A neutron Brillouin scattering study of Mg 70 Zn 30

I. Phys.: Condens. Matter 7 (1995) 4775-4785. Printed in the UK A neutron BriUouin scattering study of Mg70Zn30 C J Benmaret, B J Oliver$T, J-B Sucks, R A Robinson$ and P A Egelstafft t Depamnent of Physics, University of Guelph Guelph, Ontaio NIG 2W1.Canada t Los Alamos National Laboratory. LANSCE. Los Alamos, NM 87545. USA lmtitut Laue-Langevin, BP156 F-38M2 Grenoble C6dex 9. France Received 3 FebNW 1995 Abstract. Inelastic neutron scattering measurements have been made with momentum uansfers within the first pseudoBrillourn zone ofa magnesiumlzinc glass af 100 K and 297 K.We describe the applieaaon of a new spectrometer (PHAROS) for these studies. Because the longitudinal velocity of sound is about 4300 m s-', a high incident neutron energy (0.187 eV) was required. Data have been obtained down to Q z 8 nm-' and h o = 25 meV well inside the first pseudoBrillouin zone of the glass. Evidence for two predominmtly longitudinal excitations in the Mg,oZn,o glass is presented. Their positions agree appmximately with theoretical predictions. 1. Introduction By analogy with the Brillouin scattering of light, neutron Brillouin scattering (NBS)describes neutron inelastic scattering measurements made at momentum transfers hQ within the first Brillouin zone. Whereas conventional inelastic scattering experiments on crystals are generally performed in higher-order Brillouin zones to gain intensity and the results are presented after reduction to the first zone, the investigation of collective excitations (such as vibrations or magnetic excitations for example) in disordered matter is considerably more difficult, as no reciprocal lattice exists. However, provided these systems have short- and/or medium-range order, low-order pseudo-Brillouin zones can be defined close to the origin of reciprocal space. Here, the first maximum Qp in the static structure factor S ( Q ) can be regarded as the first, broadened reciprocal lattice point, such that the first pseudo-Brillouin zone extends up to Qp/2 in reciprocal space [I]. An important role of NBS experiments is the investigation of the collective dynamics, by determining the dynamic structure factor S(Q. w ) , for liquids and amorphous materials and the extraction of the single-excitation part S , ( Q , w ) of S ( Q , w ) , where hw is the experimentally observed energy transfer. This procedure is useful because the peaks in Sl(Q. w). which contain information on the collective dynamical modes of the system, are more easily accessible to theory and more reliably interpreted. As Q increases these peaks become less pronounced, and at Q > 2Qp the structure of S(Q, o)is dominated by the vibrational density of states, describing single-particle motion [1,21. A unique advantage of NBS experiments is their ability to distinguish between longitudinal and transverse excitations. as whilst neutrons couple directly to longitudinal vibrations they only couple to transverse vibrations via Umklapp-type processes outside the first Brillouin zone. 7 Present address: Department of Physics. Oklahoma State University. 0953-8984/95N4775t11$l9.50 @ 1995 IOP Publishing Ltd Stillwater, OK 74078, USA. 4175 4116 C J Benmore et a1 Such experiments place stringent kinematic constraints on the spectrometer, since they require a large w range at the smallest Q values. A full picture of the dispersion curve necessitates the use of high neutron energies at scattering angles close to the straight-through beam, and there is a large amount of associated background and multiple scattering with few single scattering events [3]. The metallic glass Mg7oZnso can be melt-spun in continuous ribbons, from which a suitable neutron scattering specimen may be made. Since the scattering amplitudes for Mg and Zn are similar (5.38and 5.69 fm respectively) and the coherent scattering is about 98% of the total, it is a suitable sample for the investigation of collective excitations. We may represent, therefore, the scattering cross-section by an approximate formula where E is the average scattering amplitude for Mg and Zn. Moreover, this glass has been the subject of investigation in previous experimental studies down to momentum transfers of 12 nm-' at energy transfers of 28 meV. just inside the boundary of the fvst pseudo-Brillouin zone (Qp/2 = 13 nm-I) [4,5].In