The vibrational motions in vitreous silica at high temperatures

Journal of Non-Crystalline Solids 109 (1989) 295-310 North-Holland, Amsterdam

295

T H E VIBRATIONAL M O T I O N S IN VITREOUS SILICA AT H I G H T E M P E R A T U R E S Bj/3rn GRANI~LI and Ulf D A H L B O R G Department of Reactor Physics, Royal Institute of Technology, S-100 44 Stockholm, Sweden Received 25 June 1988 Revised manuscript received 4 January 1989

Vitreous silica was studied at the temperatures 293 K, 673 K and 1073 K using inelastic neutron scattering. The zeroeth moment of the dynamic scattering function S(Q, E) agrees well with neutron diffraction results. It was found that the structure factors for elastic and inelastic scattering change with temperature in a Q region which reveals mostly long-range atomic interactions rather than those from nearest and next-nearest neighbours. The temperature variation was interpreted as if the structural and dynamical correlations between neighbouring SiO4 polyhedra decrease when the temperature increases. Effective Debye-Waller factors were derived and found to increase linearly with temperature. It was also found that the vibrational amplitudes of the atoms in silica, within the experimental errors, are the same as those in high cristobalite. Through extrapolation to zero wave vector transfer of the spectrum of the longitudinal current-current correlation function, effective frequency distribution functions were obtained as well as generalized structure factors for inelastic scattering.

I. Introduction

Vitreous silica is a typical representative of the AX2-glasses and has in the past been the subject of several experimental and theoretical studies of structure and vibrational properties. In the case of the latter, inelastic neutron scattering and techniques employing the scattering and absorption of light have proven to be valuable tools. An understanding of glasses has also been gained from modelling work on continuous random networks as well as molecular dynamics simulation methods. Below, a brief account is given of some aspects of the structure of silica glass and its dynamical behaviour together with some relevant references. It is commonly believed that the structure of silica glass should contain SiO4 tetrahedra similar to those of the different forms of crystalline SiO 2. The matter of short and medium-range order is subject to intense debate. Two of the more common views are identified as the micro-crystalline model [1-4] and the continuous random network model [5,6]. In the more direct view of Behnke et al. [7], a description of the structure of vitreous 0022-3093/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

silica in terms of crystalline polymorphs of silica is suggested. It is only fair to point out here that none of the existing models is in complete accordance with results from X-ray [8] and neutron diffraction work [9], the level of consistency being roughly the same for all of them. In many cases the experimental technique itself restricts the validity of the results, as for instance when the available range of Q-vector transfer is too limited, or when the energy resolution is insufficient. Essential characteristics of the silica structure may be identified in several different models, but as yet none of these give a fair complete view of the vitreous state. These matters are discussed in detail by Wright [101. Molecular dynamics (MD) simulations [11-14] fail to give good agreement with diffraction results even at short distances mainly because of the strongly distorted SiO 2 tetrahedra. The structure of the simulated glass was found to resemble a randomly distorted high cristobalite structure with a broad distribution of the S i - O - S i angles, forruing a cage-like assembly of tetrahedra connected at the oxygen nodes.

