Neutron scattering from vitreous silica

Journal of Non-Crystalline Solids 76 (1985) 351-368 North-Holland, Amsterdam

351

N E U T R O N SCATTERING F R O M VITREOUS SILICA Ili. Elastic diffraction Adrian C. W R I G H T J.J. Thomson Physical Laboratory, Whiteknights, Reading, Berks, RG6 2AF, UK

Roger N. SINCLAIR Materials Physics Division, AERE, Harwell, Didcot, Oxon., O X l l ORA, UK Received 15 March 1985

The differences between elastic and total neutron diffraction experiments on amorphous solids are discussed quantitatively using data for vitreous SiO 2. It is shown that in the former case early peaks in the real space correlation function T(r) are broader due to the inclusion of the effects of acoustic phonons and that for vitreous silica the atomic thermal vibrations, relative to the Si-O bond direction, are approximately isotropic. The two techniques yield the same S i - O bond length, indicating that recoil corrections are unnecessary in total diffraction studies of amorphous solids. Supplementary constant Q scans across 0~-Q space at high Q are qualitatively in agreement with a simple twin Gaussian model for the scattering law S(Q, ~0), which may prove useful in studying the effects of static approximation distortions.

1. Introduction

It has been shown in part II [1] that difficulties arise in total diffraction experiments when making Placzek corrections for departures from the static approximation, particularly in respect of the distinct scattering. An alternative approach is to measure the elastic diffraction pattern, which is directly related to the time-averaged correlation function ~ , usually denoted G(r, or) [2]. It has been suggested (e.g. see refs. [3] and [4]) that for an amorphous solid the inference function Qei(Qe) and the correlation function T(r) from an elastic diffraction experiment should be closely related to the equivalent functions from total diffraction and that this might provide an extremely valuable check on the validity of the above corrections. In addition it is possible to extract from an elastic diffraction pattern values for the directionally-averaged rms atomic thermal displacement amplitudes which can be combined with the rms bond length variation to give information about the eccentricity of the atomic thermal vibration ellipsoids. The disadvantage of elastic diffraction is that it requires an extra monochromator and this leads to reduced counting rates. Elastic diffraction experiments are usually performed with a reactor based triple-axis spectrometer and hence the maximum value of Qe obtainable is 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

A.C. Wright, R.N. Sinclair / Neutron scatteringfrorn vitreous silica 111

352

limited by the fact that it is necessary to work at a wavelength closer to the peak of the moderator Maxwellian spectrum than for total diffraction measurements. An alternative possibility [5,6], however, employs correlation techniques on a pulsed neutron source and allows the simultaneous measurement of both elastic and total diffraction patterns. In understanding the limitation of the Placzek method it would be extremely valuable to know the form of the scattering law S(Q, co) for at least one amorphous solid over a wide range of Q and ~0. It would then be possible to investigate in detail the consequence of various combinations of detector law and integration path in ~0-Q space. Thus, in addition to the present elastic diffraction measurements, constant Q scans across ~0-Q space were performed in the region of Q above 10 A ~, in order to supplement existing inelastic scattering data at low Q [7-10]. Very recently, however, a more comprehensive higher resolution study of S(Q, ~o) in the same region has been reported by Carpenter and Price [11] and so the analysis of these constant Q scans will be limited to comparison with a simple, first order free atom model which may prove useful in the analysis of the effects of the static approximation. Elastic diffraction patterns are reported for vitreous silica using two separate triple-axis spectrometers with widely different Q range and energy resolution. The data obtained are of much higher accuracy than obtained previously by Lorch [3] and allow a quantitative comparison with the total diffraction results of part II.

2. Theory 2.1. Elastic diffraction For any real elastic diffraction experiment the detected radiation falls within some ~o resolution window R(~o) which may be conveniently approximated by the Gaussian function [12,13] R(o~) = R(0) e - 'Am°/aEl'-

(1)

in which A E defines the instrumental resolution and R ( 0 ) = 1. The differential scattering cross-sections thus become

doS=F~f~oSS(Q' d~

_

.