addition, there are extensive computer simulations using apriori pair potentials and other theoretical work [6]. Our previous experimental study of Mg7021130 on the HET spectrometer at the Rutherford Appleton Laboratory [4,5]used 0.18 eV neutrons, and an angular range of 3.3" to 7". The sample was enclosed in a multicellular cadmium grid, in which each cell was 6 mm square and 2 cm deep. This arrangement enhances the first scattering compared with multiple scattering processes. Data were taken near the boundary of the first pseudo-Brillouin zone. In order to extend the data further into this zone it is necessary to use an instrument that offers significantly lower scattering angles, and also to make an improved sample geometry. These were the technical objectives of the new experiment described here: in addition, we have begun a study of the temperature variations. The intention of this paper is to present out NBS results on glassy Mg7oZn30 in which the accessible Q-range has been extended down to momentum transfers of 8.0 nm-'. The ncw experimental conditions could be met by the new instrument [71, PHAROS,at the Los Alamos pulsed, spallation source facility (LANSCE). We employed a relatively high incident energy (0.187 eV or a neutron velocity of 6000 m s-' compared with the velocity of longitudinal sound in the sample of 4300 m s-'), and scattered angles down to 1'. (However, due to the low counting rates, we averaged several detectors giving a practical minimum of 2.W.) Moreover, we have used a more sophisticated analysis of the data than was possible in the earlier experiments, since the majority of measurements were taken well inside the first Brillouin zone. These new data are compared to existing theoretical predictions [6]. A further goal of this experiment is to examine the variation of the spectral shape with temperature: for example, at room temperature the neutron scattering cross-section is higher than at low temperature (e.g. LOO K) where the excitation peaks may become sharper and better defined. In section 2, the theoretical background will be presented briefly, and in section 3, the experimental samples and procedures will be described. This will be followed by sections on the data analysis and interpretation, and their comparison with published theoretical predictions. 2. Theoretical remarks The structure (peaks and shoulders) of the single excitation part &(e,w) of the dynamic structure factor S ( Q , w ) contains the information about the collective dynamics of the A neutron Brillouin scattering study of Mg7,,Znlo 4777 atoms. Thus we separate this latter function into three parts as: S(Q. U) =SdQ. + Si(Q, + S d Q , 0) 0) W) (2) where So represents elastic scattering and S,,,represents the multi-excitation component. The (smallest) Q values which can be reached in an NBS experiment at a given o depend on the scattering angle 0 and the incident energy EO or incident (k,) and scattered (k) wavevectors necessary for the excitation of the modes. These quantities are related by the conditions for conservation of momentum and energy, namely, Q2=kz+k2-2kk,cosB ' h o=-(k:-k2) 2m (3) where m is the neutron mass. They may be combined into the equation: h Q = [ Z m ( E o - h w ) - 2 ~ ~ ~ 0 ~ 0 ] " ~ . (4) In this section we assume that these equations yield values in the first pseudo-Brillouin zone of the Mg702nw glass. We start from SI(Q, w ) for an elementary polycrystal with isotropic force constanls and an atomic mass M. and scattering in the first Brillouin zone [SI i.e. 2 W hwJZksT SI(Q,4 = e e [~6 ( Q - W'(o- o q ) p ( o q ) ] (5) where A= BhQ2 4Mosin h(liwf2k~T) and e-2Wis the Debye-Waller factor, (q,oq)are the wavenumber and frequency of a phonon in the first Brillouin zone, p(o,) is the density of states normalized to unit area and B is a dimensional factor. For a linear branch, with a sound velocity c, we would have U,, = cq. It is clear that for low values of Q the amplitude factor A is small and the density of states (which varies initially as 0,") is also small. However, in this limit the product Ap(w) is independent of o.Because these modes in a crystal are expected to be long-lived, they are represented as 6 functions. For damped modes in an amorphous material, they might be represented as Lorentzian functions, with a peak width r proportional to Qz in the (small-Q) hydrodynamic limit. Provided that they have sufficient intensity, these peaks are therefore easiest to detect, and their width l' is most reliably measured at the smallest momentum transfers. But in practice, the present experimental limitations have restricted our work to the peak positions only. To do this we need to subtract the multiphonon and multiple scattering intensities, by methods we shall describe in section 4. 