of inelastic intensity of non-acoustical origin. On semiquantitative grounds, an analogous exercise to that carried for the polycrystal can be performed for the melt, and a normal sound propagation curve can be recovered if the position of the low-frequency shoulder in J,(Q, w) is plotted instead that corresponding to the global maxima. In order to test the validity of the present predictions, neutron experiments on polycrystalline ice Ih were also carried out [9], and the results also shown schematically in the figure, constitute a nice confirmation of the predictions made from the calculation, and as can be seen from the graph, the same microscopic phenomena giving rise to the steep dispersion in polycrystalline ice are operative in liquid water. It seems then clear that some caution has to be exercised when analysing neutron or MD results of complicated systems such are most molecular liquids, since the relatively large number of excitations present in such samples can easily lead to infer the existence of strongly anomalous behaviours, when, as exemplified in this case, such apparent anomalies simply arise from geometrical (orientational averaging) effects.
3. Separation of nuclear and magnetic responses in magnetically disordered systems The magnetic dynamics of some insulators such as the condensed phases of molecular oxygen have resisted any truly quantitative analysis since the pioneering work carried out at Leiden near the turn of the century (for studies on the magnetic susceptibility, see [lo]). Even in the highest temperature (liquid) phases (the magnetic
NEUTRON
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DATA ANALYSIS
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dynamics of the liquid is discussed in Europhys. Lett. [ll], and that of the plastic crystal in J. Phys.: Condens. Matter [ 1l]), experimental evidence from polarized diffraction has shown that the spin dynamics substantially departs from the idealized isotropic paramagnet behaviour, and as a matter of fact, the characteristic spin correlation lengths inferred from such studies turned out to be not too different from those found in the plastic crystalline phase which forms upon freezing the liquid. Although a separation of magnetic and structural responses was achieved by means of polarized neutron diffraction [12], the reported attempts to separate the dynamical responses [13], even in the lowest temperature, magnetically ordered phase, have encountered a number of difficulties as a result of the relatively low count rates and the difficulty in preparing single crystals, which lead to results somewhat at odds with others derived from different experimental (optical spectroscopy) sources. For those phases where the coupling between the magnetic and nuclear dynamics can be deemed as weak, (liquid, plastic crystal and /.I phases), an approximate separation between both dynamical responses was attempted. To proceed, the total dynamical structure factors were measured by means of cold neutron spectroscopy and the measured intensity was analysed as t-11, 141,
A glance to the referred figure evidences that such a separation can be considered as reliable at least up to transfers of x 5 meV, thus encompassing the most significant part of the spectrum. The analysis of &,,(Q,w) was subsequently carried out in terms of a model for an exchange-coupled Heisenberg magnet, and relevant information regarding the spin dynamics in this phase as well as the plastic crystal and liquid phases were then derived [ll, 141. As an indication about the reliability of the separation procedure, the magnetic single-differential cross-section calculated from integration of Smag(Q,w), that is, (f*(Q)>
cc
s
%ax do La,(Q,4
L,(Q)
(4)
- %i”
is compared with the quantity measured by polarized diffraction in Fig. 2. As can be seen the calculated and measured curves show an acceptable agreement if the statistical errors in the latter and the effects from the cutoffs in the integration of the former, which arise from the finite range of energy-transfers accessible to experiment, are taken into accou$. In particula! for wave vectors within the range 0.6 A-’ < Q < 1.5 A-‘, where the kinematic range accessible to cold-neutron spectroscopy comprises the most significant part of the spectrum, the agreement between the measured cross-section and its estimate can be considered as good. (2) From the isolation of the magnetic response, estimates for the contribution of magnetic fluctuations to the speci&,,,(Q,4=dCl -exp(-RwlkeT)l~~F(Q,w), (3) fic heat were calculated and found to follow a T - ‘I2 law characteristic of spin waves (antiferromagnons) with a value for the energy gap (i.e. Q = 0 magnons) of where A is a global scaling constant and R(Q,w) the z 4.6 meV which comes rather close to that predicted for instrumental resolution function, the nuclear dynamics encompassed in S,,, (Q,w) was modelled from computer molecular and lattice dynamics calculations, the magnetic scattering intensity is specified in terms of the a 0.20 Smag(Q, o) spin dynamics correlations and the prefactors E stand for the neutron-magnetic system coupling, Lande 3 0.15 2 $4 factor and the magnetic form-factor for the oxygen ii P PO molecule,f(Q), which can be modelled from knowledge e 0.10 P 1; its spin density distribution. Notice that both static and % dynamic spin correlations are comprised within the ,@ f .p 0.05 E product xo F(Q, W) of the static susceptibility times the 8 GE; spectral weight function, and that it is precisely this the 0.00 LmII quantity to be determined. 0.0 0.5 1.0 1.5 2.0 The form of S,,, (Q, o) was then inferred upon subQ (;4-‘) traction of S,,, (Q, w) from the total experimental intensity. A comparison between the inelastic intensities for Fig. 2. A comparison between the magnetic single-differential fixed values of the energy transfer is shown in [ 14, Fig. 31 cross-section for j%oxygen as measured by polarized neutron for the orientationally ordered /I phase, where the specdiffraction [12] (filled lozenges with error bars) and that caltrum of lattice excitations was calculated by means of culated as integrals over the dynamic structure factor for magnetic excitations (open circles with a dot). a molecular Born von Karman approximation [14].
