langford ph880173

Aust. J. Phys., 1988,41,113-81

Profile Analysis for Microcrystalline Properties by the Fourier and Other Methods

J. L Langford,A R. Delhez,B Th.H. de KeijserB and E. J. MittemeijerB

A Department of Physics, University of Birmingham, Birmingham B15 2IT, U.K. B Laboratory of Metallurgy, Delft University of Technology, 2628 AL Delft, The Netherlands.

Abstract In the 1960s the Fourier and variance methods superseded the use of the FWHM and integral breadth in detailed studies of microcrystalline properties. Provided that due allowance is made in the analysis for systematic errors, particularly the effects of truncation of diffraction line profiles at a finite range, these remain the best methods for characterising crystallite size and shape, microstrains and other imperfections in cases where accuracy is important. However, the application of the Fourier, variance and related methods in general requires that the diffraction lines are well resolved and it is thus restricted to materials with high symmetry or which exhibit a high degree of preferred orientation. Most materials, on the other hand, including many of technological importance, have complex patterns with severe overlapping of peaks. The introduction of pattern-decomposition methods, whereby a suitable model is fitted to the total diffraction pattern to give profile parameters for individual lines, means that microcrystalline properties can now be studied for any crystalline material or mixture of substances. The use of the FWHM and integral breadth has been given a new lease of life; though the information is less detailed than is given by the Fourier and variance methods and systematic errors are in general greater, self-consistent estimates of crystallite size and microstrains are obtained.

At present the most promising technique for analysing the breadths obtained from pattern decomposition is the Voigt method applied to all lines of a diffraction pattern. When reliable data for two or more orders of reflections are available, a multiple-line analysis can be used to separate the contributions to line breadths from crystallite size and strain. For other reflections a single-line approach is used, though this can introduce an angle-dependent systematic error.

1. Introduction: A Historical Overview The use of diffraction line profiles to study microcrystalline properties is almost

as old as powder diffraction itself. As long ago as 1918 Scherrer noted that the line breadth varied inversely as the size E of crystallites in the sample, leading to the Scherrer equation

/3 = AlE cos 8 for breadths on a 28 scale (la)

or /3 = liE for breadths in reciprocal units, (lb)

where A is the wavelength, 8 is the Bragg angle and /3 is some measure of line Paper presented at the International Symposium on X-ray Powder Diffractometry, held at Fremantle, Australia, 20-23 August 1981.

0004-9506/88/020113$03.00

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b r e a d t h . I n ( 1 ) E i s a n a p p a r e n t s i z e w h i c h d e p e n d s o n t h e m e a s u r e o f b r e a d t h u s e d ,

t h e s h a p e o f t h e c r y s t a l l i t e s a n d t h e d i r e c t i o n o f t h e h k l p l a n e s , a n d i t i s r e l a t e d

t o t h e t r u e s i z e t h r o u g h t h e S c h e r r e r c o n s t a n t . F r o m ( 1 ) i t i s e v i d e n t t h a t s i z e

b r o a d e n i n g i s i n d e p e n d e n t o f t h e o r d e r o f a r e f l e c t i o n . T h e b r o a d e n i n g w h i c h a r i s e s

f r o m m i c r o s t r a i n s w a s r e p o r t e d b y V a n A r k e l i n 1 9 2 5 , b u t i t s c a u s e w a s t h e s u b j e c t

o f c o n t r o v e r s y f o r m a n y y e a r s a n d t h e d e s c r i p t i o n o f l i n e - p r o f i l e s h a p e f o r a s p e c i m e n

c o n t a i n i n g a d i s t r i b u t i o n o f m i c r o s t r a i n s i s s t i l l a m a t t e r o f d e b a t e ( D e l h e z e t a l . 1 9 8 2 ;

D e l h e z e t a l . 1 9 8 8 , p r e s e n t i s s u e p . 2 1 3 ) . V a r i o u s e a r l y w o r k e r s d e d u c e d t h e a n g u l a r

d e p e n d e n c e o f s t r a i n b r o a d e n i n g a n d t h e f o r m s n o r m a l l y u s e d a r e a s f o l l o w s :