~0)R(¢o) dw,

J

do D

df2 = Z E1)ibk o~ k_.Sj~(Q, ¢o)R(¢0) dw j

k

" £ o ~ ko

(2)

and in order to proceed it is necessary to assume that k -- k 0 and S(Q, co) is independent of Q over the range of o~ for which R(¢0) is finite. Substituting for S(Q, ¢o) from eq. I(7) of part I [14], it is then possible to perform the ,0

A.C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111

353

integration to give

d[2 d°S

-- Eb7 f 0 ~ GS(r, t)

e ioL''r

(3)

dr

)

and do

i)

_

d$2 - ~ ~ bJ)k f ./

oz

G,~( D r , t) ei~2L'r dr.

(4)

k

The quantities 1

; f a(r,

t) e-'A,/a,l-" dt

are the correlation functions obtained by averaging characterised by

(5)

G(r, t) over a time interval

At = h/aE (6) and are frequently given the symbol G(r, ~) [2], since the time averaging is usually dominated by times which are much larger than the period of an atomic vibration. For an idealised elastic diffraction experiment (AE = 0) ,At becomes infinite. The corresponding full widths at half maximum height (FWHM) are related by At,/2 = (8 In 2)h/AE,/2. (7) The same result is obtained qualitatively from the Uncertainty Principle. Namely, in a perfect elastic diffraction experiment AE is zero and so t is unspecified, whereas in a total diffraction experiment A E is effectively infinite which fixes t at zero. For an amorphous solid, unlike a liquid, the equilibrium atomic positions do not change with time and in a large system, obeying the normal laws of statistical physics, there is no correlation between the instantaneous thermal displacements of atoms separated by either long time intervals or large distances. It is therefore possible as t--* ~ to replace the average of the product of the &functions in I(8) and I(9) by the product of the averages [2]

GS(r, t)=-~

rm(O)-r'])(8[r'-

(t)]) dr'

N, N~ @D(r,t)=~1 £ E ff ~6[r+r,,(O)-r'])(6[r'-r,,(t)])dr',

(8)

. rn=l n=l

GfS(r, t) and G~(r, t) may then be replaced by correlation functions for the equilibrium atomic positions together with probability functions p(u), describing the probability of an atomic displacement u, both of which are time-independent. Hence if At is large eqs. (3) and (4) become do s )) d~2 J × e iq~'l*-",(°'+~,~',l d r du/(O ) d u j ( t )

(9)

A.C. Wright, R.N. Sinclair / Neutron scattering frorn vitreous silica 111

354

and

do°

j k

fo fo fo

p~k(r)

(u.I(O))P

× e `q°'lr-u,(°)+"*`'>] d r du; (0) d u k ( t ).

(10)

However [6]

fo°~fo°~P(u;(O))p(u,.(t)) e-i¢,~.-[~,(o) u,(,,, d u , ( O ) d u k ( t ) = ( e - iO°'[u,(°,-"''',l )

(11)

and, if the displacements are small and p ( u ) = p ( - u ) , approximation

then to a good

(e-iq,.-[u,(0) -~(,)1)= e Q~(,~+,,b/6

(12)

This result is exact even for large displacements if p(u) is a Gaussian distribution [15]. Thus finally do S

d~? = y'~b2 e-Q:u~/3

(13)

J

and, after averaging over all orientations of Qe with respect to r, do _ _ . f ~ d)~(z) e -q~(u~+u~)/6 sin Qer d~2D - y" ~bjbk j

k

~

Q----f-- dr - I°(Q~),

(14)

I°(Q~) is the average density term and u 2 the mean square atomic displacement. The component correlation functions for the equilibrium atom positions d)~ (r) are related to the normal component correlation functions djk (r) by the convolution

dj~(r)=[~] ffd)~(r)exp ~-~(.,~:)

dr'

(15)

in which r' is a dummy convolution variable and the term in (r + r') has been neglected (c.f. eq. II(11)). It is extremely instructive to perform the same separation into an equilibrium position and a thermal displacement in the case of total diffraction. If the thermal motions of different atoms are uncorrelated then, within the approximations discussed in parts I and II, the expression for the distinct scattering is identical to eq. (14) whereas the self scattering remains as predicted by II(4). In this case, the distinct scattering is insensitive to the extent of any energy discrimination in the scattered beam and the interference functions obtained from total and elastic diffraction experiments should be identical at scattering angles where the former is unaffected by static approximation distortions [1]. However, as will be shown in section 4, the

A.C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111

355

assumption of uncorrelated thermal motion breaks down, particularly at small interatomic separations. In an elastic diffraction experiment the self scattering depends on the detailed shape of R(co). The limiting value of u 2 will only be obtained from the self scattering if At is large compared to the period of any atomic vibrations or, expressed in an alternative way, R(co) is narrow enough not to include any significant inelastic contribution.