F.J. Bemejo et al. /Nucl. Instr. and Meth. in Phys. Rex A 354 (1995) 59-65
for a distorted triangular lattice [15]. Finally, the analysis of the strictly elastic S (Q, w = 0) intensity revealed the presence of two weak Bragg reflections of magnetic origin, the assignment of which was later confirmed in a polarized neutron experiment, which evidence the presence of static, short-range magnetic order, not evidenced in diffraction experiments, since they are masked in any total scattering experiment by the strong background of paramagnetic fluctuations which dominates the spectrum.
4. Microscopic origin of the ‘Boson’ peak in structurally disordered matter The present of a low-frequency (0.5-5 meV) broad inelastic peak in the neutron and Raman spectra of a wide variety of glass-forming materials (the Boson peak) has been considered for years as one of the fingerprints of glassy dynamics [16]. The relevance of such a spectral feature stems from the fact that excitations within the same range of frequencies can give due account of the thermal (bump in plots of C(T)/ T 3 versus temperature at 5-20 K) and transport (plateau in the thermal conductivity within a comparable range of temperatures) anomalies in most of the cases examined so far. However, recent studies on materials as diverse as fullernes [17] or metallic alloys at very low temperatures [lS], have evidenced the presence of closely related phenomena in solids which by no means can be considered as topologically disordered. Under such circumstances, efforts towards the understanding of the microscopic origin of the low-frequency peaks in structural glasses seem indicated. Our approach consisted in relating first the excitation spectrum of the glass to that of the harmonic (polycrystal) ground state, either calculated by LD or MD means or accessed experimentally by neutron or Raman scattering, so that an absolute scale of frequencies is available. From comparison between the neutron spectra of samples in both, polycrystal and glass forms, unambiguous information regarding the microscopic origin of the lowfrequency vibrations in the glass can be gained. For the polycrystal, the analysis of the mode eigenvectors giving rise to intense peaks in the vibrational density of states (generalized frequency distribution), enables to establish the character (translational, rotational internal-mode or mixtures of them) of the excitations in a quantitative manner. As seen for a limited number of cases [19-211, the lowest-frequency peak in the incoherent approximation to the neutron spectrum of the crystal, (i.e. S(Q, o) a Z(w)n(o)/w, where Z(w) is the generalized frequency distribution and n(w) a Bose factor), corresponds for molecular materials to excitations strongly mixed in character, where roto-translational contribu-
63
tions are intermingled with those arising from lowfrequency peak internal modes such are chain torsions in selenium or internal pivoting motions of the benzne rings in ortho-terphenyl (OTP). In counterposition, the glass shows a far broader peak located at somewhat lower frequencies than that of the crystal [19-211, being in some cases such as OTP barely separable from the elastic peak. Even if a complete model assignment of the excitations contributing to the ‘Boson’ peak in glasses along the lines followed for the polycrystal cannot be achieved, a partial separation of the components giving rise to intensities in Z(o), into center-of-mass (COM), rotational (ROT) internal-mode (IM) contributions and coupling terms of all of them can be achieved from results of MD calculations. The interesting point to note regards the fact that after such a decomposition is performed, the available evidence for glasses as distinct as methanol, selenium or ortho-terphenyl [19-211 suggests, as exemplified in Fig. 3, that the precise location in frequency of the Boson peak is mainly governed by the COM (in case of methanol or OTP) or van der Waals (for Se) components. In consequence [22], the bump in C(T)/T 3 appears in Fig. 3, at a temperature corresponding to the maxima of the COM contribution to the Boson peak, and most of the spectral intensity arising from other kinds of excitations will contribute to this property as a rather flat background [19-211. In counterposition, as recently evidenced from the analysis of data regarding glassy Se, the presence of low-frequency internal modes of the Se chains constitute and efficient phonon scattering mechanism, which can quantitatively account for the observed behaviour with temperature of the thermal conductivity c231. A related topic, although seldom addressed concerns the characterization of such a peak as arising from mostly single-particle or mostly collective motions. The relevance of a proper answer to such a question stems from its implications towards the confirmation of widely different scenarios proposed to account for the most significant features of glassy dynamics. From the limited experience available [24], the wave vector dependence of the position of the Boson peak evidences a rather strong collective character, something which seems to be in agreement with the fact that mostly COM motions are involved in these modes. In summary, the study of low-energy excitations in disordered matter constitutes one of the prime examples where a close knit connection between experiment and simulation is a clear prerequisite before any fully quantitative advance takes place. The basic building blocks for any comprehensive theoretical approach still rest to be laid upon, and the combination of neutron scattering with microscopic simulations seem to be decisive in this endeavour.
NEUTRON
SCATTERING
DATA ANALYSIS
F.J. Bermejo et al. /Nucl. I&r.
0
5
10 15 20 ilw
25
30
35
and Meth. in Phys. Res. A 354 (1995) 59-65
40
lh
(meV)
(meV)
8. (4 6.
4,
2.