296

B. Gran~li, U. Dahlborg / Vibrational motions in v-Si at high temperatures

In the case of the dynamic properties of vitreous silica, the efforts of theory and experiment may not seem to be on as many divergent paths as is the case for the structure work. The fundamental properties to a large extent, however, still remain to be investigated and are by no means less challenging. Early inelastic neutron experiments [15,16] showed, using the incoherent approximation, that the effective frequency spectrum of vitreous silica is almost identical to the spectrum obtained from low cristobalite for energies between 6 and 30 meV, indicating that the vibrational characteristics are not particularly sensitive to the details in the long-range order. According to phonon dispersion curve measurements on crystalline quartz (see Barron et al. [17] and refs. therein) well defined acoustic phonons with energies in that region exist. The region of low energy transfer (between 1 and 10 meV) of vitreous silica was recently investigated by Buchenau et al. [18], and it was found in three measurements from 50 K to 290 K, that only a relatively small part of the total scattering could be accounted for by acoustic phonons. A fair account of the strong additional scattering observed was given through a model calculation with coupled rotational motion of five SiO4 tetrahedra. This experiment, however, might suffer from a less convincingly made correction for the contribution from multiple scattering events in the measured spectra. A fundamental problem with amorphous solids lies in the understanding of the specific heat anomaly [19]. At temperatures of a few K a significantly higher specific heat is observed than is expected from the Debye model and this, given the correct dynamic model, should enter into the vibrational density of states. The suggested rotational model, despite its crudeness, is successful in a limited temperature range. Carpenter and Price [20,21] reported extensive measurements at 33 K using the Intense Pulsed Neutron Source at Argonne, of the dynamic scattering function S(Q, E) of vitreous silica for energies larger than 20 meV. Data between Q values of 6 ,~-a and 13 ,~-a were averaged, a procedure which will level out coherence effects in the scattering function, and a one-phonon density

of states was derived in relatively good agreement, at least for large energies, with earlier neutron results [16] obtained using the incoherent approximation. With regard to the vibrational dynamics of vitreous silica, the MD simulations have been less successful and the agreement between experimental and simulated results is hardly satisfactory in the low-energy transfer region [13,14]. Generally the high energy region also seems to be better described than the low energy end in the continuous random network model calculations of Bell and Dean [22], and Guttman and Rahman [23], while the cluster or microcrystalline model seems to give better results [24]. The aim of this experimental work was to investigate the region of low energy transfer at different temperatures and fill the gap between the studies of Carpenter and Price [20] and Buchenau et al. [18]. Of particular interest were similarities in the frequency function for the amorphous and crystalline state, and further to find quantitatively the temperature variation of the correlations of structure and dynamics. Reported in this work are measurements of the dynamic scattering function S(Q, E) for mode rate energy transfer at three temperatures (293 K, 673 K and 1073 K).

2. Basic theory The double differential scattering cross section for a coherently isotropically scattering assembly consisting of identical atoms is given by

oe

S(Q, e),

(1)

where k and k 0 are the wave vectors of the scattered and incident neutrons, respectively, dI2 is the solid angle, a the bound atomic cross section, and hQ = h I k 0 - k [ the momentum transferred to the system in the scattering process. The energy transfer is given by E = h2(k 2 - k 2 ) / 2 m , where m is the neutron mass. S(Q, E) is the scattering function or, equivalently, the dynamic structure factor from which information about the molecular motion may be obtained.

B. Grankli, U. Dahlborg / Vibrational motions in v-Si at high temperatures

The energy moments of S(Q, tomic system are defined by (E"(Q)> =

fE"s(o,

E)

for a mona-

E) dE.

(2)

For a classical system the odd moments vanish and the zeroth and the second moments are given by the theoretically exact expressions

(E°(Q)) = S(Q),

(3a)

2kB T 2

( E 2 ( Q ) ) = h ---M--Q ,

(3b)

where S(Q) is the static structure factor, kBT the temperature in energy units and M is the mass of the scattering atom. For a binary system, such as silica, which has negligible incoherent scattering, three partial dynamic structure factors are effective in the total scattering. In this case [25]

aS(Q, E)= 4~r(CAb~SAA(Q, E) + 2(cAcBbAbBSAB(Q, E) +c,bZS,s(Q, E)).

(4)

Here bA, b B and c A, e B are the scattering lengths averaged over possible variations from one nucleus to another, and the concentrations of the elements A and B respectively, and ~ is the effective scattering cross section. The classical energy moments are now given by [25]

(E°n(Q)) = SA,(Q ), 2

2

(5a) ~AB

(E2.(Q))=h Q k.T-~-~A.