3. Experimental details The magnesidzinc samples were fabricated from high-purity elements by melt-spinning. The sample container comprised a flat aluminium alloy frame container of volume 8.0 (height) x 5.5 (width) x 2.0 (depth) cm3 intersected by a 6 mm x 6 nun square grid of 0.5 mm thick cadmium blades. Strips of the Mg70%3o melt-spun glass (of mass = 55.4 g 4778 C J Benmore et a1 and mean density = 1.72 g ~ m - were ~ ) pressed into each of the 96 cells in the grid. Once loaded, the sample container was flushed for 30 minutes with helium gas before sealing, to avoid any additional background scattering from air left in the container, Two experiments were performed on the Mg70Zn3o glass, at temperatures of 100 K and 297 K, using epithermal neutrons. An incident beam energy of 0.187 eV (close to the absorption maximum for neutrons in Cd) was chosen to provide a neutron velocity greater than the velocity of longitudinal sound, and to maximize the absorption of the grid and thereby reduce the large background of multiply-scattered neutrons. An identical grid was also used for the empty sample container run, and two halves of the same grid construction were placed either side of a 3 mm vanadium plate at room temperature for the calibration run. All sets of measurements were carried out under identical experimental conditions and temperature fluctuations were kept to within 0.5" between runs. The fast Fermi chopper (see [7]) for its characteristics) was run at 240 Hz for these experiments, giving a measured resolution of 10.5 meV FWHM at the elastic line, which was constant over the entire angular range. This frequency is slower than the optimum, giving broader than optimum resolution, a choice that was made to increase the observed intensity. We note that 5 meV corresponds to a timescale, given by % / w , of 0.1 ps. PHAROS [7] is a direct-geometry inelastic chopper spectrometer, purpose-built for NBS experiments requiring high-o-resolution, low-angle work (in another mode it can also be used for high-angle, high-resolution work). It has a water moderator poisoned with gadolinium places near the spallation target, and a 20 m incident flight path (see figure 1). Two mechanical choppers are installed in the incident beam. At 14 m from the moderator, a double-bladed chopper, rotating at 120 Hz (the fundamental frequency of the linear accelerator) is employed to suppress the high-energy neutron background. Downstream (at 18 m) the high-speed Fermi chopper (referred to above) is phased with the accelerator pulse to select the required incident neutron energy. A boron carbide mask defined the incident beam of 7.5 cm x 5.0 cm at the sample position, and the scattered neutrons travelled a further distance of 5.75 m along the secondary path before reaching the detector plane. The detector grid comprised five banks of linear position-sensitive 3He detectors, each consisting of eight, one inch diameter and 36 inch long detectors mounted vertically. For convenience, this arrangement was asymmetric in the horizontal plane with two banks located on each side of the incident beam (covering angles from 1.3" to 5.4" and 3.5" to 6.9", separately), with an additional single bank covering the angles between 5.8" to 8.8" on one side. In total, this gave an angular range of 1.3" to 8.8" in the forward scattering direction and with the vertical position sensitivity gave an x-y distribution of data points. The detector distribution for the experiment is illustrated in figure 2. Detector calibration measurements performed before and after the experiment showed a high degree of detector stability throughout the experiment. Due to the relatively low flux at high incident energies, low scattering angles and attenuation of the scattered beam by the cadmium grid, sample run times of 9 days at 100 K and 4.5 days at 297 K were required to obtain useful statistics on the summed data. (A longer run at 297 K would have been desirable