0. 0
5
‘UK)
10 15 20 25 30 35 40 'I'W
Fig. 3. (a) The vibrational density of states (solid line) for glassy methanol at T = 35 K as calculated from molecular dynamics simulations. T’he dash-dot line shown the COM component, the dotted line the rotational (ROT) component, and the dash-dots the coupling (roto-translation) terms. (b) A comparison of the calorimetric specific heat [19-211 (open circles) with that derived from neutron spectra (filled circles), and that calculated from MD results corresponding to the Z(w) shown above (solid line). The contribution from COM motions is shown by the dots, and that from ROT by dashes. The dashdots line represents the sum of both contributions (excluding the coupling terms. The line with vertical bars and the one with asterisks, show the same magnitude for the polycrystal as measured calorimetrically and calculated from LD results. (c) The vibrational density of states as calculated for glassy Se (solid line) compared with two sets of experimental measurements [23] (open and filled lozenges. The dotted line shows the van der Waals contribution. (d) A comparison between the specific heat for glassy Se (solid lie with filled circles) with that calculated from the MD results (thick solid line), and that from the van der Waals contribution alone (dotted line). Filled triangles and circles represent experimental measurements for trigonal Se, and are compared with the calculation from LD results (line with vertical bars).
5. Conclusions Our aim in the present communication has been to illustrate the capability of a combined approach, experiment plus molecular simulations, to extract quantitative information from experiment in systems where the sole
analysis of the inelastic intensities cannot provide unambiguous answers. Notice that on the other hand, simulation data, by its very nature, also fails to provide quantitative answers, unless its results are continuously confronted with experiment. As shown in the three cases discussed above, relevant, albeit semiquantitative microscopic information can be
derived following such an approach. In counterposition, attempts to discuss experimental data of the kind considered here without recourse to information arising from simulations have to be confined to discussions on phenomenological (meso- or macroscopic) grounds, thus being unable to exploit the advantages,of exploring matter with a truly microscopic (in time and length-scales) probe such that provided by neutron scattering. Acknowledgements
This work has been supported in part by DGICYT grant No. PB92-0114-C03.
F.J. Bermejo et al. /Nucl. Instr. and Meth. in Phys. Res. A 354 (1995) 59-65
References [l] A. Rahman, Phys. Rev. 136 (1964) A405. [Z] R.J. McGreevy, these Proceedings (Workshop on Neutron Scattering Data Analysis (WONSDA’ 94), Chilton, Didcot, UK, 1994) Nucl. Instr. and Meth. A. 354 (1995) 1. [3] A. Campa and E.G.D. Cohen, Phys. Rev. Lett. 61 (1988) 853. [4] J. Bosse, G. Jacucci, M. Ronchetti and W. Schirmacher, Phys. Rev. Lett. 57 (1986) 5277. [S] R.W. Impey, P.A. Madden and I.R. McDonald, Mol. Phys. 46 (1982) 513; M.A. Ricci, D. Rocca, G. Ruocco and R. Vallauri, Phys. Rev. A40 (1989) 7226. [6] J. Alonso, F.J. Bermejo, M. Garcia-Hernandez, J.L. Martinez, A, Criado and W.S. Howells, J. Chem. Phys. 96 (1992) 7696, fully quantitative comparisons between neutron data and MD results are given in J. Alonso, F.J. Bermejo, M. Garcia-Hernandez, J.L. Martinez, W.S. Howells and A. Criado, Phys. Lett. Al72 (1992) 177. [7] J. Teixeira, M.C. Bellisent-Funel, S.H. Chen and B. Dorner, Phys. Rev. Lett. 54 (1985) 2681. [S] A. Criado, F.J. Bermejo, M. Garcia-Hernandez and J.L. Martinez, Phys. Rev. E47 (1993) 3516. [9] F.J. Bermejo, E. Frikkee, M. Garcia-Hernandez, J.L. Martinez and A. Criado, Phys. Rev. E48 (1993) 2300. [lo] H.K. Onnes and A. Perrier, Leiden Common. 116 (1910) 1; G.C. De Fotis, Phys. Rev. B23 (1981) 4714. [ll] A. Chahid, F.J. Bermejo, J.L. Martinez, M. Garcia-Hernandez, E. Enciso and F.J. Mompean, Europhys. Lett. 20 (1992) 71; A. Chahid, F.J. Bermejo, E. Enciso, M. Garcia-Hernandez and J.L. Martinez, J. Phys.: Cond. Matter 5 (1993) 423.
65
[12] F. Dunstetter, Ph.D. Thesis, Universitt de Paris-Sud (1988). [13] P.W. Stephens and CF. Majkrzak, Phys. Rev. B31 (1986) 1. [14] A. Chahid, F.J. Bermejo, A. Criado, J.L. Martinez and M. Garcia-Hernandez, J. Phys.: Cond. Matter 5 (1993) 6295. [15] R.J. Meier, Phys. Lett. Al 12 (1985) 341. [16] J. Jkkle, in: Amorphous Solids, Low-Temperature Properties, ed. W.A. Phillips (Springer, Berlin 1981) p. 135. [17] B. Renker, F. Gompf, R. Heid, P. Adelman, A. Heiming, W. Reichart, G. Roth, H. Schober and H. Rietschel, Zeit. Phys. B90 (1993) 325. [18] P. Esquinazi, R. Konig and F. Pobell, Zeit. Phys. B87 (1992) 305. Cl93 F.J. Bermejo, J. Alonso, A. Criado, F.J. Mompean, J.L. Martinez and M. Garcia-Hernandez, Phys. Rev. B46 (1992) 6173; M. Garcia-Hernandez, R. Burriel, F.J. Bermejo, C. Pique and J.L. Martinez, J. Phys.: Cond. Matter 4 (1992) 9581. [20] M. Garcia-Hernandez, F.J. Bermejo, B. Fik, J.L. Martinez N.G. Almarza and E. Enciso, Phys. Lett. Al75 (1993) 217; ibid. Phys. Rev. B48 (1993), 149. [21] A. Criado, F.J. Bermejo, M. Garcia-Hernandez and J.L. Martinez, Mol. Phys. 80 (1993) 1263; ibid. Mol. Phys., 82 (1994) 787. [22] F.J. Bermejo, A. Criado, J.L. Martinez, E. Enciso and B. F&, Physica A 201 (1993) 286. [23] F.J. Bermejo, E. Enciso, A. Criado, M. Garcia-Hemandez and J.L. Martinez, Phys. Rev. B 49 (1994) 8689. [24] F.J. Bermejo, A. Criado, M. Garcia-Hemandez, J. Alonso, C. Prieto and J.L. Martinez, J. Phys.: Cond. Matter 6 (1994) 405 and references therein.
