Diffraction studies of glass structure

JOURNAL OF NON-CRYSTALLINE SOLIDS 7 (1972) 37-52 © North-Holland Publishing Co.

DIFFRACTION STUDIES OF GLASS STRUCTURE II. THE STRUCTURE OF VITREOUS GERMANIA A. J. LEADBETTER and A. C. WRIGHT* School of Chemistry, University of Bristol, Bristol, U.K. Received 24 May 1971; revised manuscript received 19 July 1971 The effects of pile irradiation on the polymorphs of germania are described and it is shown that the structural effects are small. For the glass, density changes on irradiation are smaller than those resulting from different methods of preparation. The X-ray diffraction pattern of vitreous germania has been measured for Q between 0.8 and 16/~ 1. The experimental method and data reduction procedures are described. The structure of vitreous GeO2 is discussed, using the correlation function Dx (r) obtained by fourier transformation of the X-ray intensity data. This is compared with correlation functions derived from neutron diffraction experiments and from quasi-crystalline models based on the various crystalline polymorphs of GeO2. The structure is shown to be built of GeO4 tetrahedra: the coordination number of oxygen about germanium is 4.0 to within one or two per cent, the Ge-O distance is 1.74/~ and the average Ge-O-Ge angle is 133°. The distribution of Ge-O-Ge angles is fairly narrow but cannot be determined quantitatively with data of present resolution. The average structure of the glass closely resembles that of the quartz modification as smeared out in the quasi-crystalline model. The remaining differences between model and experiment may be accounted for in terms of an unwinding (and disordering) of the crystal structure by relative rotation of tetrahedra, keeping the average intertetrahedral angle of 13Y'.

I. Introduction Three m a j o r crystalline p o l y m o r p h s of G e O 2 are k n o w n in a d d i t i o n to the glass. At temperatures below 1049 °C the t h e r m o d y n a m i c a l l y stable phase is a rutile form, with octahedral c o o r d i n a t i o n of oxygen a b o u t g e r m a n i u m . This is replaced by a [3-quartz modification between 1049°C and the melting p o i n t (1116°C). The rutile to [3-quartz t r a n s f o r m a t i o n is very sluggish a n d although it will proceed in a matter of hours on heating, the reverse transf o r m a t i o n occurs only extremely slowly, if at all, in the pure material. The [3-quartz modification exists metastably d o w n to 1020°C when it transforms to an a - q u a r t z form which exists indefinitely at lower temperatures. The liquid readily supercools to give a glass a n d the glass t r a n s i t i o n temperature is a b o u t 550°C (ref. 1). The glass readily devitrifies at temperatures greater * Now at J. J. Thomson Physical Laboratory, Whiteknights, Reading, RG6 2AF, U.K. 37

38

A . J . LEADBETTER AND A. C. W R I G H T

than about 700°C to give the quartz form but in the presence of a flux (e.g. NaC1) will give the rutile form for T < 1049°C. Treatment for long periods at relatively low temperatures (T~600°C) will give a mixture of cristobalite and a-quartz modifications2). However, experiments in this laboratory have so far failed to establish the conditions necessary for the preparation of the cristobalite form, which has only been obtained in irreproducible experiments. All the low pressure forms of silica are susceptible to pile irradiation damagea, 4) and at high dose tend to the same vitreous phase with a density about three per cent higher than that of normal vitreous silica. The radiation induced vitrification usually proceeds via a structure resembling but not identical with the high temperature (13-) modification of the phase in questionS). This process is of interest in the study of the vitreous state because it represents a means of gradually producing a glass from a crystal by a solid state process. We have therefore investigated the effects of pile irradiation on the various polymorphs of germania and find that in contrast to silica they are very resistant to radiation damage, and that none undergo any gross structural change at fast neutron doses up to about 102°/cm 2. The glass structure was first investigated using X-ray diffraction by Warren 6, 7) but the data were not fourier transformed. The only other X-ray study is that of Zarzycki s) whose data were limited to Q < 9 A - 1 thus giving rather poor structural resolution. Two total neutron scattering experiments on vitreous GeO2 have been reported 9,10) and although there are significant differences between the data they are both consistent with a structure based on GeO4 tetrahedra, in agreement with the X-ray work. In both cases, however, the neutron radial distribution functions gN(r) * were interpreted in such a way as to suggest a marked discrepancy with the X-ray results. Thus, a peak in gN(r) at about 3.45 A was attributed to the first G e - G e pair and this, combined with the G e - O distance of ,-~ 1.72 A, indicated a G e - O - G e angle of ~ 180 °, whereas the X-ray data shows a peak attributed to G e - G e pairs at 3.15 A, with a G e - O peak at 1.70 A, indicating a G e - O - G e angle of 125-152 °. This discrepancy will be shown not to be real and due simply to the different relative scattering powers of Ge and O for X-rays and neutrons which causes DN(r) and DX(r)* to differ significantly. New X-ray di~raction results on vitreous GeO 2 will be presented and the interpretation of these and the total neutron scattering experiments will be discussed in terms of the quasi-crystalline model presented in part 1. * See Part I (preceding paper) for definitions of these and other terms.

