Diffraction studies of glass structure III. Limitations of the fourier method for polyatomic glasses

JOURNAL OF NoN-CRYSTALLINE SOLIDS 7 (1972) 141-155 © North-Holland Publishing Co,

DIFFRACTION STUDIES OF GLASS STRUCTURE III. LIMITATIONS OF THE FOURIER METHOD FOR POLYATOMIC GLASSES A. J. LEADBETTER and A. C. WRIGHT* School of Chemistry, University of Bristol, Bristol, England

Received 24 August 1971 Using the scattering intensity calculated for a simple quasi-crystalline model based on the high-cristobalite structure, for which the structural correlation function is known exactly, the fourier transformation of X-ray and neutron intensity data for a polyatomic glass has been investigated. The effects of termination of the data at finite Q, quadrature interval, data normalisation and various kinds of errors are discussed. The resulting form of the distortions on the derived correlation functions, of errors in peak positions and in peak areas are described and it is shown that a variety of errors give rise to pronounced structure in the correlation function at low values of r, below the first true peak. 1. Introduction

In fourier transforming X-ray or neutron intensity data it is important to know the effect on the resultant correlation function o f various kinds o f experimental error, o f the multiplication o f the intensity by modification functions and o f the use o f any theoretical approximations, such as the socalled K approximation for X-rays. A n u m b e r o f discussions have appeared o f some o f these effectsl, 2) but no full, detailed treatment has been given for polyatomic systems, which have some features not found in the m o n atomic case, nor one including special features found for neutron scattering. Furthermore, the use o f the scattering intensity for a simple model, such as the quasi-crystalline model described in Parts 13) and II4), provides an ideal basis for such an investigation because in this case the correlation functions are k n o w n exactly. Hence, errors in the correlation functions m a y be determined exactly in results which accurately simulate the experimental situation. 2. The model

The model chosen for this investigation was the quasi-crystalline model * Now at J. J. Thomson Physical Laboratory, Whiteknights, Reading, England. 141

A. J. L E A D B E T T E R A N D A. C . W R I G H T

142

described previously, based on Wyckoff's symmetrical 13-cristobalite structure for silicaS). Although not a good model for vitreous silica, this has the advantage of relatively few interatomic spacings at low r, with no close doublets. This enables the widths of peaks to be easily measured. In addition, a low correlation length of 8.2 A was chosen so that distortions at higher r are readily visible. Inversions were performed using Filon's 6) quadrature with basic intervals of 0.05 A - 1 for Q and 0.02 A for r. Except where otherwise mentioned the effect of the finite upper limit in Q was reduced using the resolution function due to Lorch 7), and the X-ray form factors were those for the free atoms. The intensity functions Qi(Q)were calculated for the model using eqs. (3.3)

(a) ~ !

Neutron

::

il

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•

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143

D I F F R A C T I O N STUDIES OF GLASS S T R U C T U R E . III

and (3.4) of Part I. Fourier transformation then gives the correlation functions OX(r), Dr(r) and di~(r ) (see Part I) s h o w n in fig. la. For the X-ray case the sharpening function fo2 (Q) introduces distortions into the correlation functions. This is demonstrated by comparing the X-ray correlation functions o f fig. la with those calculated from the same model using form factors for which the K-approximation (form factors simply proportional to each other) is exact (fig. lb) i.e.

f E(Q) = K,fo (Q),

i

k

(2.2)

il il

~ooo

(b)

Ii i!

t~

i l

?

i

o<

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% I

i

i

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(c) 1.2

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1.O

0.8 =

I

10

Q/j,-'

~

I

20

Fig. 1. (a) X-ray and neutron correlation functions derived from the model intensities (Qm = 22/~-1). (b) X-ray correlation functions for the model, using atom form factors for which the K-approximation is valid. (c) Deviation of the form factors for Si and O from the K-approximation. For (a) and (b): ( - - - ) Si-Si, x t s = l ; ( - . - . - ) Si-O, x ~ = 2 ; ( ...... ) O-O, x,s = 2; ( - - ) total.

144

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

where

fo (Q) = E f~ (Q)[E z,. i

(2.3)

i

fo (Q) was calculated from the normal scattering factors and the values of K used (Ksl = 14.85, Ko = 7.57) were the average values /~ found for the same form factors [cf. Zarzycki s)] according to

Q,.

,f

X, = ~

K~(Q) dQ,

(2.4)

0

with

K~(Q)=f(Q)/fo(Q)

and

Q,,=22A-'.