but was not possible.) The pixels of the x-y detector were grouped into four rings centred at zero degrees. The profiles for these constant-@ rings are shown in figure 3, together with theoretical curves for the two longitudinal modes in Mg&kro glass (see Hafner [6], figure 6). From an experimental viewpoint the situation is very favourable as the U-Q rings intersect with the theoretical dispersion curves at an angle close to 90". In addition, the width of each ring is only spread over a 4 meV range when intersecting either of the two dispersion (we) A neutron Brillouin scattering study of Mg70Znao 4779 Bulk shield Building wall Figure 1. A schematic diagnm of the initial sac-up for [he PHAROS spectrometer at Los Alamos. Full details wn be found in [TI. 0 A 2 Ll 0 4 6 10 Figure 2. The angular distnbution of the detector pixels on PHAROS for this eaperiment. The decreae beyond 5' is due to the limited size of the reclangular detector a m y . curves shown in figure 3. The ringed data sets were converted from time-of-flight onto an 4780 C I Benmore et al 60 40 -40 -60 0 5 10 15 20 25 Q b-') Figure 3. The four cuts in u-Q space along which x-y data were collected into constantangle data (solid lines). The dashed lines conespond to the theoretical c u ~ y e sfor the two longitudinal modes (seetext) calculatfd by Hafner (1-51 figure 6) for the metallic glass Mg.loZnw (u=longitudinal acoustic, V)=longiludinal optic). energy scale, the backgrounds subtracted and analysed using a two-stage process. 4. Analysis of data In the first stage of the analysis, the ringed data were rebinned in energy onto a linear scale using 2.0 meV bins. A correction for neutron capture efficiency in cylindrical detectors was made 191, and the data sets normalized to the ringed vanadium spectra using the number of counts under the vanadium elastic peak. An energy-independent correction of the type described in [IO] was applied to the data and empty container scattering using the equation where I,,, is the corrected intensity. Is+c and IC denote the measured intensities of the sample plus container and smoothed empty container runs respectively. As.sc, Ac.sc and Ac.c are the corresponding absorption correction factors calculated for Bat-plate geometry, and M is the monitor count ratio of the two runs. For the second stage, a two-channel maximum entropy method [I I] was used to separate out the relatively sharp, single-excitation scattering events from the large multiply-scattered and multi-excitation background. The maximum entropy algorithm creates two channels; one is designed to accept only sharp features i.e. the peaks (see equation 2) corresponding to the required So(@, U ) S l ( 0 , o),whilst the other accepts only broad structure i.e. S,,,(0, U ) . Our experimental data will be presented as constant0 results, and the relationship + A neutron Brillouin scarrering study of Mg,oZnso 4781 energy transfer (mev) Figure 4. Step in data handling for Q = 5.35‘ at temperatures of 100 K ( ( a ) to ( d ) ) and 297 K ((e) to (k)). respectively. (U)and (e): initial spectm corrected for detector efficiency and normalized to vanadium. (b)and rebinned energy spectra for sample plus can and smoothed empty can rum. (c) and ( 8 ) :sampleonly spectra and broad maxtmum entropy mmponent. ( d ) and (h): SI(0. o)t SI (Q, w ) for the Mg.ioZnle glass (error bars) and So@, 0 ) (line) obtained from the vanadium s p t m v): between SI(0, o)and SI(Q, o)for example, may be obtained readily from equations such as (4) and (5). It was expected that the multiple-scattering and multi-excitation contributions (which include both multiple and single scattered multiphonon terms) would dominate the broad part of the spectra. To aid in the analysis, the following constraints were imposed on the two channels: the broad component should be smoothly varying with the ends of the spectra decreasing to a constant value, and the excitation peak area in the sharp spectra should increase as Q2 (see section 2). Thus the lowest angular spectra taken at an average angle of 0 = 2.87” were fitted assuming virtually all the inelastic scattering was due to multiple scattering. The width of the angular rings ranged from A Q = 1.5 nm-I for the middle two angular rings to about 3.9 nm-’ for the highest angular ring. At each angle, a comparison of the broad components at both