(5b)

In order to completely determine the molecular dynamics in silica, it would be necessary to perform three experiments with different relative isotope concentrations to obtain the three partial dynamic scattering functions. The large costs for some of the isotopes and the relatively small differences in scattering lengths that can be achieved render such an experiment rather impractical at present. The interpretation of the neutron scattering data in this investigation will be made in terms of the effective scattering function S(Q, E ) as defined in eq. (4).

297

As has been demonstrated in molecular dynamics simulation [26], it is appropriate to consider the concept of normal modes also for an amorphous solid, even without any assumption about local order of the kind discussed in the introduction. In a harmonic crystal all phonons have well defined wave vectors for every mode, contrary to the case for an amorphous solid, because of its lack of long-range order. In spite of this, it still seems appropriate to apply the phonon concept to disordered systems, since the motions of neighbouring atoms must be correlated and the excitations therefore can be considered as non-local [26]. The scattering cross section may quite formally be written as an expansion in terms related to the respective order of the phonons,

S(Q, E) =

E) + + S~m'(O, E), S~m)(Q, E) contains

E) (6)

where all phonon terms of second order and higher (i.e. the multi-phonon terms). The first term in eq. (6) represents the elastic scattering and is generally given as

Sm)(Q, E)= E-bib~ e -(w,+w~).e iO'R'-R,).

(7)

i.j

W, is a Debye-Waller factor, proportional to the mean square displacement (fi~) of an atom about its equilibrium site R i. The one-phonon term can be written as [21]

S°)(Q, E) = E biby e-(W'+ wj). eiO(R,-R~) i.j

× Eh (Q" eT)(Q" e") :
E.), (8)

where e," is the displacement vector of the normal mode with energy E, and ( a , ) the population factor of that mode and M, atomic masses. Equation (8) can be rewritten in the form of an effective one-phonon scattering function of the much simpler form

h2Q2 kBT SIn(Q, E)= 2Ma, E 2 .F(Q, E).e -2w,

(9)

B. Gran$li, U. Dahlborg / Vibrational motions in v-Si at high temperatures

298

where F(Q, E)= M(Q, E). Z(E). M(Q, E)can be considered to be a structure factor for coherent one-phonon scattering, Meff is the effective atomic mass and Z(E) is the phonon energy distribution. Z(E) includes all types of atomic motions and is an effective vibrational density of states (VDOS) weighted for the different components of the system. For an amorphous solid the energy dependence of M(Q, E) is expected to be weak (Buchenau et al. [18]). Disregarding the energy dependence it is however still different from the static structure factor S(Q). It should be noted that M(Q, E ) = 1 gives the incoherent approximation, and an expression similar to eq. (9) was accordingly derived by Leadbetter and Stringfel-

correlation function, through

J(Q, E)= (EZ/Q2)S(Q, E). (10) If now M(Q, E) approaches unity when Q goes to zero, there will be a close connection between

J(Q, E) and Z(E). As was pointed out by de Wette and Rahman [29], F(Q, E) for a polycrystal has a direct physical meaning, as it is proportional to the response of the system to longitudinal current fluctuations of wave length

2~r/Q. 3. Experimental details

The measurements were performed on the thermal neutron time-of-flight spectrometer [30] at the Studsvik reactor R2. The energy of the incident neutrons was 48.1 meV and the 31 observation angles ranged from 12 ° to 100 ° corresponding to Q values in elastic scattering ranging from 1,1 ,~-1 to 6.9 ,A-]. The Q resolution was about 0.08 A - I , while the energy resolution varied from 1.3 meV at the smallest scattering angle to 1.6 meV at the largest one. These resolution figures, which include all instrumental effects, were obtained through vanadium calibration measurements. The specimens were cylindrical rods of transparent pure vitreous silica (Vitreosil). The water content was according to specifications less than 5

low [15]. Carpenter and Pelizzari [27] developed a formalism that allowed the calculation of the inelastic scattering from sound waves using the elastic scattering w i t h o u t a d j u s t a b l e p a r a m e t e r s . Buchenau [28] recently gave an approximate reformulation valid in the regime beyond sound waves without any assumptions about the detailed character of the vibrational modes. These two formulations give comparatively easy access to model calculations. Equation (9) still retains the general validity of the derivations leading to eq. (8). It is in this conjunction convenient to introduce J(Q, E) for the spectrum of the current-current