NEUTRON SCATTERING DATA ANALYSIS
NUCLEAR INSTNUMENTS
BMETNODS IN PHYSICS
=i!EEY”
EISEVIER
Molecular simulations-aided analysis of inelastic neutron scattering spectra F.J. Bermejo”p*, E. Encisob, A. Criado” “Institute de Estructura de la Maten’a, Consejo Superior de Investigaciones Cientijicas, Serrano 123, E-28006 Madrid, Spain “Departamento de Quimica-Fisica I, Universidad Complutense de Madrid, E-28040 Madrid, Spain ‘Institute de Ciencia de Materiales. Consejo Superior de Investigaciones Cientijicas, and Departamento de Fisica de la Materia Condensada, Universidad de Sevilla, P. 0. Box 1065. E-41080 Sevilla, Spain
Abstract The present communication illustrates, by means of three case examples, a way for extracting quantitative information from inelastic neutron spectra on samples characterized by complicated structural and/or magnetic dynamics, where the sole approach employing either experiment or computer simulation fails to provide results free of considerable ambiguities.
1. Introduction The information comprised in the S(Q,w) dynamic structure factors as derived from experiment or model calculations for simple systems such as Bravais crystals or monoatomic non-crystalline matter, suffices to specify their dynamical states, and fully quantitative comparisons between experiment and calculation are now common. However, detailed information regarding the microscopic dynamics of a vast class of materials cannot be derived directly from the measured structure factors without recourse to strong, simplifying assumptions. In particular, studies on the dynamics of non-crystalline multicomponent systems, those composed by particles of non-spherical shape, which can exhibit rather intricate dynamics as far as their internal degrees of freedom are considered, or those evidencing an interplay between magnetic and structural excitations, very often encounter substantial difficulties in specifying the microscopic origin of the observed excitations. To exploit fully the capabilities of present day neutron instrumentation when used to investigate systems belonging to the abovereferred classes, the need to recourse to various kinds of computer modelling was early recognized (to our knowledge the study of the Rahman
*Corresponding author.
constitutes the first attempt to calculate quantities directly amenable to experimental scrutiny Cl]), and for some particular applications such as disentangling the different contributions to the measured total S(Q) structure factors (or to g(r)) they now constitute a widely accepted tool (see for instance [2]). The situation is, however, rather less advanced when S(Q,o) is considered instead. Routine use of computer codes adequate for the study of time-dependent properties strongly relies upon the physical soundness of the model interparticle potentials employed for the calculation, something which in a good number of cases pre-empts any truly quantitative comparison between calculation and experiment. However, such approaches after an adequate parameterization of the model potentials, can serve, in some selected cases to separate the various contributions to the observed response, and therefore to provide a microscopic insight into the dynamics. Three case examples are discussed comprising: (a) the experimental verification of the existence of kinetic sound modes in multicomponent liquids, (b) an approximate separation of the magnetic and structural responses in magnetically disordered systems, and (c) some insight into the microscopic origin of the low-frequency excitations appearing in glassy materials. Since most of the ensuing discussion refers to work which has already appeared, the interested reader is referred to the original references for further details.
0168~9002/95/$09.500 1995 Elsevier Science B.V. All rights reserved 0168-9002(94)00934-l
SSDI
NEUTRON SCATTERING DATA ANALYSIS
60
F.J. Bennejo
et al. /Nucl. Instr. and Meth. in Phys. Res. A 354 (1995) 59-65
8.0 lo'* 7.0 1c2 6.0 1C’
Q = 0.25 A-’
5.0 lo’* 4.0 1w2 3.0 lO-’ 2.0 10.’ 1.0 10’ 5.0 w
;
1
4.0 lo’*3.0 UP0.0
0.5
1.0
2.0 UT’-
Q 1.0 w-
1.5
2.0
(i’)
12
I
I
,
1
3.5 3.0 2.5 2.0 1.5 1.0 5.0 2.5 0
ito
( THz)
Fig. 1. (a) The left side shows plots of the longitudinal current correlation function J, (Q, w) of liquid (T = 200 K) methanol as calculated from MD simulations [6]. Values for the wave vectors are given in the inset. The arrows show the values of the average frequencies derived from analysis of the experimental neutron intensities.(b) Frequency positions of the main peaks in Ji (Q, o) from a polycrystalline ide sample as calculated from LD results. The solid and dash-dots lines represent hydrodynamic, orientational averaged longitudinal and transverse sound velocities. Open squares depict the frequency dependence of the most intense peak in J, (Q, w), and the lines with vertical bar and filled lozenges represent the dispersion of transverse and longitudinal sound, respectively, as seen by the dispersion of less intense peaks. (c) A comparison of the dispersion of the main peak in Ji(Q, o) (dashed line with vertical bars) with that measured experimentally in polycrystalline ice Ih [9] (open circle with error bars). Open lozenges and squares with a cross display MD data for liquid water as calculated from the position of the current-current correlations [S]. The solid and dashed straight lines passing through the origin represent longitudinal hydrodynamics sound for the solid and liquid, respectively.