The extent of the deviations from the K-approximation for the free atom form factors is shown in fig. lc.

Qm

o

o ? o<

i ~

_i o

'15

I 5 r//,~

I 10

Fig.2. NeutroncorrelationfunctionfordifferentvaluesofQm.

145

DIFFRACTION STUDIES OF GLASS S T R U C T U R E . I l l

3. Termination effects

The resolution o f the final correlation function is governed by the m a x i m u m value of Q to which reliable intensity data are available. This is illustrated in fig. 2 where neutron correlation functions are shown for a series o f integration limits. Similar results are obtained f r o m the corresponding X-ray data. The various methods for dealing with the finite upper limit of Q are c o m p a r e d in fig. 3, for the X - r a y case. The resolution of the resulting correlation functions may be specified by the height and width [at D x ( r ) = 0] o f the first peak, and these two parameters are plotted as functions o f Qm in

2000

1000

~-

Z

o

~

o,~

°F -lOOO

o

4

I

Fig. 3. Different methods of dealing with the finite upper limit of Q, shown for X-ray correlation functions (Qm = 20 A,-1). (1) Unmodified, M(Q) is a step function. (2) Electronic, no sharpening function lfo(Q)= 1]. (3) Resolution function, M(Q)= (sinr~Q/QnO/(~Q/Qm). (4) Exponential modification function M(Q)~exp(-BQ2), B chosen such that M(Q,n) = 0.1.

146

A. J. LEADBETTER

AND A. C. WRIGHT

fig. 4. The resolution of the electronic distribution (no sharpening function) is very poor, while that of the unmodified atomic distribution is easily the best. There is little to choose between the resolution function and the conventional exponential modification function but, for the same resolution, the former is more effective in reducing subsidiary features and has been used throughout this work. Very similar results are obtained for the neutron case, except that there is here no equivalent of the electronic correlation function.

~3000 I

0 0 0

I

0 0

-

o o

1000 I

~

0 0

, I

oII

0.8~

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rl I

~c x'-" 121 0.6

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i~

o

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[]

o

0

o o 2S

e

o

~

o

0.4

o

r-

0

~

0.2 i

1'o O,m ,//~ -2

Fig. 4. The height and width of the first peak in DX(r) for different modification functions: (©) unmodified; (D) electronic; (0) resolution function; (x) exponential modification function [ M ( Q m ) = 0.1].

Although the effect of the finite upper limit in Q is a convolution, which directly affects only the peak shape, changes in peak positions may be caused by interaction between adjacent features. This is particularly true for the total radial distribution function, where peak shapes are also affected by the extra factor of r. The variations in the positions of two of the peaks as functions of Qm, are shown in fig. 5a for the three functions: differential correlation, D' (r); total correlation, D' (r) + D~ (r) and total radial distribution, 9' (r) = r [O' (r) + O~ (r)]. The absence of points for low Q,, reflects the fact that some peaks may not be observed at all under low resolution. Also shown in fig. 5b is the variation with Q,. of the position of the 1.55 A peak in D'(r) for the unmodified correlation function. This shows that,

147

DIFFRACTION STUDIES OF GLASS STRUCTURE. II1

despite having the highest resolution, the unmodified correlation function will not necessarily give the best values for the atom separations unless Q= is sufficiently large, again because of the effect of termination ripples from neighbouring peaks. These results show that for Q,, < 20 A - z the best values of atomic separations are obtained from the differential correlation function D (r) and that significant errors may be introduced by using the total radial distribution function g' (r).

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20

Fig. 5. Peak positions from different functions. (a) Variation with Qm of the positions of peaks at 1.550 and 3.100/~: (×) X-ray; (N) neutron; (O) D'(r); (©) D'(r)÷Do'(r); (El) g'(r) = r[D'(r) ÷ Do'(r)]. (b) Position of the 1.550 A peak in D'(r) as a function of Qm, for the unmodified correlation functions D'(r).