( i ) T h e m i c r o s t r a i n e i s c o n c e i v e d b y c o n s i d e r i n g t w o e x t r e m e v a l u e s o f t h e

l a t t i c e s p a c i n g d , n a m e l y d + l l . d a n d d - I l . d , w i t h e = I l . d / d . B y e q u a t i n g t h e

i n t e g r a l b r e a d t h { 3 w i t h t h e a n g u l a r r a n g e c o r r e s p o n d i n g t o d + I l . d a n d d - I l . d ,

a n d b y a s s u m i n g t h a t B r a g g ' s l a w h o l d s o v e r t h i s r a n g e ( i m p l y i n g t h a t p a r t s o f t h e

s p e c i m e n w i t h s p a c i n g s d - I l . d a n d d + l l . d d i f f r a c t i n d e p e n d e n t l y , i . e ; i n c o h e r e n t l y ) ,

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i n r e c i p r o c a l u n i t s ( 1 / d s c a l e ) .

( 2 b )

( i i ) T h e c o m p o n e n t o f s t r a i n e ( n ) d e n o t e s t h e a v e r a g e s t r a i n b e t w e e n t w o u n i t

c e l l s , n c e l l s a p a r t , i n a c o l u m n o f c e l l s p e r p e n d i c u l a r t o t h e d i f f r a c t i n g p l a n e s . F o r a

G a u s s i a n d i s t r i b u t i o n o f e ( n ) , i n d e p e n d e n t o f t h e s e p a r a t i o n o f c e l l s n [ i m p l y i n g t h a t

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r e g i o n w h i c h i n c l u d e s t h e v a r i a t i o n i n l a t t i c e s p a c i n g i s n o w c o n s i d e r e d a s d i f f r a c t i n g

c o h e r e n t l y a n d ( S t o k e s a n d W i l s o n 1 9 4 4 )

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( 3 = 2 ( 2 1 7 ) i ( e 2 ) i / 2 d = 5 ( e 2 ) i / 2 d i n r e c i p r o c a l u n i t s . ( 2 d )

E q u a t i o n s ( 2 ) i n d i c a t e t h a t s t r a i n b r o a d e n i n g d e p e n d s o n t h e o r d e r o f a r e f l e c t i o n .

I n 1 9 4 9 H a l l p r o p o s e d a m e t h o d f o r s e p a r a t i n g s i z e a n d s t r a i n e f f e c t s b y p l o t t i n g

t h e b r e a d t h s o f r e c i p r o c a l l a t t i c e p o i n t s a g a i n s t t h e i r d i s t a n c e f r o m t h e o r i g i n . T h e

i n t e r c e p t i s t h e n 1 / E a n d t h e s l o p e i s p r o p o r t i o n a l t o t h e s t r a i n . H e f u r t h e r d e v e l o p e d

t h i s a p p r o a c h w i t h W i l l i a m s o n t o f o r m t h e b a s i s o f t h e W i l l i a m s o n - H a l l ( 1 9 5 3 ) p l o t .

E x p e r i m e n t a l h l i n e p r o f i l e s a r e t h e c o n v o l u t i o n o f t h e s p e c i m e n f a n d i n s t r u m e n t a l

9 c o n t r i b u t i o n s , a n d t h e f p r o f i l e s i n t u m c a n b e t h e c o n v o l u t i o n o f s e v e r a l f u n c t i o n s

w h i c h a r e g r o u p e d t o g e t h e r a s t h e o r d e r - i n d e p e n d e n t ( ' s i z e ' ) a n d o r d e r - d e p e n d e n t

( ' s t r a i n ' ) c o n t r i b u t i o n s , o r

h ( x ) = f s ( x ) * f D ( x ) * g ( x ) ,

( 3 )

w h e r e S a n d D d e n o t e ' s i z e ' a n d ' s t r a i n ' . F o r e a s e o f d e c o n v o l u t i o n o f t h e v a r i o u s

c o m p o n e n t s , e a r l y w o r k o n l i n e - b r o a d e n i n g a n a l y s i s w a s b a s e d o n t h e a s s u m p t i o n

t h a t t h e c o n s t i t u e n t l i n e p r o f i l e s a r e e i t h e r C a u c h y ( L o r e n t z i a n ) o r G a u s s i a n . I t

w a s e v i d e n t t h a t n e i t h e r f u n c t i o n a c c u r a t e l y m o d e l s e x p e r i m e n t a l p r o f i l e s a n d t h a t a

Profile Analysis for Microcrystalline Properties 175

more rigorous approach would be to use the multiplicative property of the Fourier transforms of convoluted functions, or

H(t) = F(t) G(t), (4)

where F, G, H are Fourier transforms of /, g, h. This milestone in the development of line-broadening analysis led to the Stokes (1948) correction for instrumental broadening and the Warren-Averbach (1949, 1950) method for the separation of size and strain effects by using the Fourier coefficients of the pure diffraction profile. An alternative procedure to overcome the problems associated with deconvolution, suggested by Tournarie (1956) and developed by Wilson (1962) and bngford (1965), is to use the additive property of the variances (second central moments) of convoluted functions.