F.J. Bermejo et al. /Nucl. Instr. and Meth. in Phys. Rex A 354 (1995) 59-65
2. Origin of the anomalous dispersion in two componbnt liquids The existence of propagating short-wavelength modes with velocities well above that of hydrodynamic sound exhibiting a dispersion behaviour akin to ordinary sonic waves, in fluids composed by particles with disparate mass-ratios was inferred from results of kinetic theory calculations [3] some time ago. The experimental evidence regarding such phenomena is rather controversial, and in particular, no results have yet been obtained for an archetypal system such is the alloy Li,Pb, where the existence of such anomalous dispersion was inferred from computer molecular dynamics (MD) results [4]. In the search for systems exhibiting such phenomenology, liquid water appeared as an ideal candidate since the large difference in masses of its atomic components and the easiness in carrying out experimental measurements [4]. As a matter of fact, results from MD simulations have repeatedly evidenced the presence of such an anomalous dispersion from plots of the maxima of the longitudinal current correlations Ji (Q, o) = 0’ S(Q, o)/Q’ versus wave vector [S]. However, previous experience with neutron experiments as well as with MD simulations carried out on a related liquid (fully deuterated methanol) [6], had evidenced the presence in JI (Q, o) of noticeable structure (see Fig. l(a)), something which introduces substantial uncertainties when assigning a sonic character to some of the peaks. As a matter of fact, the experimental curves giving the wave vector dependence of the excitations evidenced a dispersion steeper than that characteristic of normal sound [6], reaching values some 2.5-3 times larger than what could be expected from values of the hydrodynamic sound as well as from knowledge of the dispersion of acoustic branches along the main symmetry directions of the single crystal. The microscopic origin of such a complicated structure in JI (Q, w) was then unveiled by comparison of the functions calculated for the liquid with those for a polycrystalline solid derived from Lattice dynamics (LD) computations using the same model potential, and the complicated peak structure in J1 (Q, w) was found to have a clear correlate in the harmonic polycrystal. Following an analogous line of reasoning, and from consideration of the neutron data of Teixeira et al. [7], a LD calculation was carried out for one of the ice polymorphs [S], and some of the most representative results are shown in Fig. l(b). As can be seen, at least three different ‘dispersion’ curves can be traced following peaks of the polycrystalline ice spectra. The peaks assignable to sound propagation can be identified in the crystal spectra and show a standard dispersion behaviour approaching the correct, orientation-averaged hydrodynamic sound limits [8]. On the other hand, the steep dispersion curve which can be traced from frequencies corresponding to the maxima of Jl(Q,o) evidences its
61
non-acoustic character by the fact that the spectrum gets exhausted just past the zone boundary (i.e. near Q,/2 where Q, stands for the wave vector corresponding to the strongest Bragg reflection, Qp z 1.7 A- ‘). A clear correlate of the excitations manifest in the polycrystalline solid was also seen in the functions calculated for the melt by MD procedures, albeit as expected, most of the peaks are now somewhat displaced to lower frequencies as well as considerably broadened. As can be seen in Fig. l(c), the shape of the steep dispersion curve for the ice polycrystal is reproduced at somewhat lower frequencies in the MD calculation of the liquid. If a value for a sound velocity is inferred from the slope at low-Q of these curves, as done in MD works [S] or from analysis of experimental data [7], a speed far higher than that of hydrodynamic sound is found (notice that the ratio of such a slope to that of normal sound is approximately the same in both crystal and liquid). However, reference to the polycrystalline solid evidences that the peaks characteristic of sound propagation are the less intense ones located at lower frequencies, and upon melting, to follow the dispersion of the sound-mode peaks becomes extremely difficult since these are buried within broad bands of inelastic intensity of non-acoustical origin. On semiquantitative grounds, an analogous exercise to that carried for the polycrystal can be performed for the melt, and a normal sound propagation curve can be recovered if the position of the low-frequency shoulder in J,(Q, w) is plotted instead that corresponding to the global maxima. In order to test the validity of the present predictions, neutron experiments on polycrystalline ice Ih were also carried out [9], and the results also shown schematically in the figure, constitute a nice confirmation of the predictions made from the calculation, and as can be seen from the graph, the same microscopic phenomena giving rise to the steep dispersion in polycrystalline ice are operative in liquid water. It seems then clear that some caution has to be exercised when analysing neutron or MD results of complicated systems such are most molecular liquids, since the relatively large number of excitations present in such samples can easily lead to infer the existence of strongly anomalous behaviours, when, as exemplified in this case, such apparent anomalies simply arise from geometrical (orientational averaging) effects.