For a perfect total radial distribution function the area under the first peak is a direct measure of the coordination number, in this case of oxygen about silicon. In practice, however, the ideal peak shape will be modified by convolution with the function Pij (r), which is simply the cosine transform of the resolution function for neutrons, but for X-rays contains additionally the sharpening function and the form factors [see eqs. (2.30) and (2.31) of Part I]. It is important to know the effect of this convolution on the determination of coordination numbers. Fig. 6 shows the variation with Qm of the coordination number of O about Si, expressed as a percentage difference from the correct value predicted by the model parameters. For the X-ray

148

A. J. L E A D B E T T E R A N D A. C . W R I G H T

case the results were calculated using the K-approximation and since the values of 2( [eq. (2.4)] vary with Qm the coordination number will vary accordingly. Values were, therefore, also calculated for the model using form factors for which the K-approximation is correct (c.f. section 2). For the neutron case, the coordination number is accurate to better than 1~, for Qm~ 15 A-1, and the same is true for X-rays when the K-approximation is good. This will be the situation for heavy elements of closely similar atomic

:I

o o

In 0 0

[] o

~-0

tILl

!

10

, D

-2

o

ore/i-'

9 D []

D

-4

Fig. 6. Co-ordination number of O about Si as a function of Q,~: (0) neutron; (D) X-ray, normal form factors; ((3) X-ray, form factors obeying the K-approximation.

number e.g. As-Se systems. The inadequacy of the K-approximation in the general case will lead to errors in the coordination number [e.g. 11~ for GeO2 - see Part 114)] and for silica the values for higher terminations are consistently low by about 2~o. Thus, for accurate results it will be necessary either to calculate the actual peak shape functions Pij(r) or to compare experimental distribution functions with those obtained from the intensities for models (see Part II). It is worth noting that errors in experimental intensities often give structure in the correlation function below the first peak which may hinder the accurate determination of peak areas since this involves integration over satellite ripples arising from Pij (r). However, the use of a resolution function gives subsidiary features which damp down very quickly and so minimise this difficulty. In the present case, for example, a lower limit of integration at 1.0 A only introduces an error of 1~ into the area of the peak at 1.55/k (Qm=20/~-1).

149

DIFFRACTION STUDIES OF GLASS STRUCTURE. III

4. Quadrature Ino 9) has suggested that the quadrature interval required to obtain an accurate representation of the correlation function from a given intensity function should not exceed ~/rmax, where rmax is the value of r beyond which the correlation function, including effects due to termination, is effectively zero. For the present case rmax may be set equal to the correlation length, which means A Q ~ 0.4 A -1. Fig. 7 shows neutron correlation functions obtained with AQ values of 0.05, 0.4 and 0.8 A -1. Use of AQ = 0.4 A -1 results in a reduction in amplitude of structure at higher r and the introduction of spurious features at r greater than ~ / A Q . At A Q = 0 . 8 A -~ the correlation function has become severely distorted. A simple quantitative illustration of the above effects is provided by the height of the peak observed at r,-~ 5.2 A in both X-ray and neutron correlation functions, as a function of AQ: 0.05 485 0.945

AQIA -1

D x (5.2)/.~-2 DN (5.2)/barns A -2

0.1 484 0.944

0.2 476 0.928

0.4 383 0.783

2

A Q / ~ -1

o

0.05

--g

b

C

I

b VX kJ

,.4

~

O.B

v i

o

I

I

5

~/~

I

I

I

I

I

0

Fig. 7. Effect of quadrature interval on the neutron correlation function for Qm = 16 •-1.

150

A. J. LEADBETTER AND A. C. WRIGHT

There is already a reduction in the height of this peak by nearly 2% at AQ=0.2 A -1 and it is 15-20% low at AQ=0.4 A -1 so that Ino's criterion appears to be too optimistic. 5. NormaHsation of intensity data The experimental intensity data throughout this work have been normalised using the integration method4). The accuracy of this procedure has been investigated as a function of Qm, using the model intensities for which the scaling factor a is unity and the incoherent intensity is zero. For Qm> 13 A - 1, is always within 0.1% of unity, while for 8 ( Q / A - l ( 1 3 it differs from unity by less than 0.5% and at lower Q still, errors in a of several percent may be obtained. If the intensity is internally inconsistent, or if a wrong value for pO is used, a systematic variation of scaling factor with Qm will be obtained. It is worth noting that this normalisation method is equivalent to making the pij(r) functions go to zero at r--0. Any error in an experimentally measured intensity will in general affect i(Q) both additively and multiplicatively, as can be seen by considering the special case of an error in the scaling factor for a model intensity [cf. Bienenstockl0)]. If the original model intensity is multiplied by a factor (l + e) the resulting correlation function D"(r) will be related to the original function D' (r) by Qm

D" (r) = (l + .) D' (r) + --

Of~2 (Q) M (Q) sin Qr dQ i

(5.1)

0

= (1 + e) D'(r) + cAD'(r).