By the late 19608 the use of line breadths had largely been discarded in favour of the more powerful Fourier (Warren-Averbach), variance and related methods. Provided that the data are of high quality and due allowance is made for systematic errors (Section 2), these methods give a wealth of detailed and accurate information about microcrystalline properties. Size broadening gives the mean dimension of crystallites or domains in various directions and hence provides information about their shape, and the distribution of size can be obtained. The 'size' contribution can also be interpreted in terms of various types of mistake--deformation and twin faults in metals, arising from a misplacement of successive layers of atoms, mistakes due to the more or less random rotation of layers in turbostratic structures, and due to a change in the interlayer spacing at random intervals-and information on the nature and density of dislocations can be extracted. The variance method gives the r.m.s. strain (cf. equation 2) and details of the variation of strain within crystallites or domains can be found from the Fourier coefficients. A comprehensive survey of the procedures for applying both methods and the interpretation of the results has been carried out by Klug and Alexander (1974, ch. 9; see also Warren 1969). Of the two methods, the Fourier approach is the more rigorous, since no assumptions are made about the nature of the intensity distribution when correcting for instrumental effects. In practice the distinction may be unimportant, due to the presence of systematic errors (Section 2), and in any event the two methods can be regarded as complimentary. The application of the Fourier and variance methods in general requires that the diffraction lines are well resolved and is thus restricted to materials with high symmetry or which exhibit a high degree of preferred orientation. Most materials, on the other hand, including many of technological importance, such as ceramics, some catalysts and polymers, have complex patterns with severe overlapping of peaks. Significant advances during the last decade or so in pattern-decomposition methods, whereby a suitable model is fitted to the total diffraction pattern to give the characteristics of individual lines, means that a study of any crystalline material or mixture of substances is now feasible. Analyses are based on the FWHM and integral breadth of each line and, with modifications, the methods of early workers, based on assumed line shapes, have been given a new lease of life (Section 3). The Fourier and variance methods remain the best technique in cases where they are applicable, but otherwise line-broadening analysis by means of total pattern fitting can give valuable, if less accurate, information on crystal imperfections.

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F i g . 1 . C o m p a r i s o n b e t w e e n e s t i m a t e s o f t h e a p p a r e n t c r y s t a l l i t e s i z e

d e t e r m i n e d b y t h e F o u r i e r ( W a r r e n - A v e r b a c h ) a n a l y s i s ( D ) a n d t h e

v a r i a n c e m e t h o d ( E k ) : c i r c l e s , L a n g f o r d ( 1 9 6 5 , 1 9 6 8 ) , L o u e r e t a l .

( 1 9 7 2 ) ; t r i a n g l e s , H a l d e r a n d W a g n e r ( 1 9 6 6 ) ; c r o s s e s , G u i l l a t t a n d

B r e t t ( 1 9 7 0 , 1 9 7 1 ) ; s q u a r e s , M i g n o t a n d R o n d o t ( 1 9 7 3 ) ; p l u s e s , N i e p c e

e t a l . ( 1 9 7 8 ) ; a n d s t a r s , L e B a i l a n d L o u e r ( 1 9 8 0 ) .

2 . S o u r c e s o f E r r o r i n L i n e P r o f i l e A n a l y s i s

A s n o t e d a b o v e , t h e F o u r i e r a n d v a r i a n c e m e t h o d s f o r l i n e - b r o a d e n i n g a n a l y s i s o n l y

g i v e r e l i a b l e a n d a c c u r a t e i n f o r m a t i o n a b o u t s t r u c t u r a l i m p e r f e c t i o n s i f t h e e f f e c t s o f

r a n d o m a n d s y s t e m a t i c e r r o r s a r e t a k e n i n t o a c c o u n t ; e i t h e r t h e y m u s t b e m i n i m i s e d

o r a c o r r e c t i o n m u s t b e m a d e . T h e r e i s e v i d e n c e t o s u g g e s t t h a t t h i s h a s n o t a l w a y s