DIFFRACTION STUDIES OF GLASS STRUCTURE. II

39

2. Pile irradiation of GeO2 Irradiations were carried out in sealed silica ampoules in the Herald reactor at A.W.R.E. Aldermaston and doses monitored by means of cobalt and nickel wires included in the aluminium irradiation cans. The fast neutron doses given below refer to an idealised fission spectrum based on a nickel cross section (28Ni58(n, p) 27Co 58) of 107 mbarns and are believed to be accurate to within about 10 per cent. At doses up to 8.3 x 1019 fast neutrons/cm z there is no significant change in lattice parameter of the a-quartz modification. The powder diffraction peaks of the most heavily irradiated material were sharper than those of the starting material but some at least of this sharpening occurred during the time the X-ray specimen was made up and its diffraction pattern measured. Some of the a-quartz GeO 2 starting material contained a small amount of the rutile form and irradiation of this mixture resulted in an increase in the amount of rutile form. Thus the ratio (R) of the heights of rutile 110 and c~-quartz 101 powder diffraction peaks as a function of dose was as follows fast n e u t r o n s / c m 2 R

0 0.016

1.2 X 1019 0.087

6.6 × 1019 0.160

Pure a-quartz samples showed no trace of rutile formation on irradiation so that the effect of the irradiation is to promote growth of existing rutile nuclei or crystallites at the expense of the thermodynamically less stable a-quartz form. The density of the vitreous material changed only very slightly on irradiation to 9 x 1019 fast neutrons/cm z (see table 1) showing that there were no gross structural changes, but the irradiated samples are very dark brown due to a shift and broadening of the UV absorption edge to lower energies. Both quartz and vitreous forms are considerably more reactive to atmospheric moisture after irradiation. The densities of a variety of samples of vitreous germania have been measured and the characterisation of the samples, with their density, is given in table 1. In all cases the starting material was 99.999~o pure crystal powder and the samples were melted in platinum. Densities were measured by weighing in air and in diethyl phthalate. Changes on irradiation are less than differences due to different preparation conditions of the glass but the densities may be tending to a limiting value of 3.647 g/cm 3. Melting conditions appear to have the largest effect, vacuum melting producing glasses of higher density, but all the samples are considerably less dense than

Unknown Ex-sample 1

Ex-sample 5

Ex-sample 1

5 6

7

8

1565 Air 1430 Air 32 hr 1400 Dry N2 23 hr 1400 Dry N2, 2 0 h r 1150 Air 168 hr

Refining (oc)

520 Air 24 hr

520 Air 16 hr 520 Air 24 hr 520 N~, 16 hr

Annealing (oc)

1.2 2.5 5.6 1.1 9.3 1.8

× x × × x ×

1019 1019 1019 102o 1019 1020

Irradiation fast neutrons/cm z thermal neutrons/cm 2

3.6465

3.6457

3.6524 3.6489

3.6590

3.6573

3.6443 3.6452

Density (g/cm a) ± 0.0003

* We are very grateful to Dr. S. W. Barber, Dr. Vergano and to the Owens-Illinois Corporation for supplying samples 1, 2, 3, 4, 6 and 8. Also to Dr. C. R. Kurkjian and the Bell Telephone Laboratories for supplying samples 5 and 7. We t h a n k Mr. A. J. Apling for measuring the densities of samples 2, 3, and 4.