(5.2)

Neutron correlation functions are shown in fig. 8a for scaling factors of 1.00, 1.01 and 1.05. The contribution from the additive term cAD' (r) which has the form of a rapidly damping oscillation (cf. ref. 2) is much more prominent than the change in the magnitude of D' (r) and it is the former which may reveal the presence of a scaling factor error in the experimental situation. Lorch 7) has used this contribution to determine the optimum scaling factor for his experimental neutron data by plotting the magnitude of the first ripple in D N(r) as a function of ~ and choosing the value giving zero ripple. This is very effective, provided that no other distortions are present at low r (e.g. due to static approximation effects or experimental error), since the magnitude of the ripple obtained is proportional to e [eq. (5.2)]. This is illustrated in figure 8b for both X-rays and neutrons. It is also noteworthy that the change in the area under the first peak in the total radial

DIFFRACTION

STUDIES OF GLASS STRUCTURE.

151

III

(a) 2

0< t_

#o

I

10

t. v

z~3 - ,

t-.,i

(b)

~

1

400

f,_ m

O

..e.eee e'~"

g

*~ t~ t.I

-400 x Z I

I

I

0.96

I

1.00

I

1.04

ct

Fig. 8. (a) Effect of scaling factor errors o n the n e u t r o n correlation function: ( - ) a = 1.00 (correct); ( - - - ) ct = 1.01; ( . . . . . . ) ct = 1.05. (b) A m p l i t u d e of first ripple in D'(r) as a function of scaling factor (a).

distribution function, determined with a lower integration limit of 0.5 A, is only 0.270 for a scaling factor error of 570. 6. Errors in intensities Errors in experimentally measured intensities may formally be described by writing Qi'(Q) = Qi(Q) [1 + e(Q)] + QE(Q), (6.1) where E(Q) is some additive error, e is now Q-dependent and the simple multiplication of D' (r) by (1 + e) in eq. (5.1) is replaced by a convolution, giving 09

O" (r) = .t" O' (u) P' (r - u) du + AD' (r),

(6.2)

-oo

where the peak function is the cosine transform of [1 + e (Q)]. For most errors, particularly those which vary only slowly with Q, it will again be AD'(r) which is most apparent in the experimental correlation

152

A. J. L E A D t l E T T E R A N D A. C . W R I G H T

function, appearing as structure below the first real peak in D' (r). The size, shape and period of the structure at low r will, of course, depend on the nature of E(Q) but for errors which are slowly varying functions of Q this structure will be qualitatively similar to that shown in fig. 8. Also, the normalisation procedure will tend to reduce the effect of any error in the intensity. An interesting example of this type of error is provided by an estimate of the effects of inadequacy in the static approximation on the neutron scattering results for vitreous BeF z 11). A contribution of the form ~,,i [~2(BiQ2 _ A i) was added to a model intensity, with values of the constants such that the correction amounted to about 107o of I(Q) at Q = Q,, _ 13 A - 1. The best scaling factor was found for the perturbed data and the resulting net effect on DN(r) was broadly similar to that obtained for a simple scaling factor error of one or two percent (cf. fig. 8). This is discussed more fully in Part IVll). The simplest error to treat is one which occurs at a single point (Q') in the intensity. Provided that the error is not large enough to affect significantly the scaling factor, e (Q) is zero for all Q and E(Q) becomes a 6-function, and gives AD' (r) = const, x sin Q'r,

(6.3)

which is a ripple, of frequency 2~/Q', and constant amplitude. An example of such an error, with the point in IM (Q) at 10.0 A-1 set to zero, is shown in fig. 9. Statistical errors are always present in experimental intensity data and, although these are usually reduced by smoothing procedures, it is worthwhile investigating their effect on D' (r). This is simply a sum of terms of the type given by eq. (6.3), one for each point used in the quadrature, and the result is a noise waveform added to the correlation function. Results for a gaussian error distribution with root mean square deviations of 1 and 4 ~ are shown for the neutron correlation function in fig. 9. Analogous results are, of course, obtained for the X-ray case. Errors may also arise because it is experimentally not possible to collect data below a certain minimum value of Q, which for both X-ray and neutron experiments is typically less than 1 A -1. Distortions in I(Q) for Q < 1 A - i will in general lead to low amplitude, long wavelength ripples in D'(r). This is illustrated in fig. 9 which shows the neutron correlation function obtained by fourier transforming the intensity with Qr, = 1 A - i and a resolution function appropriate to Qm= 17 A -~. This represents the total information obtained from the intensity at Q < 1 A - ~ and any errors arising from faulty extrapolation to zero Q would be much less than this, showing that the details of the extrapolation are not important. For X-rays I(0) is usually taken to be zero but for neutrons this is not the case due to incoherent and

DIFFRACTION

STUDIES OF GLASS STRUCTURE.