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B. Gran$li, U. Dahlborg / Vibrational motions in v-Si at high temperatures

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300

B. Gran$li, U. Dahlborg / Vibrationalmotions in v-Si at high temperatures

p p m and the diameter of the sample was 8 mm. The scattering from the silica was measured at three different temperatures (293 K, 673 K, and 1073 K) and the temperature was controlled to within 5 K. The measured spectra were corrected for trivial instrumental effects (background, self-absorption, etc) and normalized to absolute cross section units with vanadium calibration measurements. The multiple scattering was calculated in an iterative way and subtracted from the data. The normalized effective scattering function S(Q, E), corrected for the contribution from multiple scattering (MS), is shown in fig. 1 for Q = 4.6 A -] for the three measured temperatures. The statistical errors are of the same size as the points

when not visible as vertical lines. In order to demonstrate the magnitude of the MS this is also included in fig. 1 as full curves. It is obvious that the MS does not have any significant influence on the accuracy of S(Q, E ) at the highest temperatures while at large energies and low temperatures, it might introduce an extra uncertainty in the results.

4. Results 4.1. General

The fully corrected scattering functions S(Q, E ) for vitreous silica at three temperatures

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B. Grankli, U. Dahlborg / Vibrational motions in t~-Si at high temperatures

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301

1

Q (/~-1) Fig. 4. The zeroeth energy m o m e n t (E°(Q)) = S(Q) of the measured S(Q, E). The full line is the static structure factor obtained by Johnson et al. [7]. Circles: 1073 K; crosses: 673 K; triangles: 293 K.

are shown in fig. 2 for some selected constant Q values and in fig. 3 as a function of Q for some constant energies. The error bars, visible in the figure only for Q = 1.6 A 1, here include, besides the statistical errors, also the standard deviation in the multiple scattering calculation. Values of S(Q, E) for energies smaller than 2.5 meV were

discarded, as in this energy region the shapes of the measured spectra are entirely determined by the resolution function. The smooth behaviour of the experimental data points for Q > 3.6 ,~-] resuits from an applied averaging and interpolation procedure, which was not used for smaller Q values because of the large number of detectors.

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302

B. Granbli, U. Dahlborg / Vibrationalmotions in o-Si at high temperatures

Figure 3 indicates that S(Q, E ) at different temperatures and for fixed energy transfers, is roughly consistent with the variation of the Bose factor. A more detailed discussion of this fact will be given below. In m a n y respects the structure of S(Q, E ) seen in fig. 3 resembles that obtained from a coherently scattering liquid [31], the main difference being that the structure is more pronounced in the amorphous case. A convenient measure of the reliability of the obtained scattering functions is obtained by investigating to what degree the lowest energy moments, for which exact theoretical values exist, are satisfied. The average cross section in eq. (4) is CAbA 2 + cBb ~ which ensures that the total static structure factor approaches unity for increasing Q. The measured zeroeth moment for amorphous SiO2 is shown in fig. 4 together with the neutron diffraction results obtained at 295 K by Johnson et al. [9]. The agreement between the two sets of data is indeed very satisfactory. There is, however, a small difference for Q > 4 ,~-1, the reason of which is unknown. For Q > 6 ,~-1 the reason may be related to the limited energy integration range, but the differences on both sides of the peak near 5 , ~ - ] cannot be attributed to this effect.