4

3

2

1565 Air 1510 Air 22 hr 1150 Vacuum 18 hr 1150 Vacuum 21 hr

Melting (°C)

1

Sample

TABLE 1

Characterisation and density of vitreous GeOa specimens*

4~

DIFFRACTION STUDIES OF GLASS STRUCTURE. II

41

those reported by Mackenzie 11) who found 3.667+__0.002 g/cm 3 for a glass melted at 1500°C and annealed at 520°C. Samples 3 and 4 were prepared to investigate differences between reduced and stoichiometric samples. The reduced sample (3) has a faint yellow colouration because of the long wavelength tail of the absorption band at 244.5 nm which appears in the reduced materiaP 2) and its density is slightly lower than the fully oxidised sample (4) as expected. The X-ray diffraction specimen was a 2.5 mm thick plate similar to sample (1). Infrared absorption measurements showed similar samples to contain about 7 x 10 -4 OH groups per Ge atom.

3. Experimental The X-ray diffraction apparatus is based on a Philips vertical goniometer and is similar to that described by Levy et a1.13). Molybdenum radiation was used over the angular range 5 to 150° in 20 (0.8 < Q/A-' < 17). A curved LiF crystal monochromator was employed in the diffracted beam and X-rays detected with a scintillation counter, in conjunction with a pulse height analyser. Reflection geometry was used throughout, although a comparison of reflection and transmission experiments at low angles (20<20 ° ) using a BeF z sample showed excellent agreement. Four separate angular ranges with different slit widths were used and nine separate scans were carried out over each range in order to minimise the effects of generator and electronic drift. This was shown to be less than 0.1 ~o over the time required for a single scan over any of the four angular intervals. For each scan 4 x 10 3 counts per point were accumulated. After applying various corrections to the raw intensities the four ranges were internormalised empirically. The data were first corrected for background and, at low angles, for air scatter (the counting system paralysis correction was negligible). The correction for the penetration of X-rays into the sample was calculated in the parallel beam approximation, using Milberg's 14) formulae for the geometry where the counter sees more than the irradiated surface of the sample. The correction showed negligible angular variation in each of the four angular ranges. Data were corrected for polarisation at the sample and monochromator using the factor ½ (1 "{-COS220 COS2 20.1) where 0m is the Bragg angle for the monochromator. The corrected intensity data were normalised and the intensity function computed according to

42

A.J. LEADBETTER AND A. C. WRIGHT

The normalisation constant ~ was found by means of the integration method of Krogh-Moe 15) and Norman 16) which gives (2.,

f[ ~ ", :f~~L2 - - , - . Q 2 (Q)+

Ii.~(Q)]M(Q)dQ,_ 2~2pO( ~' zi )

2

0 Qm

; QI(Q)M(Q)dQ f2(Q) 0

Here I[,¢ (Q) represents the incoherent Compton-modified radiation which is transmitted by the diffracted beam monochromator and the independent intensity was calculated from the form factors tabulated by Hanson and Pohler 17) with dispersion corrections for Ge given by CromerlS). In this and in the model calculations dispersion corrected form factors for Ge 2 ÷ and Owere used, in an attempt to recognise approximately the partially ionic bonding in GeOz, but the results are in fact very insensitive to the use of ionic as opposed to atomic form factors. The correction for Compton scattering represents the greatest difficulty in deriving accurate intensity functions for vitreous materials. The Comptonmodified intensity transmitted by the monochromator with a pass band P(2) may be written in terms of the Cornpton scattering of the individual atoms as

[ ol

=

.