2.2. The scattering law S(Q, co) at high Q At low and intermediate Q the scattering law S(Q, co) depends on the exact shape of the frequency distribution g(co) and is complicated by the presence of coherence effects. However at high incident energies the scattering cross-section o(Eo) tends to the free atom value [14] and under these circumstances (Q >__10 A -1) it might be expected that the form of the self scattering law would similarly approach that for an assembly of free atoms in statistical equilibrium [16]

co)=( Mj ),j2 2~rQ2kBT

exp

\ 2~7k- ~

J

(16)

Eq. (16) comprises a Gaussian distribution, with rms deviation (h2Q2kBT/ M~) 1/2, centred at hco = hZQ2/2Mj rather than at the energy for elastic scattering Eo(hw = 0). This type of behaviour has been observed for liquid nitrogen [17]. The distinct scattering, on the other hand, involves interference effects between a large number of covalently bound atoms within some coherence volume all of which must recoil together if coherence is to be maintained. Thus the effective mass for the distinct scattering is very much larger than for the self contribution with the result that the former will remain effectively at hw = 0. A simple model for the total scattering law S(Q, w) therefore, is an elastic line plus the free atom contribution of eq. (16), both broadened by the instrumental energy resolution function. Note that, for the constant Q scans of section 3.2, the contribution at hw = 0 will also include some self scattering since the Debye-Waller factor in eq. (13) will not have decayed effectively to zero.

3. Experimental method

3.1. Elastic diffraction Two separate elastic diffraction experiments were performed. The first employed the DIDO triple-axis spectrometer (fig. 1), a relatively long wave-

356

A. C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica l I l

Lce

,rum

Mono( Cryst,

¢ ~

'

,

,

Beam Stop

Fig. 1. The DIDO triple-axis spectrometer.

length (1.4043 A) and a narrow energy resolution. The second set of measurements was made on the IN1 triple-axis spectrometer at ILL Grenoble which is situated on the neutron hot source. This enables data to be obtained at a shorter wavelength, (0.6276 A) and hence higher Qe (Qmax = 15.8 ,~-l), but with poorer energy resolution. The layout of IN1 was similar to that of the D I D O machine except that it used a translating monochromator so that the absolute position of the sample was fixed and did not change as E 0 was varied. The analyser setting for elastic diffraction was found for each instrument by performing constant Q scans at low values of Q where the "elastic" peak is still symmetrical about h~0--0 (c.f. section 3.2). In the case of the D I D O spectrometer scans were performed for vanadium at Q -- 2, 4 and 6 A - I . The position of the elastic line was invariant for the three scans as was the energy

A. C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111 10

1

I - -

I

I

r

i

----T

r

357

I

8

7 c :D 6 7,

5 •

->" 4

•

•

gm,~

%

~•

.-

•

3

o.

g

;

°

..-

-

;

*"

o•

o: ~

"~.....

o

\ L

I

I

10

20

30

Z, 0

50

60

l

I

70

80

~

:)°°°

90

100

2e °

Fig. 2. Raw data from the DIDO triple-axis spectrometer. • sample. © background and • vanadium (displaced vertically 3 units). resolution AE1/2 of 1.96 meV. Only one constant Q vanadium scan was obtained with IN1 at Q = 5.26 .~-1. The energy resolution was 27.7 meV. Once the analyser had been accurately set at h~0 = 0 the experimental technique closely followed that described in part I1 except that individual scans were performed with points equally spaced in Q. The raw data are shown in figs. 2 and 3. On the D I D O spectrometer it was possible to use the fully focused configuration (fig. 4A) for the whole experiment but IN1 is constructed in such a manner that it was necessary to work mainly on the defocused side of the monochromatic beam (analyser focused: fig. 4B). At low scattering angles a large background was experienced in this mode as the detector shielding moved into the beam from the monochromator, so that the low Q data were obtained on the focused side (analyser defocused: fig. 4C), using a smaller interval AQ to improve the accuracy of the first peak. Experimental corrections were applied as discussed in part II except that it was not possible to subtract the contribution due to "water". In correcting for multiple scattering it was assumed that the D e b y e - W a l l e r factor for the multiply scattered component was on average appropriate to two scattering events each with a scattering angle of 90 °. The absolute intensity from the vanadium standard is given by

Iv ( Qe) =b~ve- Q~.,,~/3

(17)

and hence the scaling factor a (c.f. part II) can be found by plotting ln(IE(Qe)) against Q2 and extrapolating to zero Qe.