3. Separation of nuclear and magnetic responses in magnetically disordered systems The magnetic dynamics of some insulators such as the condensed phases of molecular oxygen have resisted any truly quantitative analysis since the pioneering work carried out at Leiden near the turn of the century (for studies on the magnetic susceptibility, see [lo]). Even in the highest temperature (liquid) phases (the magnetic
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dynamics of the liquid is discussed in Europhys. Lett. [ll], and that of the plastic crystal in J. Phys.: Condens. Matter [ 1l]), experimental evidence from polarized diffraction has shown that the spin dynamics substantially departs from the idealized isotropic paramagnet behaviour, and as a matter of fact, the characteristic spin correlation lengths inferred from such studies turned out to be not too different from those found in the plastic crystalline phase which forms upon freezing the liquid. Although a separation of magnetic and structural responses was achieved by means of polarized neutron diffraction [12], the reported attempts to separate the dynamical responses [13], even in the lowest temperature, magnetically ordered phase, have encountered a number of difficulties as a result of the relatively low count rates and the difficulty in preparing single crystals, which lead to results somewhat at odds with others derived from different experimental (optical spectroscopy) sources. For those phases where the coupling between the magnetic and nuclear dynamics can be deemed as weak, (liquid, plastic crystal and /.I phases), an approximate separation between both dynamical responses was attempted. To proceed, the total dynamical structure factors were measured by means of cold neutron spectroscopy and the measured intensity was analysed as t-11, 141,
A glance to the referred figure evidences that such a separation can be considered as reliable at least up to transfers of x 5 meV, thus encompassing the most significant part of the spectrum. The analysis of &,,(Q,w) was subsequently carried out in terms of a model for an exchange-coupled Heisenberg magnet, and relevant information regarding the spin dynamics in this phase as well as the plastic crystal and liquid phases were then derived [ll, 141. As an indication about the reliability of the separation procedure, the magnetic single-differential cross-section calculated from integration of Smag(Q,w), that is, (f*(Q)>
cc
s
%ax do La,(Q,4
L,(Q)
(4)
- %i”
is compared with the quantity measured by polarized diffraction in Fig. 2. As can be seen the calculated and measured curves show an acceptable agreement if the statistical errors in the latter and the effects from the cutoffs in the integration of the former, which arise from the finite range of energy-transfers accessible to experiment, are taken into accou$. In particula! for wave vectors within the range 0.6 A-’ < Q < 1.5 A-‘, where the kinematic range accessible to cold-neutron spectroscopy comprises the most significant part of the spectrum, the agreement between the measured cross-section and its estimate can be considered as good. (2) From the isolation of the magnetic response, estimates for the contribution of magnetic fluctuations to the speci&,,,(Q,4=dCl -exp(-RwlkeT)l~~F(Q,w), (3) fic heat were calculated and found to follow a T - ‘I2 law characteristic of spin waves (antiferromagnons) with a value for the energy gap (i.e. Q = 0 magnons) of where A is a global scaling constant and R(Q,w) the z 4.6 meV which comes rather close to that predicted for instrumental resolution function, the nuclear dynamics encompassed in S,,, (Q,w) was modelled from computer molecular and lattice dynamics calculations, the magnetic scattering intensity is specified in terms of the a 0.20 Smag(Q, o) spin dynamics correlations and the prefactors E stand for the neutron-magnetic system coupling, Lande 3 0.15 2 $4 factor and the magnetic form-factor for the oxygen ii P PO molecule,f(Q), which can be modelled from knowledge e 0.10 P 1; its spin density distribution. Notice that both static and % dynamic spin correlations are comprised within the ,@ f .p 0.05 E product xo F(Q, W) of the static susceptibility times the 8 GE; spectral weight function, and that it is precisely this the 0.00 LmII quantity to be determined. 0.0 0.5 1.0 1.5 2.0 The form of S,,, (Q, o) was then inferred upon subQ (;4-‘) traction of S,,, (Q, w) from the total experimental intensity. A comparison between the inelastic intensities for Fig. 2. A comparison between the magnetic single-differential fixed values of the energy transfer is shown in [ 14, Fig. 31 cross-section for j%oxygen as measured by polarized neutron for the orientationally ordered /I phase, where the specdiffraction [12] (filled lozenges with error bars) and that caltrum of lattice excitations was calculated by means of culated as integrals over the dynamic structure factor for magnetic excitations (open circles with a dot). a molecular Born von Karman approximation [14].
F.J. Bemejo et al. /Nucl. Instr. and Meth. in Phys. Rex A 354 (1995) 59-65
for a distorted triangular lattice [15]. Finally, the analysis of the strictly elastic S (Q, w = 0) intensity revealed the presence of two weak Bragg reflections of magnetic origin, the assignment of which was later confirmed in a polarized neutron experiment, which evidence the presence of static, short-range magnetic order, not evidenced in diffraction experiments, since they are masked in any total scattering experiment by the strong background of paramagnetic fluctuations which dominates the spectrum.