153

In

2

1

0

0

c_

.~ 0

%

0

~

I

0

5

i

i

J

10

Fig. 9. The effect of statistical errors on the neutron correlation function. (1) No error. (2) Single error at Q = 10 A-1. (3) Gaussian error distribution with a r.m.s, deviation (a) of 1 ~. (4) Gaussian errors with tr = 4 ~. (5) Transform of Qi(Q)for Q < I/~-1.

multiple scattering. The latter is usually assumed to be isotropicl~), so that I ( 0 ) also represents a fiat b a c k g r o u n d to be subtracted f r o m the measured intensity. Thus, in addition to extrapolation errors analogous to those for the X-ray case, I ( 0 ) itself m a y be in error for neutron scattering. To a g o o d approximation this will only affect the scale o f i(Q) and hence that o f the resulting correlation function, and will give errors in co-ordination numbers but not interatomic distances. F o r X-ray experiments, inaccuracies can arise from errors in the tabulated

154

A. J. LEADBETTER

AND

A. C. WRIGHT

form factors and Compton scattering intensities, since these are free atom or free ion values neither of which are strictly appropriate for a glass, particularly for very light atoms such as Be where 50% of the electrons are involved in bonding. Errors in Compton-modified intensities will result in an additive component plus a scaling factor error e, independent of Q. F o r m factor deviations are rather complex, and in addition to an additive part due to wrong independent scattering, there will be a Q-dependent multiplicative error which will, in general, be different for each component of i(Q). The transform of the additive part giving ADX(r), will again be much more obvious in the correlation function than distortions due to the multiplicative error. This is illustrated in fig. 10 where the results are shown of analysing the intensity for a model computed with free atom form factors, using form factors for the free ions Si 4+ and O 2-, and vice versa. The analysis is not sensitive to the ionicity of the form factors used; distortions

1

8O0

400

0

1

2

3

o\

4

-400 i

o

I

,

~

i

i

I

5

10

ri/~ Fig. 10. The effect of errors in X-ray form factors

(Qm=

17 A -1)

Curve

1

2

3

4

Form factors used for model intensity FormfactorsusedfortransformtoDX (r)

Si, O Si, O

Si4+, 02Sir O

Si, O Si4+, 02-

Si4+, 02Si4÷ 02-

DIFFRACTIONSTUDIESOF GLASSSTRUCTURE.lI1

155

o f D X ( r ) are very small except below the first peak, where they are clearly apparent and therefore not important. In conclusion, the important result for the treatment o f experimental data is that errors o f various kinds will c o m m o n l y reveal themselves by giving rise to structure in the correlation functions at low r, below the first true peak. This m a y even allow the nature o f the error to be recognised and corrections to be made. A c c o m p a n y i n g distortions o f the correlation function will not generally be directly recognisable in the experimental situation, but these generally appear to be small for quite large ripples at low r.

Acknowledgment We are grateful to the Science Research Council for the award o f a maintenance grant to A.C.W.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

J. Waser and V. Schomaker, Rev. Mod. Phys. 25 (1953) 671. R. Kaplow, S. L. Strong and B. L. Averbach, Phys. Rev. 138A (1965) 1336. A. J. Leadbetter and A C. Wright, J. Non-Crystalline Solids 7 (1972) 23. A. J. Leadbetter and A. C. Wright, J. Non-Crystalline Solids 7 (1972) 37. R. W. G. Wyckoff, Crystal Structures, Vol. l, 2nd ed. (Interscience, New York, 1963). L. N. G. Filon, Proc. Roy. Soc. (Edinburgh) 49 (1929) 38. E. A. Lorch, J. Phys. C 2 (1969) 229. J. Zarzycki, in: Travaux du IVe Congrds International du Verre, Paris, (1956) p. 323. T. Ino, J. Phys. Soc. Japan 12 (1957) 495. A. Bienenstock, J. Chem. Phys. 31 0959) 570. A. J. Leadbetter and A. C. Wright, J. Non-Crystalline Solids 7 (1972) 156. J. E. Enderby, in: Physics of Simple Liquids, Eds. H. N. V. Temperley, J. S. Rowlinson and G. S. Rushbrooke (North-Holland, Amsterdam, 1968) ch. 14.