358

A.C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111

10

l

I

I

I

I

I

~ - -

9

*~~AA%%A~IIAA4~A~#%II%~jAAI%%AI~%~%~,~,#~ ~

g7 • x~ 5

A

•. . . .

g

.,.

<

"

. -...~

:,4

"

.-

•

=

,~"

3

o°

%

%

~.,~,.,...-"

"--

.

oz

-

•

•

2

2

4

6

8 O~(,~1)

10

12

14

16

Fig. 3. Raw data from IN1. • sample, O background and • vanadium (displaced vertically 3 units) with spectrometer configuration as in fig. 4B. • sample and [] background as in fig. 4C.

The sample self scattering is of the form IS(Qe) = y ' b 2.! e Q;.j3.

(18)

)

In order to obtain initial values of t,,s~jr"21~/2and [u~] t/2 from each experiment, the corrected, vanadium-normalised intensity, at the values of Qe for which the I

on~

b Ion

v

a

O"

I

on~ v

C

. j2o \

\

Fig. 4. Spectrometer configuration, a fully focused, b monochromator defocused, analyser focused and c monochromator focused, analyser defocused.

A.C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica l II

359

Table 1 R m s thermal d i s p l a c e m e n t a m p l i t u d e s [u~] 1/2 Instrument

Si

O

DID() IN1

0.160 0.083

0.233 0.121

total diffraction interference function from part lI (fig. I1 13) is zero, were read off and fitted to eq. (18). The contribution to the self scattering from the Si 2 atoms is small (bsJ2b o2 = 0.2555) and the quality of the data is limited both in the number of intensity points and in their statistical accuracy. There is also a small Q¢ dependent distortion due to "water". Hence the quantity [us,/ u~] ~1"2 was kept constant at 0.69, an average value for the various crystalline silica polymorphs. The data were then renormalised using the K r o g h - M o e - N o r m a n method described in part II and Fourier transformed to give the real space correlation function T(r). The interference function Qci(Qe) contains many contributions, with different Debye-Waller factors, and if thermal motions are correlated there is no reason why the zero points should be the same as for total diffraction. The rms displacements were therefore optimised, keeping the ratio t-s~/-oJ [, 2 / , 211/2 constant, to give minimum noise in T(r) below the first peak, the data being renormalised before each Fourier transformation. Optimum parameters for both experiments are given in table 1 and the final data are illustrated in figs. 5 and 6.

1,6r

1.2

I_.

°

,", 08 0

,.~ 0.4

0.0 0 Q e ( A -~ )

Fig. 5. N o r m a l i s e d diffraction pattern from the D I D O triple-axis spectrometer. • e x p e r i m e n t a l points, - s m o o t h line a n d . . . . . . self scattering.

A.C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111

360

16r

,

I

[ o8

' --

-

"

0"00

2

.4

~

6

8

10

12

14

16

Q e ( A -1 ) Fig. 6. Normalised diffraction pattern from INI. Key as fig. 5.