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B. Grankli, U. Dahlborg / Vibrational motions in v-Si at high temperatures 300

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4.2. Elastic scattering

the inelastic scattering was represented by two displaced gaussian functions. Disregarding any physical interpretation of the two methods of evaluation, the difference between them was found to be negligible. The Q-dependence of the total elastic scattering, denoted by F(Q), is displayed in fig. 5. The agreement between F(Q) at 293 K and the results of Buchenau et al. [18] and Dianoux et al. [34] for small Q values is satisfactory. A corn-

In order to separate the inelastic scattering components from the elastic, two different methods were tried. The first involved a least-squares fitting of one gaussian function, representing the elastic scattering, and two maxwellian functions, representing the inelastic scattering contributions from energy gain and loss. In the second method

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304

B. Gran61i, U. Dahlborg / Vibrational motions in v-Si at high temperatures

parison of S(Q) with F(Q) shows that only the latter has a marked temperature dependence outside the FSDP. Generally for a given Q, F(Q) decreases with temperature, and with a higher rate for larger Q values. At least outside the FSDP, the rate of decrease also rises with temperature. This behaviour indicates, as is also expected from eq. (7), that the temperature dependence of the crosssection is of the kind expected from a DebyeWaller factor. The ratio F(Q)/S(Q) was then calculated in order to determine this, see fig. 6, which shows that F(Q) and S(Q) do not have the same Q dependence. A difference would, however, be expected, since S(Q) depends on the instantaE=5 meV

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305

B. Gran$li, U. Dahlborg / Vibrational motions in v-Si at high temperatures

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Q2(~-2) Fig. 9 (b).

between silicon and oxygen atoms and the expected dependence of 2 2 o n atomic mass. The earlier result [36] seems to be substantially different from these. The vibrational motion of the oxygen atoms will, because of their smaller mass as well as larger scattering length and higher concentration, have a much larger weight than the silicon atoms in an effective fi2. It is interesting to note that the values of (fi2)1/2 almost exactly agree with the ones found for oxygen in high cristobalite [37] and also that they are proportional to the temperature, which is characteristic for harmonic oscillators. This observation certainly supports the use of the formalism of eq. (6)

to extract a vibrational density of states from a measured S(Q, E) for a glass.

4.3. Inelastic scattering The second energy m o m e n t of the effective

S(Q, E), ( E E ( Q ) ) (fig. 7), can serve as a measure of the reliability of the experimental determination. The data exhibit the oscillatory b e h a v i o u r , which is expected from a coherent scatterer. Further, a strong dependence on temperature could be anticipated from the theoretical expression eq. (5b). According to the definition of S(Q, E), the

B. Gran$lL U. Dahlborg / Vibrational motions in v-Si at high temperatures

306

second energy moment can be written as

4---~(E2(Q)>=¢ab2+cBL,2B. G

(11) The theoretical curves shown in fig. 7 were calculated using the known values of the scattering lengths for silicon and oxygen. It is obvious that the experimental data satisfy the second moment only for small Q and at the lowest temperature. For other conditions the measured second moi

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ment is too small, which means that the measured S(Q, E) does not extend to large enough energy transfer, as is obvious from a comparison with the measurement of Price and Carpenter [21]. As fig. 1 shows, the measured S(Q, E) has a maximum at an energy transfer of about E = 5 meV at all measured temperatures for not too small Q values. A similar behaviour was found in the experiment by Buchenau et al. [18] and shown to be a temperature-independent feature of the scattering function at low temperatures. The position of the peak for the higher temperatures in this experiment does however vary with temperature, as is obvious from fig. 8. For Q larger than about 3 ,~-~ it shows a rather distinct variation with Q, which is similar for all temperatures. There are certainly some multiphonon effects in this energy region which might influence the shape of the peak. It is, however, improbable that the smooth shape of a multiphonon contribution should to any larger extent change the position of the peak. When comparing the oscillation with that of the second moment of S(Q, E), as shown in fig. 7, it can be seen that the two sets of data are almost exactly of opposite phases which is indicative of a de Gennes narrowing effect. There are two principally different routes available to extract Z(E) from experimental data via eq. (6), as was pointed out by Suck and Rudin [38]. The subtraction method implies that the multiphonon contribution is calculated by some approximate means, for example using the phonon expansion for incoherent scattering. The calculated contribution is then subtracted from the measured scattering function together with the elastic contribution. In performing this operation, the finite Q range covered in the experiment would have to be taken account. Another possibility emanates from the close connection between the spectrum of the longitudinal current-current correlation function J(Q, E) and the vibrational density of states (VDOS), Z(E). Since the multiphonon terms include powers of Q larger than 2, an extrapolation of J(Q, E) to Q = 0 along a mean value will, neglecting some trivial factors, give Z(E) as shown by eq. (9). The second method requires a proper extrapolation to be found for the measured J(Q, E), as