L2-,-I # r°°(o).

where (20/2,)2 is the Breit-Dirac recoil factor for a counter 23) and to and 2, are respectively the incident and scattered wavelengths. The monochromator pass-band was measured using a quartz crystal and white radiation from a copper tube. The resultant intensity function contained errors, which also show up as ripples in the correlation function at low r(below the first peak). They almost certainly arise from distortions caused by the diffracted beam monochromator limiting the acceptance angle of the detector system, which would give errors increasing with sample penetration by the X-ray beam 19, 20) as observed in this work, where data for GeO 2 were much less affected than those for BeF 2. Analagous errors were observed in the similar apparatus of Levy et al. and the effective pass-band for the monochromator was therefore determined by an empirical method similar to that described by these workers. This was done using data for silica for which recent very accurate results are available for comparison and the results then applied to other materials with final refinement of the intensity function by Fourier transforming residual ripples in at low r. Because of the empirical removal

D(r)

DIFFRACTION STUDIES OF GLASS STRUCTURE, n

43

in this way of spurious contributions to the intensity, no additional correction was made for multiple scattering. It should be emphasized that these errors and their empirical removal will not affect the positions of peaks in D(r) but only their shape and area.

1200

1000

800

600

400 I

2OO

Q / A -1

la0

115

Fig. 1. The normalised intensity of X-ray scattering from vitreous Ge02 in absolute units (electron units). The results are shown in fig. 1 as the normalised intensity, in absolute units (electron units), and in fig. 3 as the intensity function Qi(Q). The data of Zarzycki are in good agreement with those in fig. 1 but the new results cover a much wider range of Q-values. 4. D i s c u s s i o n

Fourier transformation of the data using eq. (2.21) of Part I gives D x (r), shown in fig. 4, where

D x (r) = E E dx. (,), i j

and co

d x = f d U(u) P x ( r - u ) d u , 0

44

A.J. LEADBETTER AND A. C. WRIGHT

with ~m

P~j (r) = 1 f f~ (Q) f j (Q) sin nQ/Q,, cos Qr dQ ;

r~

fo2 (Q)

rtQ/Qm

0

dij(r ) are the true component correlation functions and the problem is essentially to extract from D x ( r ) as much information as possible about these functions. As a first step an approximate check may be made as to whether the present data are consistent with the tetrahedral coordination of first neighbour oxygens about germanium indicated by previous X-ray and neutron scattering experiments. The first peak in the total radial distribution function 9 X ( r ) has an area of 1980 electron units, in excellent agreement with Zarzycki's value of 2000 e.u. This agreement is gratifying as the termination effect (including any modification function used to reduce satellite ripples) should not affect the peak area by more than a few per cent for the range of termination values involved in this and Zarzycki's experiment 21). An approximate coordination number of oxygen about germanium is obtained by assuming that the first peak in gx (r), at ,,; 1.74 A is due solely to Ge-O atom pairs, and using the approximation that the atom form factors are proportional to each other and may be writtenf~ = KiJo withfo = ~ f ~ / ~ zi. Average values over the Q-range of the experiment are K~e=35.5 and K'o=6.23, which give a coordination number of 4.46 oxygens round each germanium. The so-called K-approximation is very poor for GeO2: the K values for Ge 2 ÷ and O - vary from 30 and 9 respectively at Q = O through extrema at Q = 7.6 A - i of 37.5 and 5.25 to 36.0 and 6.0 at Q = 16 A - 1. In view of these theoretical approximations and experimental errors, the result of 4.46 cannot be regarded as differing significantly from 4. The agreement with the earlier X-ray data, together with the coordination numbers of 3.8 and 3.9 obtained in the two neutron scattering experiments, suggest that the high value from the X-ray data is due primarily to the use of the K-approximation. It will be shown later that this is indeed the case. The use of this approximation is not essential as it is possible to calculate the peak function Pix (r), and hence the expected shape and area of a peak in D x (r), for an assumed number of Ge-O (or any other) atom pairs and to compare this with experiment. This has not been done here because the same information may be obtained using the quasi-crystalline model described in part I (see below) and furthermore at higher r, where there will be many overlapping peaks, the use of the quasi-crystalline model will be simpler. An advantage of the direct calculation of peak functions might arise in the calculation of distributions of distances between particular atom types, e.g.