4. Microscopic origin of the ‘Boson’ peak in structurally disordered matter The present of a low-frequency (0.5-5 meV) broad inelastic peak in the neutron and Raman spectra of a wide variety of glass-forming materials (the Boson peak) has been considered for years as one of the fingerprints of glassy dynamics [16]. The relevance of such a spectral feature stems from the fact that excitations within the same range of frequencies can give due account of the thermal (bump in plots of C(T)/ T 3 versus temperature at 5-20 K) and transport (plateau in the thermal conductivity within a comparable range of temperatures) anomalies in most of the cases examined so far. However, recent studies on materials as diverse as fullernes [17] or metallic alloys at very low temperatures [lS], have evidenced the presence of closely related phenomena in solids which by no means can be considered as topologically disordered. Under such circumstances, efforts towards the understanding of the microscopic origin of the low-frequency peaks in structural glasses seem indicated. Our approach consisted in relating first the excitation spectrum of the glass to that of the harmonic (polycrystal) ground state, either calculated by LD or MD means or accessed experimentally by neutron or Raman scattering, so that an absolute scale of frequencies is available. From comparison between the neutron spectra of samples in both, polycrystal and glass forms, unambiguous information regarding the microscopic origin of the lowfrequency vibrations in the glass can be gained. For the polycrystal, the analysis of the mode eigenvectors giving rise to intense peaks in the vibrational density of states (generalized frequency distribution), enables to establish the character (translational, rotational internal-mode or mixtures of them) of the excitations in a quantitative manner. As seen for a limited number of cases [19-211, the lowest-frequency peak in the incoherent approximation to the neutron spectrum of the crystal, (i.e. S(Q, o) a Z(w)n(o)/w, where Z(w) is the generalized frequency distribution and n(w) a Bose factor), corresponds for molecular materials to excitations strongly mixed in character, where roto-translational contribu-
63
tions are intermingled with those arising from lowfrequency peak internal modes such are chain torsions in selenium or internal pivoting motions of the benzne rings in ortho-terphenyl (OTP). In counterposition, the glass shows a far broader peak located at somewhat lower frequencies than that of the crystal [19-211, being in some cases such as OTP barely separable from the elastic peak. Even if a complete model assignment of the excitations contributing to the ‘Boson’ peak in glasses along the lines followed for the polycrystal cannot be achieved, a partial separation of the components giving rise to intensities in Z(o), into center-of-mass (COM), rotational (ROT) internal-mode (IM) contributions and coupling terms of all of them can be achieved from results of MD calculations. The interesting point to note regards the fact that after such a decomposition is performed, the available evidence for glasses as distinct as methanol, selenium or ortho-terphenyl [19-211 suggests, as exemplified in Fig. 3, that the precise location in frequency of the Boson peak is mainly governed by the COM (in case of methanol or OTP) or van der Waals (for Se) components. In consequence [22], the bump in C(T)/T 3 appears in Fig. 3, at a temperature corresponding to the maxima of the COM contribution to the Boson peak, and most of the spectral intensity arising from other kinds of excitations will contribute to this property as a rather flat background [19-211. In counterposition, as recently evidenced from the analysis of data regarding glassy Se, the presence of low-frequency internal modes of the Se chains constitute and efficient phonon scattering mechanism, which can quantitatively account for the observed behaviour with temperature of the thermal conductivity c231. A related topic, although seldom addressed concerns the characterization of such a peak as arising from mostly single-particle or mostly collective motions. The relevance of a proper answer to such a question stems from its implications towards the confirmation of widely different scenarios proposed to account for the most significant features of glassy dynamics. From the limited experience available [24], the wave vector dependence of the position of the Boson peak evidences a rather strong collective character, something which seems to be in agreement with the fact that mostly COM motions are involved in these modes. In summary, the study of low-energy excitations in disordered matter constitutes one of the prime examples where a close knit connection between experiment and simulation is a clear prerequisite before any fully quantitative advance takes place. The basic building blocks for any comprehensive theoretical approach still rest to be laid upon, and the combination of neutron scattering with microscopic simulations seem to be decisive in this endeavour.
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0
5
10 15 20 ilw
25
30
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40
lh
(meV)
(meV)
8. (4 6.
4,
2.
0. 0
5
‘UK)
10 15 20 25 30 35 40 'I'W
Fig. 3. (a) The vibrational density of states (solid line) for glassy methanol at T = 35 K as calculated from molecular dynamics simulations. T’he dash-dot line shown the COM component, the dotted line the rotational (ROT) component, and the dash-dots the coupling (roto-translation) terms. (b) A comparison of the calorimetric specific heat [19-211 (open circles) with that derived from neutron spectra (filled circles), and that calculated from MD results corresponding to the Z(w) shown above (solid line). The contribution from COM motions is shown by the dots, and that from ROT by dashes. The dashdots line represents the sum of both contributions (excluding the coupling terms. The line with vertical bars and the one with asterisks, show the same magnitude for the polycrystal as measured calorimetrically and calculated from LD results. (c) The vibrational density of states as calculated for glassy Se (solid line) compared with two sets of experimental measurements [23] (open and filled lozenges. The dotted line shows the van der Waals contribution. (d) A comparison between the specific heat for glassy Se (solid lie with filled circles) with that calculated from the MD results (thick solid line), and that from the van der Waals contribution alone (dotted line). Filled triangles and circles represent experimental measurements for trigonal Se, and are compared with the calculation from LD results (line with vertical bars).