B. Granbli, U. Dahlborg / Vibrational motions in v-Si at high temperatures

shown in fig. 9 for some constant energies. In order to perform the extrapolation objectively, the J(Q, E) functions were fitted by a least squares method to the analytical expression

J(O, E) = c,A(O). (1 + c2O 2) e -c~&,

absolute units and the methods of extracting them are different, it is important to note that a normalisation was arbitrarily made to the last point available in the data of Buchenau et al. The overall agreement with the results of Leadbetter and Stringfellow [15] at the same temperature is acceptable in view of the fact that that experiment was performed at fixed scattering angle, which has Q varying over the covered energy region. The use of the incoherent approximation was expected to give accurate results only for very high energies. No comparison is made in fig. 11 with the results of Price and Carpenter [21] since these data are claimed to be valid only above 15 meV. The agreement with the low energy part of this measurement is relatively poor because of the Considerably better energy resolution in the present experiment. Figure 3 shows that with rising temperature S(Q, E) apparently increases more slowly for large energies than the linear relationship predicted by eq. (9). This observation is still valid when account is taken of the comparatively large experimental errors for the measurement at 293 K, which indicates that the nature of the vibrational motions changes with temperature. At energies smaller than 5 meV, however, the VDOS is very well described

(12)

where A(Q) was chosen quite arbitrarily in two different ways. Firstly as the measured zeroeth moment, shown in fig. 4, for any Q dependent function, or secondly, simply equal to unity. The results are shown in fig. 9 as solid and broken curves. The solid curve describes the data rather well except around Q = 3 ,~-], where there is a definite disagreement. A complete agreement cannot be expected since the effective scattering functions are used in this work. The parameter c I in eq. (12) is proportional to the VDOS and shown in fig. 10. The length of the vertical lines indicates the difference between the two different choices for A(Q) and is taken as a measure of the error. The shape of the room temperature VDOS is for small energies similar to the one reported by Dianoux et al. [34]. In fig. 11 a comparison is made between the results for the VDOS at 290 K from the measurements by Buchenau et al. [18], from Leadbetter and Stringfellow [15] and from the present one. Since not all results were available in

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308

B. Gran$li, U. Dahlborg / Vibrational motions in v-Si at high temperatures

by a parabolic function at all temperatures. There is also a linear temperature variation of the VDOS in this energy region in accordance with eq. (9) and with the findings of Dianoux et al. [34] for temperatures below 290 K. Derivation of an effective Debye energy yields E D = (23 + 2) meV as an average for all temperatures. It should also be noted that the shoulder in the VDOS at about 10 meV closely corresponds to the maximum energy of acoustic phonons in alpha-quartz (Barron et al. [17]). The parameter c 2 was found to increase rather drastically with temperature when eq. (12) was fitted to the experimental data, which is due to multiphonon terms incorporated in this parameter. The errors in the calculated values of c 2 were, however, too large to allow any specific interpretation. The last factor in eq. (12), c3, can be considered to be an effective D e b y e - W a l l e r factor for inelastic scattering, and as such it can be used to derive a mean square atomic displacement 22 for inelastic scattering in the same way as the total mean square atomic displacement was derived from the elastic structure factor F(Q). A necessary but not sufficient condition for this interpretation to have any meaning is that c a is independent of energy, and this is actually the case, as is obvious from fig. 12, where the full line is the average of the values for all energies. For comparison the broken line showing the values of (22 ) obtained f r o m F(Q) is included. The average values of (h2)t/2 were found to be 0.27, 0.29 and 0.35 ,~ at the three temperatures 293, 673 and 1073 K, respectively. In addition to the relatively poor precision in the calculation, a significant contribution to the difference between the two sets of values for (h2)t/2 can be attributed to the averaging process applied to the partial scattering functions in the two cases. The weighting factors on the scattering lengths are different, as eqs. (7) and (8) show. Values of the atomic displacement derived from elastic scattering will depend strongly on the experiment if inelastic contribution is allowed to enter into the elastic part through poor energy resolution. It is not expected that the number of excitations in the energy region of the resolution function of the present experiment should be very large.