DIFFRACTION STUDIES OF GLASS STRUCTURE. II

45

distributions of first G e - G e distances. However, such a calculation will only be meaningful if the peak in question is not seriously overlapped by other peaks and with present resolution this is only so for the first (Ge-O) peak (see below). Even here, however, nearly all the peak width is due to the peak

500

7< 5 4

'I J

500

0

-500

1'0 Q A -1

1'5

2'0

Fig. 2. Intensity functions for quasi-crystalline models of vitreous GeO2 with a correlation length of 10.0 ,~ and based on the crystal modifications (1) c~-cristobalite(2) ~x-quartz (3) rutile. function PXe_o ( r ) itself so that an accurate determination of the (small) bond length distribution is not possible without higher resolution. The results of preliminary quasi-crystalline model calculations of the intensity function are shown in fig. 2. These were computed using eq. (3.3) of Part I for the three major crystalline modifications of GeO2, and a correlation length of 10 A. Comparison with experiment shows that the model based on the a-quartz structure 22) is better than that for the a-cristobalite structure, while the rutile structure is obviously very different from that of the glass. Comparison of the correlation functions confirms the quartz model as being better than that from cristobalite since for the latter the G e - O bond length is only 1.62 A compared with the experimental value of 1.74/~. This is because no positional parameters are available for the ct-cristobalite structme of GeOz and it was necessary to use those for the corresponding silica

46

A.J. LEADBETTER AND A. C. W R I G H T

polymorph. It is, however, possible that when the correct parameters are available this crystal form will give a better representation of the glass structure. At this stage, further refinement was limited to the quasi-crystalline model based on the a-quartz structure. A best fit to the intensity function

1000 I,

'<

~oo~ ~, / IA~

5

I

"~-500

-oooL/i

f~

ii! i

o

Jil i

•

J i~

"7<

"-~' ~ b !~,

5

~I

"v'

•

i v

'~ Ill

r'k

oY',

3

L

I

I

•:-.--" 0

4

o :,.....,/ "........y,

...........................................................................

0

I

10

I

20

a / A -~

Fig. 3. Intensity functions for vitreous GeO2. (l) Total intensity: solid line, experimental

results; broken line, best model (a-quartz structure, correlation length L = 10.5 .~.). Curves 2, 3 and 4 are the total contributions to the intensity function from Ge--Ge, Ge-O and O-O pairs, for which x~j is 1, 2 and 2, respectively. was obtained with L = 10.5 A and this is shown in fig. 3 together with the three c o m p o n e n t contributions xq Qiu (Q), which illustrate how little of the scattering arises from oxygen atoms. There is a slight difference in peak positions between model and experiment for the first two peaks, which arises because the model density is higher than that of the glass, but otherwise the agreement is excellent.

47

DIFFRACTION STUDIES OF GLASS STRUCTURE. II

2O00

7

~I000 %

t,,i I',

.,, ,,

1

0

-100(

i

i i i 100(

^ I~ I!

o -. "E"

I

ii

I

i

t~ /I

i i i i

li ii iiii i ii i V i.~. •

:'

..":

/,.\ ;~../

2 ' ~ . / ' ' - ....

..

fl

4

0 ~

5

10

r/A Fig. 4. X-ray correlation functions. (!) Total DX(r); solid line, experimental; broken line, best model (a-quartz structure, L = 10.5 .~). 2, 3 and 4 are the total contributions to DX(r) from Ge-Ge, Ge-O and O-O pairs, for which x~j is 1, 2 and 2, respectively. [N.B. D'(r) = ~,~Y~d'ij(r) and d ' G e - o ( r ) = 2d'o ~e(r).] The dotted lines in 1, 3 and 4 show

the modified correlation functions obtained by expansion and disordering of the basic model as described in the text. In fig. 4 are compared the experimental and model correlation functions, together with the individual components of the correlation function for the model, all obtained by Fourier inversion of the appropriate intensities under exactly the same conditions. The overall agreement is good; the main discrepancies being that the model peaks beyond r _~4 ,~ are too sharp and at too low r values, which show again that the model is too dense and not sufficiently disordered. This will be further discussed below, in conjunction