5. Conclusions Our aim in the present communication has been to illustrate the capability of a combined approach, experiment plus molecular simulations, to extract quantitative information from experiment in systems where the sole
analysis of the inelastic intensities cannot provide unambiguous answers. Notice that on the other hand, simulation data, by its very nature, also fails to provide quantitative answers, unless its results are continuously confronted with experiment. As shown in the three cases discussed above, relevant, albeit semiquantitative microscopic information can be
derived following such an approach. In counterposition, attempts to discuss experimental data of the kind considered here without recourse to information arising from simulations have to be confined to discussions on phenomenological (meso- or macroscopic) grounds, thus being unable to exploit the advantages,of exploring matter with a truly microscopic (in time and length-scales) probe such that provided by neutron scattering. Acknowledgements
This work has been supported in part by DGICYT grant No. PB92-0114-C03.
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References [l] A. Rahman, Phys. Rev. 136 (1964) A405. [Z] R.J. McGreevy, these Proceedings (Workshop on Neutron Scattering Data Analysis (WONSDA’ 94), Chilton, Didcot, UK, 1994) Nucl. Instr. and Meth. A. 354 (1995) 1. [3] A. Campa and E.G.D. Cohen, Phys. Rev. Lett. 61 (1988) 853. [4] J. Bosse, G. Jacucci, M. Ronchetti and W. Schirmacher, Phys. Rev. Lett. 57 (1986) 5277. [S] R.W. Impey, P.A. Madden and I.R. McDonald, Mol. Phys. 46 (1982) 513; M.A. Ricci, D. Rocca, G. Ruocco and R. Vallauri, Phys. Rev. A40 (1989) 7226. [6] J. Alonso, F.J. Bermejo, M. Garcia-Hernandez, J.L. Martinez, A, Criado and W.S. Howells, J. Chem. Phys. 96 (1992) 7696, fully quantitative comparisons between neutron data and MD results are given in J. Alonso, F.J. Bermejo, M. Garcia-Hernandez, J.L. Martinez, W.S. Howells and A. Criado, Phys. Lett. Al72 (1992) 177. [7] J. Teixeira, M.C. Bellisent-Funel, S.H. Chen and B. Dorner, Phys. Rev. Lett. 54 (1985) 2681. [S] A. Criado, F.J. Bermejo, M. Garcia-Hernandez and J.L. Martinez, Phys. Rev. E47 (1993) 3516. [9] F.J. Bermejo, E. Frikkee, M. Garcia-Hernandez, J.L. Martinez and A. Criado, Phys. Rev. E48 (1993) 2300. [lo] H.K. Onnes and A. Perrier, Leiden Common. 116 (1910) 1; G.C. De Fotis, Phys. Rev. B23 (1981) 4714. [ll] A. Chahid, F.J. Bermejo, J.L. Martinez, M. Garcia-Hernandez, E. Enciso and F.J. Mompean, Europhys. Lett. 20 (1992) 71; A. Chahid, F.J. Bermejo, E. Enciso, M. Garcia-Hernandez and J.L. Martinez, J. Phys.: Cond. Matter 5 (1993) 423.
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[12] F. Dunstetter, Ph.D. Thesis, Universitt de Paris-Sud (1988). [13] P.W. Stephens and CF. Majkrzak, Phys. Rev. B31 (1986) 1. [14] A. Chahid, F.J. Bermejo, A. Criado, J.L. Martinez and M. Garcia-Hernandez, J. Phys.: Cond. Matter 5 (1993) 6295. [15] R.J. Meier, Phys. Lett. Al 12 (1985) 341. [16] J. Jkkle, in: Amorphous Solids, Low-Temperature Properties, ed. W.A. Phillips (Springer, Berlin 1981) p. 135. [17] B. Renker, F. Gompf, R. Heid, P. Adelman, A. Heiming, W. Reichart, G. Roth, H. Schober and H. Rietschel, Zeit. Phys. B90 (1993) 325. [18] P. Esquinazi, R. Konig and F. Pobell, Zeit. Phys. B87 (1992) 305. Cl93 F.J. Bermejo, J. Alonso, A. Criado, F.J. Mompean, J.L. Martinez and M. Garcia-Hernandez, Phys. Rev. B46 (1992) 6173; M. Garcia-Hernandez, R. Burriel, F.J. Bermejo, C. Pique and J.L. Martinez, J. Phys.: Cond. Matter 4 (1992) 9581. [20] M. Garcia-Hernandez, F.J. Bermejo, B. Fik, J.L. Martinez N.G. Almarza and E. Enciso, Phys. Lett. Al75 (1993) 217; ibid. Phys. Rev. B48 (1993), 149. [21] A. Criado, F.J. Bermejo, M. Garcia-Hernandez and J.L. Martinez, Mol. Phys. 80 (1993) 1263; ibid. Mol. Phys., 82 (1994) 787. [22] F.J. Bermejo, A. Criado, J.L. Martinez, E. Enciso and B. F&, Physica A 201 (1993) 286. [23] F.J. Bermejo, E. Enciso, A. Criado, M. Garcia-Hemandez and J.L. Martinez, Phys. Rev. B 49 (1994) 8689. [24] F.J. Bermejo, A. Criado, M. Garcia-Hemandez, J. Alonso, C. Prieto and J.L. Martinez, J. Phys.: Cond. Matter 6 (1994) 405 and references therein.
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