3.2. Constant Q scans The form of the scattering law S(Q, co) at high Q was investigated by means of constant Q scans across w-Q space at Q values of 11.26, 12.54 and 14.04 ,~ 1 corresponding to positions of minima and maxima in i(Qe). A similar measurement was made at 5.26 ,~-1 [a maximum in i(Qe)] where the scattering is still centred at hco = 0. This gave the shape of R(co) more accurately than the vanadium scan at the same Q since the latter is distorted by the presence of low frequency modes. A special sample was used for this part of the experiment in order to reduce the effects of multiple scattering and comprised 2 m m thick silica discs ("Spectrosil B") alternated with 1 m m thick sintered ~°B4C neutron absorbers. The sample diameter was 1 cm and the assembly was held together with a specially designed G clamp. A calculation, following Sears [18], predicts that the total multiple scattering for this arrangement should be less than 5%. Measurements were made of the scattering from the sample, Is(Q, co), and the empty sample holder, IB(Q, co), in 1 T H z steps (1 T H z = 4.136 meV) from - 1 2 to the instrumental maximum of + 20 T H z ( - 1 0 to + 12 T H z at 5.26 - 1). The intensities were recorded for a preset number of monitor counts and the variation in hco was achieved by changing E o at constant E. In this mode the analyser efficiency is constant and the 1 / v dependence of the incident beam monitor (a low efficiency I°BF 5 3 proportional counter) exactly compensates for the factor k / k o in eq. I(6). Each scan was repeated several times and the average intensities are shown in fig. 7. The beam incident on the sample consists not only of neutrons of energy E 0 but also includes neutrons of other energies scattered incoherently from the monochromator crystal. A further polychromatic contribution may arise from the crystal support and surrounding shielding. The total polychromatic contribution to Is( Q, co) and

A.C. Wright, R.N. Sinclair / Neutron scatteringfrorn eJtreous silica 111 0.6

,

- - ~

0.4 ] 1 8p ....

16

/

F-

, ."

, , Q =5.26~ -1

~

~

~

.''".

o 21 o

•

, q

Q=11, '26"&q

.."

361

I

"i'"'"

""#:

.... i

i

T

Q =12.54~, -~

g

12

t

0"6

"

0-4 *t

~

0.2

08 ~

Io oo

• oj"

o

•••°'o*~

0-6~

iii 0

-10

-5

0 5 v(THz)

10

"""" -10

-5

0

5 v(THz)

Fig. 7. Constant Q scans. • sample, O background and - -

t 10

15

20

smoothed background.

IB(Q, co) [IsD(Q, co) and IBD(Q, co)] were therefore determined by repeating the above measurements with the monochromator crystal (0 M) detuned by 2 ° and recording the total number of counts in a preset time interval. (The monitor counts were lost on detuning 0M. ) Both /so(Q, co) and IBD(Q, co) were found to be independent of hco and were replaced by their average values. The required sample scattering is thus

I(Q, co)= [ I s ( Q , c o ) - IsD(Q, co)] - [ I B ( Q , c o ) - / B D ( Q , CO)]

(19)

and is shown later in fig. 12 (section 4.2).

4. Discussion

4.1. Elastic diffraction In his paper comparing the elastic and total diffraction patterns of vitreous silica, Lorch [3] comments on the presence of an extra peak in the elastic diffraction pattern at 6.6 A ~ which is not present in the total diffraction pattern. There is no sign of this extra peak in the IN1 data (fig. 6), nor of the start of it in the DIDO data (fig. 5) which have a higher energy resolution than that of Lorch. It must therefore be concluded that this 6.6 A--~ peak is spurious and not a real feature. Interference functions, Qei(Q~), for the elastic diffraction experiments are

362

A.C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111

"T *< C t....

0

0_ 2 L

2

0

4 Qe(,& -1)

6

Fig. 8. Interference function from the DIDO triple-axis spectrometer. • experimental points. . . . . . . smooth line and - Qei(Q~) from total diffraction measurements (part I1).

shown in figs. 8 and 9 together with the equivalent function from part II [1]. The m a x i m u m value of Q~ (Q .... = 15.8 ,~-1) for the IN1 data is such that the total diffraction interference function should not contain significant static approximation distortions (QeX0 < 8 for X0 = 0.5 ,~). Correlation functions from the IN1 data and the total diffraction data with the same value of Qma~ are c o m p a r e d in fig. 10. A correlation function is not shown for the D I D O data due to the extremely low value of Qm,× (6.48 ,~-1). It is immediately apparent from figs. 8 and 9 that the elastic diffraction

03 t" t.-

/J' So./

()

2

&

6

8 I'0 Qe (/~-1)

1'2

Fig. 9. Interference function from IN1. Key as fig. 8.

"

t

A.C. Wright, R.N. Sinclair / Neutron scattering from eitreous silica I l l

I

T

r

i

1

,

~

I,

-i 0

s

0

~

'

o.~

2

4

6

363

'- ~+~ I..-.-

~ .....

8

10

Fig. 10. T o t a l c o r r e l a t i o n f u n c t i o n for v i t r e o u s silica. - diffraction.