b e e n d o n e i n t h e p a s t . F o r e x a m p l e , i n t h e o r y b o t h m e t h o d s g i v e t h e s a m e m e a s u r e o f

s i z e , n a m e l y t h e a r e a - w e i g h t e d m e a n t h i c k n e s s o f c r y s t a l l i t e s o r d o m a i n s m e a s u r e d i n

a d i r e c t i o n p e r p e n d i c u l a r t o t h e d i f f r a c t i n g p l a n e s ( W i l s o n 1 9 6 2 ) , b u t L a n g f o r d ( 1 9 6 8 )

n o t e d a s i g n i f i c a n t d i s c r e p a n c y b e t w e e n t h e s i z e E k g i v e n b y t h e v a r i a n c e a n d t h a t

o b t a i n e d f r o m t h e F o u r i e r c o s i n e c o e f f i c i e n t s ( D ) f o r n i c k e l p o w d e r . T h e d i f f e r e n c e

c a n o n l y a r i s e i n t h e a n a l y s i s o f t h e e x p e r i m e n t a l d a t a a n d L e B a i l ( 1 9 7 6 ) m a d e a

s u r v e y o f v a l u e s o f E k a n d D r e p o r t e d i n t h e l i t e r a t u r e b y s e v e r a l a u t h o r s a n d f o r

a w i d e r a n g e o f s a m p l e s . H i s f i n d i n g s a r e s u m m a r i s e d i n F i g . 1 , w h e r e i t c a n b e

s e e n t h a t t h e ' F o u r i e r s i z e ' i s a l w a y s g r e a t e r t h a n t h e ' v a r i a n c e s i z e ' , s o m e t i m e s b y

a f a c t o r o f t w o o r m o r e . I t i s n o w k n o w n t h a t t h i s d i s c r e p a n c y i s l a r g e l y d u e t o

t r u n c a t i o n o f l i n e p r o f i l e s a t f i n i t e r a n g e a n d t h a t i t d e p e n d s c r i t i c a l l y o n t h e m e t h o d

u s e d f o r d e t e r m i n i n g t h e i n i t i a l s l o p e o f t h e A ( n ) v e r s u s n c u r v e ( D e l h e z e t a f . 1 9 8 6 ) .


The effect of truncation is to reduce the estimate of E k' the fractional decrease being approximately equal to the fraction of intensity 'lost' by truncation, typically about 5% (Langford 1982). The size D is increased by a somewhat greater amount and there may also be an appreciable error in the strain value (Delhez et af. 1988). (a) Errors in the Fourier Method

The principal sources of error in the FourierlWarren-Averbach analysis are: (a) counting statistics; (b) standard used to obtain 9 profiles; (c) background determination; (d) truncation of profiles at finite range; (e) sampling interval (step length); (t) choice of origin; and (g) limitations of approximations used in analysis.

These affect the analysis in different ways and by differing amounts. Young et af. (1967) simulated the effects of (c), (d) and (e) and many authors have since discussed the treatment of errors in the Fourier method. For example, Delhez et af. (1980, 1982) have given corrections for (a), (b) and (t). Procedures for improving the reliability of line profile analysis by the Fourier method are presented by Zorn (1988, present issue p. 237) and Delhez et af. (1988, p. 213). (b) Errors in the Variance Method

Many of the sources of error listed above apply equally to the variance approach, except that the origin is always taken as the centroid of the peak and any error in its position has negligible effect, and the choice of step length is less critical. The method is based on the assumption that the intensity falls to zero as the inverse square of the range in the profile tails. The variance W then varies linearly with the range of integration 0" (Wilson 1962), or

W = Wo+kO". (5)

The level and slope of the background subtracted from the peak are adjusted until an optimum fit of (5) is obtained (Langford and Wilson 1963). These parameters define what may be regarded as the 'true' background under the peak for the purposes of line-broadening analysis, and they can be used to advantage for making a truncation correction in the Fourier method.