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A combination of eqs. (9), (10) and (12) leads to an expression for an energy dependent onephonon structure factor

M(Q, E) = 1 j ( Q , E) e C'Q2,

(13)

which will be given in absolute units. The M(Q, E) so calculated, appeared to be independent of energy at all temperatures within the experimental

B. Grankli, U. Dahlborg / Vibrational motions in v-Si at high temperatures

309

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limits of error, allowing the immediate presentation of average structure factors M(Q) for inelastic scattering as shown in fig. 13, where the width of the bands is a measure of the spread of the measured M(Q, E). This particular behaviour of M(Q, E) was earlier found by Dianoux et al. [34] for small energies. The results of Price and Carpenter [21] indicate a strong energy dependence for the M(Q, E) at much higher energies, but the rather different treatment of data does not allow meaningful comparisons to be made. The conclusion to be drawn from fig. 13 is that the incoherent approximation cannot be used directly to extract a phonon density of states in the Q range covered in this experiment. It is obvious that the general features of M(Q) and the static structure factor S(Q) are similar, but also that they are not identical. It is interesting to note, that the shape of M(Q) changes in a systematic way with temperature for Q smaller than 4 ,~ i while for larger Q values the difference between the different bands lies within the limits of error. From the temperature variation it may be concluded that the medium-range dynamical correlation in silica decreases when the temperature increases. This conclusion conforms with the previous discussion above concerning the sharpness

of the first sharp diffraction peak of S(Q). A forthcoming paper is intended to elaborate on these implications.

5. Conclusion In this study the temperature dependence of inelastic scattering was assessed in a temperature and energy range complementing previous work. It was found that the structure factors for elastic and inelastic scattering change with temperature in a Q region which reveals mostly long-range atomic interactions rather than those from nearest and next-nearest neighbours. The temperature variation was interpreted as a decrease in the structural and the dynamical correlations between neighbouring SiO4 polyhedra with increasing temperature. It was further observed that the vibrational amplitudes of the atoms in vitreous silica are the same as those in high cristobalite allowing for experimental errors. The temperature dependence of the VDOS indicates, contrary to the findings of Buchenau et al. [18] for lower temperatures, that the nature of the vibrational motions changes with temperature. However, at small energies the VDOS varied as

310

B. Gran$li, U. Dahlborg / Vibrational motions in v-Si at high temperatures

p r e d i c t e d b y the h a r m o n i c a p p r o x i m a t i o n , y i e l d ing a temperature-independent D e b y e energy. F r o m a c o m p a r i s o n w i t h m e a s u r e d d i s p e r s i o n rel a t i o n s in q u a r t z , a c o r r e s p o n d e n c e is i n d i c a t e d b e t w e e n a s h o u l d e r in t h e V D O S a n d the m a x i m u m e n e r g y o f the p h o n o n s in t h e crystal. T h e a u t h o r s a r e g r a t e f u l for the v a l u a b l e disc u s s i o n s w i t h D r I. E b b s j b , w h o also c r i t i c a l l y r e a d the m a n u s c r i p t .

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