0

1

2 r(,&)

3

elastic d i f f r a c t i o n a n d . . . . . .

z~

total

Fig. 11. Fit to the first three p e a k s in the c o r r e l a t i o n f u n c t i o n T(r) o b t a i n e d f r o m elastic ( u p p e r curves) a n d total (lower curves) d i f f r a c t i o n . - experiment. - ..... fit a n d . . . . . . residual.

interference function decays more rapidly with increasing Q~ than that from total diffraction and that the effect is more marked with the D I D O data which have a much higher energy resolution. This would not be expected on the basis of uncorrelated thermal motions (c.f. section 2.1) and is an indication that such an assumption is incorrect. What is happening may most easily be understood by considering the exact form of the correlation function extracted from each experiment for the atoms at either end of an S i - O bond in the limiting cases of an optic phonon and a long wavelength acoustic phonon. For both types of experiment the total measurement time is effectively infinite compared to the period of an atomic vibration. A single scattering event in a total diffraction experiment effectively takes a "snapshot" of the system and measures the instantaneous atomic separation. The complete measurement is thus a superposition of a large number of such snapshots giving the time average of G(r, 0), ~ . For an optic phonon the atoms vibrate out of phase and the instantaneous separation of the two atoms is different at different times so that a total diffraction correlation function is broadened by optic phonons. This is not the case for long wavelength acoustic phonons since the atoms vibrate in phase with a constant bond length. A single scattering event in an elastic diffraction experiment, on the other hand, measures the time averaged distri-

364

A. C. Wright, R.N. Sinclair / Neutron scattering from uitreous silica I l I

bution of the position of one atom relative to the other fixed at the origin at time t = 0, while the complete measurement averages over all possible positions of the origin atom. Elastic diffraction experiments are therefore sensitive to both optical and long wavelength acoustic phonons and the peaks in the resulting correlation function are broader than in a total diffraction experiment as may be seen from fig. 10. Note that in practice the finite energy resolution in an elastic diffraction experiment restricts the averaging time [c.f. eq. (5)] so that the broadening increases with decreasing A E. Fits to the first S i - O and O - O peaks in the correlation functions of fig. 10 are shown in fig. 11. The first Si-Si peak was fixed as in part II, except that for elastic diffraction its width was increased to 0.12 A and the O - O co-ordination number was fixed at 6 due to overlap problems. A X 2 minimisation technique was used to extract the optimum peak parameters which gives good agreement with the manual method employed in part II for the full total diffraction data (Qm~ = 23.56 ,~-1). Even with the poor energy resolution obtained with IN1 the first S i - O and O - O peaks are significantly broader than for total diffraction, the values of (uZk)1/2 being 0.083 and 0.111 A compared to 0.054 and 0.091 A respectively. The latter values were obtained from the total diffraction data with Q .... = 15.8 ,~ 1 and should be compared to values of 0.049 and 0.092 for Q .... = 23.56 ~ - 1 . In all the fits described, the position of the S i - O peak was constant to within + 0.002 A. The O - O peak for elastic diffraction is at 2.624 A compared to 2.632 ,~ for total diffraction, this difference being well within the experimental uncertainty. Whereas the value of (uZi_o) 1/2 from a total diffraction experiment is the rms bond length variation, the equivalent quantity extracted from an elastic diffraction measurement is given by the convolution of the thermal vibration ellipsoids of the atoms at either end of the bond, resolved along the bond direction. An idea of the eccentricity of the thermal vibration ellipsoids can thus be obtained by comparing the value of xg,,2 , - S i _ O,1,~1/2 from the IN1 data (0.083 A) with the quantity [~(Usi 1 z + ug)] ~/2 calculated from table 1 [c.f. eq. (15)], which is equal to 0.085 A. This suggests that the atomic thermal vibrations are approximately isotropic, although note that the effects of static disorder on d~k(r ) have been neglected. Great care must be taken in interpreting values of (U~k)1/2 from elastic diffraction experiments since they clearly depend on the energy resolution A E. The interference function for the D I D O experiment decays much more rapidly with increasing Q than does that obtained with IN1 and hence the values of (u2k) 1/2 obtained at the D I D O energy resolution would be much greater as are the rms thermal displacement amplitudes (table 1). It is also important to understand the difference between measurements on crystalline and amorphous solids. In the former case the elastic scattering occurs as sharp Bragg peaks and hence there is Q selection of the elastic scattering. This means that the thermal parameters from nominally total diffraction studies of crystals are appropriate to elastic diffraction and cannot therefore be directly compared with those obtained from total diffraction measurements on amorphous solids.