48

A.J. LEADBETYER AND A. C. W R I G H T

with the neutron data, but first it is worthwhile to consider in more detail the first two peaks in DX(r) at 1.74 and 3.18 A. The peak positions for the model are in excellent agreement with experiment and since the first peak is due entirely to Ge-O pairs, and the second mainly to Ge-Ge pairs, this shows that in the glass the average Ge-O intratetrahedral bond length (1.74 A) and the intertetrahedral Ge-O-Ge bond angle (133°) are not significantly different from those in the quartz form. Since convolution effects identical to those for the real glass are automatically included in the model calculation it is meaningful to compare also the widths of peaks in D (r). The half height widths of the first two peaks are not significantly different between model and experiment, indicating that the distribution of Ge-O and Ge-Ge distances in the glass is much the same as for the quartz form. The results indicate an average Ge-O-Ge angle of 133° and although there will in fact be a distribution of angles it is not possible to determine this accurately with experiments of the present resolution. In addition to the inherent resolution problem the difficulty is enhanced by the overlapping or peaks and by the insensitivity of the Ge-Ge distance to the angle. Nevertheless, the distribution of Ge-O-Ge angles cannot be very broad. Although the width and position of the first Ge-O peak in D x (r) for the model agrees so well with experiment, its height and area are significantly less. This is due to the effect of the damping function F(r ) used in the model (see Part I) which is partially counteracted by a higher model density. The use of the F(r) function to damp down the peaks in the crystal D(r) is, of course, the essential feature of the model but, since F (r) is unity at r--0, it will also affect the first peak in D (r) and will thus result in a difference between model and experiment if the glass has crystal-like short range correlation (e.g. GeO4 tetrahedra). However, the effect of the damping function and any density difference between model and glass may readily be calculated. Furthermore, since only a relatively small difference is involved the use of the K-approximation should be adequate. Since the coordination number of oxygen about germanium is required, these calculations have been applied to the total radial distribution function gX (r). The expected area (AE) under the first peak in gx (r) at rp (in e.u.), if the coordination number is exactly 4, is then related to the area derived from gX (r) for the model

(AM) by

peak

using the definition gX (r) = r [ D x (r) + DX(r)-I,

(2)

49

D I F F R A C T I O N S T U D I E S O F G L A S S S T R U C T U R E . II

where D x ( r ) is obtained by fourier transforming the model intensity and is related to the original crystal correlation by the factor F(r ). Dx (r) is defined by Dox ( r ) = (~i z~)2 4nrp ° (eq. (2.25) of Part I) and does not contain the factor F(r). pO is the average density of composition units for the model. In the present case pO exceeds Pglass o by 17% and (1) gives A E =2046 + 436 - 529 = 1953 e.u.,

compared with the experimental result of 1980 e.u. This result shows that the coordination of oxygen about germanium is 4.0 to within one or two percent, which is less than the experimental and computational uncertainty. The value of 4.46 obtained earlier by the approximate method thus differs from 4.0 almost entirely as a result of the use of the K-approximation. It will be useful to examine the overall correlation function in conjunction with those from the neutron data. These have been published as plots of rDN(r) so we have computed model functions in this form, using model intensity data out t o Qm = 18 A - l, as in Lorch's experiments. The comparison is shown in fig. 5, together with the model components.

0 S< t~ -o-

c ¢0 d3

5 .

3

% £.

-0

o

Z

-5

0

5

5 r/A

10

Fig. 5. Neutron distribution functions. (1) rDN(r) experimental: solid line from L o r c h % broken line from Ferguson and Hassl°). (2) Total model rD•(r). 3, 4 and 5 are respectively the component functions x,~d,~N (r) for Ge-O, O - O and G e - G e pairs with x~j as for fig. 4. The dashed lines in 2, 3 and 4 show the modified functions obtained as for those in fig. 4 and described in the text.