temperatures showed them to be similar in shape, although the T = 297 K data are greater in intensity on the neutron energy gain side 4782 C J Benmore et a1 energy transfer (meV) + Figure 5. SdQ, U ) &(e. U ) (see equation 2 ) spectra for MnoZnlo ilt ( 0 ) 100 K and ( b ) 297 K derived by wllecling x-y data into angular goups and interpreting by the maximum enmpy method. The points (with mor bur) represents he experimental data while the solid line represents the maximum envopy fit to the data (see [IO]). as expected. Moreover, both temperature runs showed a strong angular dependence of the broad component, a possible effect of the cadmium grid in reducing the multiple scattering below the level of the multi-excitation processes. These broad spectra were subtracted from + the original four data sets to give the function SO(@,w ) SI(@, w ) (measured at fixed scattering angle 0). Figure 4 shows this process at both temperatures; the original data, the rebinned data and the separation into broad and structured components are illustrated. Figure 5 shows the SO(@, w ) SI(@. w ) data points with the most likely smooth line through the data. Differences taken between detector rings (at the same and different temperatures) showed little structure above the statistical noise, which suggested the presence of a plateau in the dispersion curve which is evident when the ( W . Q) relationship for the observed peak positions is plotted in figure 6. An alternative interpretation may be made by calculating the density of states function + A neutron Brillouin scattering study of MgmZnso 4783 Figure 6. An +Q plot for the peaks in Sr(0, U ) for the melallic glass Mg$na. The results of the present investigalion are shown as solid diamonds and s q u m for the 291 K run and 100 K run, respectively. The solid circle denates the lower beak position obtained hom the fi (0.w ) function shown in figure 1. The open symbols correspond to the previous results on the same metallic glass made on the IN4, IN6 and HET spemomelers 14.51. The top line corresponds to the theoretical pzak positions in S,(B. w). calculated by J-B Suck (see Egelsraff [I21 figure 3). The bottom line corresponds 10 b e dispersion relation for the 'longitudinal acoustic' mode calculated Q by Hafner (see [6] figure 6). The error bars shown on our data are a combination of the vatialion in Q xising from the finite width of the angulv ring and the shape of the resalved peak in SI(@, w ) . f ' ( Q , w ) as in [6]. If we assume there is no incoherent contribution, then by using equation (3) of 141, this quantity may be written, where ci and M, are the concentration and mass of component i . as: where and SA6 is the single-excitation partial dynamic structure factor f ' ( Q , w ) will denote a w ) partials, and ft(Q, o)will denote the similar weighted sum weighted sum of f,s(Q, using constant angle data in place of constant Q data. These Functions are shown in figure I. For OUI case, the Debye-Waller factor was calculated to be very close to unity (using equation (3.63) in [8]). The advantages of this presentation are that it emphasizes the energy transfer intensity distribution, and that a theoretical predication is available in [6] (figure 4 of [6]). In order to separate ft(6', w), the elastic peak contribution to the spectra, S&3, w ) needed to be subtracted from the spectra shown in figure 5 . &(e, w ) was C J Benmore et a1 4784 obtained by fitting the elastic peak of the vanadium spectra with a Gaussian distribution (see figures 4(d) and 4(h)). To improve our statistics, we have added together the fl(0,o) spectra for both the 4.29" and 5.35" rings (these angles were chosen as they have the best 8 resolutions) and both temperatures. Finally, the comparison of the experimental and theoretical functions is shown at figure 7. - h 3 8 g 6 ru- ru 0 0 0 10 20 30 40 energy (mev) 50 0 IO 20 30 40 50 energy (mev) Figure 7. A comparison of the /'(e. w ) function calculated from the sum of individual /.s(b', 01 functions t&en hoin [61 and shown in 7(b).and the function f,(s,o)(full line in 7(a)) derived from our d3ta at anaes of 4 . 3 ' . 5.39 and both temperatures combined. The individual partial dynamic stmciure factors xe represented by /M&Q, 0 ) (dotled line). h(Q. w ) (dashed line), / M ~ ~ ( Q U) , (cham line) shown in (b) and the solid line in (a) is a fit to the experimental data. The two CUNS in 7(n) (from experiment) and ( b ) (from theory) have not been drawn to the same scale. 