A.C. Wright, R.N. Sinclair / Neutron scattering./~'om (;itreous silica I l l 0 6~

T---r

•

~

l

365 ~--

T

0 =1126/~ -1 118

i

i i

Q=526

A -1

116 la3

001''T - _ ~ O.B

114

c--

o L_

0.2

i

L

i

±_

T

f

~

r

Q = 1254 ,&-I 12!

06

lI0

04

0 I8

0.2

0.6

0.01 "'T

L_

<~

i

i

I

~

i

r

i

~

_

i

c i

i

0 I4

0

0.2 0.0 -10

-~5 v (THz)

,5

1'0

F i g . 12. F i t t o c o n s t a n t Q scans. • ......

.

2

-10

~

0

-5

.

0

0

5 (THz)

v experiment, -

-

t o t a l fit, - . . . . .

00 =41404 .&-i

10

15

20

individual peaks and

constant background.

Significant differences between the elastic and total diffraction correlation functions in fig. 10 are confined to values of r below the peak at 5.03 ,~. Above this peak any differences are small compared to the noise level characterised by the error ripples at low r(r < 1 A). If it is assumed that the broadening due to any vibrational mode is fully included in the time average when the period of vibration equals At then the averaging can be characterised by a minimum phonon energy AE which for the D I D O and IN1 data is 0.83 and 11.8 meV respectively. These values should be compared to the vibrational density of states for vitreous silica in refs. [9-11]. 4.2. Constant Q scans The final corrected constant Q data are shown in fig. 12 and compared to the results of a twin Gaussian fit along the lines of section 2.2. The total fit is shown as a solid line and comprises the two Gaussian components (shown dashed) and a small residual background (dotted). The qualitative form of the model yields an excellent fit to the experimental data. However, as may be seen

A.C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111

366

Table 2 Fit to constant Q scans Q(A

5.26 11.26 12.54 14.04

i)

Const.

0.000 0.042 0.050 0.048

"Elastic .

.

.

.

Free atom"

t'E(THz )

AvE(THz)

VF(THz)

A VF(THz)

- 0.30 0.00 - 0.03 0.08

2.37 3.03 3.08 3.53

3.94 6.85 7.52 7.72

3.69 4.84 4.31 5.33

from table 2, the parameters obtained for the inelastic component, while of the right order of magnitude, do not fall between the values for Si and O predicted by eq. (16). The peak is at too high an energy transfer and, except at Q = 5.26 ,~-l, is too narrow. A much more comprehensive study of S(Q, w) for vitreous silica at similar values of Q has been reported very recently by Carpenter and Price [11]. These authors used the LRMECS chopper spectrometer, on the IPNS spallation pulsed neutron source at the Argonne National Laboratory, which has a higher energy resolution and allows measurements to be made to higher energy transfers. Scattering was recorded out to beyond 40 T H z revealing further inelastic features in the tail of the component observed here. Similar features are recorded by Leadbetter and Stringfellow [9] although to compare their density of states with the present data it is necessary to include a factor which varies as - Q2/v. (Note that with a beryllium filter spectrometer each value of v is obtained at a different m o m e n t u m transfer, hQ). The behaviour of vitreous silica in fig. 12 should be contrasted with that of l i q u i d N 2 (ref. [17], fig. 1) where there is no "'elastic" component, even from the distinct scattering. The general form of the scattering law is as predicted by eq. (16), with M/ equal to the atomic mass of nitrogen [17], although this will be modulated by interference effects. As a result the average Q, at a given scattering angle, in a total diffraction experiment on a molecular liquid is not Qe, which means that, if the data are analysed in terms of Qe, the wrong bond length is obtained. In order to extract the correct bond length it is necessary to include a recoil correction. The present data show that this is not the case for an amorphous solid. A component of the scattering remains at hw = 0 and, as demonstrated by the peak fits of the previous section, the S i - O bond lengths from elastic and total diffraction experiments are in excellent agreement.