50

A.J. LEADBETTER AND A.C. WRIGHT

The rD N (r) function of Ferguson and Hass has been normalised to that of Lorch at the first peak maximum for ease of comparison. All the rD (r) functions are then given in units of barns A-1 but note that the scales are different for model and experimental functions. There are very significant differences between the two sets of neutron results. The widths of the peaks in Lorch's correlation function are much larger than those of Ferguson and Hass, despite the fact that these functions were obtained by inversion of data extending respectively to 18A -1 and 11 A -1. The first peak in Lorch's rD N (r) is also much wider than that for the model, where identical Qm values (18 A-1) were used. There is also a striking difference in the heights of the third peak at r,-~3.4 A. Figs. 4 and 5 show immediately that the published interpretations of the neutron data are erroneous in attributing the third peak in the distribution function at 3.45 A to the G e - G e pair and in the concomitant discrepancy with the X-ray results. In fact, this peak arises mainly from G e - O pairs with some contribution from O-O and very little from G e - G e pairs. The first G e - G e peak is at 3.18 A but because of the different relative scattering powers of the atoms for X-rays and neutrons it shows up directly in the X-ray results and not at all as a distinct peak in the neutron correlation function. (It is seen in rDN(r) for the simple model, but see below.) The comparison of X-ray and neutron data with each other and with the quasi-crystalline model may be used to determine the way in which the real glass structure differs from that of the model. This may readily be done quantitatively for r E 4 . 5 A and qualitatively for higher distances. Thus, between 2 and 4 A the detailed agreement between experiment and model correlations is considerably better for the X-ray than the neutron case and this shows that the main inadequacy of the model must be in the positions of the oxygen atoms. The model is certainly too dense but a straightforward lattice parameter expansion is not the correct way to reduce the density as this would also expand the basic tetrahedra and so destroy the good agreement with experiment for r < 3 A. What is required may be pictured as an expansion (and disordering) of the structure by an unwinding process involving relative rotation of tetrahedra about the intertetrahedral angle of 133 °. This process is illustrated in fig. 6 where the locus (L) of possible positions for a second Ge atom, relative to a first, with fixed Gel-O1-Gez angle is shown. The overall density and disorder in the structure will be determined by the relative orientation of tetrahedra with respect to this type of rotation, then that about Ge2-Oj and so on. It is also worth emphasising that in the real glass not all tetrahedra will be complete, nor will a tetrahedron always be linked to four others. The most important effect of the above unwinding process on the model at short distances is to increase the separa-

DIFFRACTION STUDIES OF GLASS STRUCTURE. II

51

tion of the three nearest intertetrahedral oxygen atoms from any given oxygen atom. In the quartz structure, these have separations only slightly larger than the intratetrahedral oxygen spacing. One such pair is shown by the atoms labelled A and B in fig. 6. There will also be an attendant small

L

®

k.--,, i %A

® ®

Fig. 6. Schematic representation of structural features for vitreous germania; Q: Ge atoms; encircled numbers: O-atoms; L is the locus of positions of a Ge atom (2) generated by rotation about the Gel-Or bond with a fixed Ge-O-Ge angle.

increase in some of the G e - O spacings. The result of increasing the separation of these oxygen atom pairs by an average of about 0.7/k is shown in the component and total model correlation functions of figs. 4 and 5. The resultant agreement with experiment for r ~,4/~ is very good indeed for both X-rays and neutrons. At higher distances it is obviously much more difficult to deal with the structure quantitatively in this simple manner, but the comparison between model and experiment shows that the real structure is not grossly different from that represented by the model, since features in the model correlation functions correspond closely to those for the real glass. The model is certainly too dense and the damping function F(r) does not accurately represent the disorder, particularly for r between about 4 and 6/~, where the peaks in the model correlation functions are too high and sharp by roughly similar amounts in both the X-ray and neutron cases. However, it seems clear that a rather small further disordering of the tetrahedra and an expansion of their packing along the lines described above can account for the remaining differences.

Acknowledgements We are grateful to the Science Research Council for the award of a maintenance grant to A.C.W.

52

A, J. LEADBETrER AND A. C. WRIGHT

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