5. Discussion and conclusions It can be seen in figures 5(a) and (b) that SI(e. w ) exhibits a single-excitation peak even at the lowest angle in both the 100 K and 297 K data sets. Furthermore, despite the shorter run time, the greater count rate associated with the single-excitation peaks in the room temperature experiment is clearly illustrated, especially at the largest angles. The ratio of the areas of the measured single-excitation peak at the two temperatures is also in reasonable agreement with those values calculated from an isotropic Einstein model of the glass given by equation (5). The high signalhackground ratio obtained on PHAROS and an improved data treatment process have yielded overall better data compared with our previous experiments [4,5], although the energy resolution is worse. From the data in figure 5, the peak positions (in w and Q)have been extracted and are plotted in figure 6. Also shown in this figure are our earlier data identified as a longitudinal and it can be seen that the new and old data are in good mode in the MgToZn30 glass [4,5], agreement. The new data also extend significantly further into the first zone. The solid lines in this figure are based on the calculation by Hafner [6]of the two longitudinal modes: as in our earlier work, we estimated the theoretical peak positions in SI(8, w ) from Hafner's results. Only at the lowest momentum transfers (calculated to be Q 6 3.1 nm-] in 161) A neutron Brillouin scattering study of Mg70Znao 4785 can the acoustic and optic modes (as defined by Hafner) be clearly identified. For larger Q values, towards the zone boundary ( Q 6.2 om-' in [e]) it has been predicted that the acoustic optic modes will tend to be dominated by the vibrations of the Zn and Mg atoms, respectively. The main phonon intensity measured in this study is therefore expected to be due to the longitudinal 'optic' mode which mainly corresponds to the vibrations of the lighter Mg atoms. There will also be a less intense phonon peak at lower energy transfers, close to the elastic peak, primarily due to the Zn atomic vibrations (i.e. the longitudinal 'acoustic' mode) e.g. see figure 7 ( a ) . This explanation is the simplest interpretation of the present results i.e. that two peaks are found in the spectra at around 18 meV and 30 meV. Due to the experimental set-up, there is a finite band of Q values over which the phonon peak energies were averaged: however, this phenomenon does not alter the overall results of this experiment. The broader peaks observed at higher energy transfers in figure 5 ( E 40meV) are probably due to residual multiphonon effects; a full calculation of the expected S ( Q , o) intensity is required for a detailed description of this w region. The comparison shown in figures 7 ( a ) and ( b ) suggests that a larger contribution from the Zn-Zn correlations is seen experimentally than is predicted by theory. In addition, the peak positions are not in full agreement, which suggests that the small discrepancy between theory and experiment in figure 6 may be real. It is concluded that a modern NBS instrument can extend the data on high velocity of sound materials (e.g. Mg,oZn,o) into the first Brillouin zone. In this way, the nature of the modes can be identified positively and excellent comparisons with theoretical predictions (based on fundamental information) may be attempted. > > Acknowledgments We would like to acknowledge the assistance of P S Lysaght, E Larson and J P Sandoval during the c o m e of our experiments, and the advice of Dr D S Sivia during the maximum entropy processing. We are also grateful to the NATO Collaborative Research Grants Program for a travel grant and to the NSERC of Canada for their financial support of the Canadian participants. This work was funded in part by the division of Basic Energy Sciences of the US Department of Energy. References Suck J-B, Rudin H. Giintherodt H-J and Beck H 1983 Phys. Rev. Lett. 50 49 Suck 1-B, Rudin H, Giintherodt H-J and Beck H I980 3. Phys. C:SolidState Phys. 13 LlC45. 1981 14 2305 Robinson R A 1989 Advanced Neutron Sources 1988 OOP Con! Ser. Topics in Applied Physics) p 31 I Suck J-B, Egelstaff P A, Robinson R A. 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