5. Conclusion The differences between the elastic and total diffraction patterns of vitreous silica have been investigated quantitatively and it has been shown that, in the former case, the first two peaks in the resulting correlation function, T(r), are significantly broader, due to the inclusion of the effects of acoustic phonons,

A. C Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111

(Av2F- Av~.)a/2(THz)

Vsi(THz )

Avsl (THz)

vo(THz)

Avo(THz)

2.83 3.77 3.01 3.99

0.50 2.28 2.83 3.55

2.47 5.30 5.90 6.60

0.87 4.00 4.97 6.23

3.28 7.02 7.81 8.75

367

even with the rather p o o r energy resolution of IN1. A comparison of the quantity 'z,,2 ~ S i ~ O 7~1/2 with the value expected from the directionally averaged rms atomic thermal displacements ~,~Si~' r, :~l/a and (U2o)1/2 indicates that the atomic thermal vibrations are approximately isotropic when referred to the S i - O b o n d direction. The present data have clearly demonstrated the potential of neutron elastic diffraction studies of a m o r p h o u s solids. However, to be of the m a x i m u m use, an elastic diffraction experiment should be designed to extract the values of u 7 and u)A as AE---, 0, which means working with the highest available energy resolution ( A E < 1 meV) while maintaining a high Q .... (X - 0.5-0.6 ,~). If possible data should be obtained as a function of A E , at the same X 0, by using several different analyser planes, in order to allow an extrapolation to zero zl E. In this way thermal parameters may be extracted which can be directly c o m p a r e d with those obtained for the crystalline state. The constant Q scans show that for an a m o r p h o u s solid, unlike a molecular liquid, a c o m p o n e n t of the scattering remains centred at h~0 = 0. As a result it is not necessary to apply a recoil correction to the S i - O b o n d length obtained from total diffraction measurements, as demonstrated by the agreement with the value from elastic diffraction. The simple twin Gaussian model for S(Q, w) m a y prove useful in studying the effects of static approximation distortions, although further work is required to extract more precise parameters over a much wider range of Q. The authors would like to thank the Science Research Council, A E R E Harwell and the Institut L a u e - L a n g e v i n for financial and experimental support.

References

[1] [2] [3] [4] [5]

P.A.V. Johnson, A.C. Wright and R.N. Sinclair, J. Non-Cryst. Solids 58 (1983) 109. V.F. Turchin, Slow Neutrons (Israel Program for Scientific Translations, Jerusalem, 1965). E.A. Lorch, J. Phys. C. 3 (1970) 1314. A.C. Wright, Adv. Struct. Res. Diffr. Meth. 5 (1974) 1. P. Pellionisz, Nucl. Instr. and Meth. 92 (1971) 125.

368

A. C. Wright, R.N. Sinclair / Neutron scattering from vitreous silica 111

[6] D.F.R. Mildner and A.C. Wright, J. Non-Cryst. Solids 42 (1980) 97. [7] P.A. Egelstaff, Physics of Non-Crystalline Solids, ed., J,A. Prins (North-Holland, Amsterdam, 1965) p. 127. [8] A.J. Leadbetter, J. Chem. Phys. 51 (1969) 779. [9] A.J. Leadbetter and M.W. Stringfellow, Neutron Inelastic Scattering 1972 (IAEA, Vienna, 1972) p. 501. [10] U. Buchenau, N. Niacker and A.J. Dianoux, Phys. Rev. Lett. 53 (1984) 2316. [11] J.M. Carpenter and D.L. Price, Phys. Rev. Lett. 54 (1985) 441. [12] G. Caglioti, Nuovo Cimento Suppl. 5 (1967) 1177. [13] G. Caglioti, Theory of Condensed Matter (IAEA, Vienna, 1968), p. 539. [14] R.N. Sinclair and A.C. Wright, J. Non-Cryst. Solids 57 (1983), 447. [15] B.E. Warren, X-ray Diffraction (Addison-Wesley, Reading, 1969). [16] D.E. Parks, M.S. Nelkin, J.R. Beyster and N.F. Wikner, Slow Neutron Scattering and Thermalization (Benjamin, New York, 1970). [17] R.N. Sinclair, J.H. Clarke and J.C. Dore, J. Phys. C:8 (1975) L41. [18] V.F. Sears, Adv. Phys. 24 (1975) 1.