In order to correct for the unavoidable truncation of a profile at finite range, some function must be used to allow for the missing 'tails'. In practice the same inverse-square variation as leads to (5) is used and the corrections to be subtracted from Wo and k are then Wo k/O" max and k 2 /0" max respectively. Also, the total integrated intensity S>, equivalent to the cosine coefficient A(O) multiplied by the range in the Fourier approach, is given by

S> Suma/(I-k/O"max), (6)

where S(7" is the integrated intensity of the profile at maximum range. It should be noted that in theory the intensity never falls to zero, but is a minimum half-way

1 7 8


b e t w e e n s u c c e s s i v e r e f l e c t i o n s ( D e l h e z e t a l . 1 9 8 6 ) . H o w e v e r , i n p r a c t i c e t h e a b o v e

a s s u m p t i o n i s a d e q u a t e f o r t h e p u r p o s e o f m a k i n g a c o r r e c t i o n f o r t r u n c a t i o n a n d i n

a n y e v e n t f o r a p o w d e r s a m p l e o t h e r p e a k s w i l l o c c u r b e t w e e n s u c c e s s i v e o r d e r s . A n

a n a l o g o u s a s s u m p t i o n i s m a d e w h e n c o r r e c t i n g f o r t r u n c a t i o n i n t h e F o u r i e r m e t h o d

( D e l h e z e t a l . 1 9 8 6 ) . A c o r r e c t i o n p r o c e d u r e i s c u r r e n t l y b e i n g d e v e l o p e d a n d i t

r e m a i n s t o b e s e e n h o w t h i s w i l l a f f e c t t h e d e t a i l e d i n f o r m a t i o n o n s t r a i n w h i c h c a n

b e e x t r a c t e d b y t h i s m e t h o d .

T h e r e a r e t w o o t h e r c o r r e c t i o n s , f o r n o n - a d d i t i v i t y a n d c u r v a t u r e , w h i c h d o n o t

o c c u r i n t h e F o u r i e r a p p r o a c h . W h i l e t h e a d d i t i v e p r o p e r t y o f v a r i a n c e s h o l d s i f t h e

c o n s t i t u e n t p r o f i l e s e x t e n d t o i n f i n i t y , s o m e i n s t r u m e n t a l f u n c t i o n s f a l l t o z e r o a t a

f i n i t e r a n g e a n d a s m a l l n o n - a d d i t i v i t y c o r r e c t i o n i s r e q u i r e d . A l s o , t h e r e i s a s l i g h t

r e s i d u a l c u r v a t u r e o f t h e v a r i a n c e - r a n g e f u n c t i o n ( 5 ) , f o r w h i c h a c o r r e c t i o n i s a g a i n

m a d e . S y s t e m a t i c e r r o r s i n t h e v a r i a n c e m e t h o d h a v e b e e n s u m m a r i s e d b y L a n g f o r d

( 1 9 8 2 ) .

3 . L i n e P r o f i l e A n a l y s i s f r o m T o t a l P a t t e r n F i t t i n g

T h e r e s o l u t i o n o f a p o w d e r p a t t e r n i n t o i t s c o n s t i t u e n t B r a g g r e f l e c t i o n s i s c a r r i e d

o u t i n t w o s t a g e s . F i r s t a p e a k s e a r c h i s c a r r i e d o u t t o a s c e r t a i n t h e p o s i t i o n s o f a l l

l i n e s o f t h e p a t t e r n , e l i m i n a t i n g ' f a l s e ' p e a k s d u e t o s t a t i s t i c a l n o i s e ( H u a n g 1 9 8 8 ,

p r e s e n t i s s u e p . 2 0 1 ) , a n d t h e n t h e p a r a m e t e r s f o r i n d i v i d u a l p r o f i l e s a r e o b t a i n e d

a n d r e f i n e d . V a r i o u s m o d e l s h a v e b e e n u s e d f o r t h e l a t t e r . F o r e x a m p l e , t h e

c o n v o l u t i o n o f s e v e r a l C a u c h y f u n c t i o n s w a s p r e f e r r e d b y P a r r i s h e t a l . ( 1 9 7 6 ) , t o g i v e

d e c o n v o l u t e d 1 p r o f i l e s . L a n g f o r d e t a l . ( 1 9 8 6 ) u s e d t h e p s e u d o - V o i g t o r P e a r s o n V I I

f u n c t i o n s , w i t h a l l o w a n c e f o r a s y m m e t r y o f t h e e x p e r i m e n t a l p r o f i l e s . A p r o m i s i n g

a p p r o a c h , i n t e n d e d f o r R i e t v e l d r e f i n e m e n t b u t e q u a l l y a p p l i c a b l e t o l i n e - b r o a d e n i n g

a n a l y s i s , i s t h e n o n - a n a l y t i c a l ' l e a r n e d p e a k - s h a p e f u n c t i o n ' i n t r o d u c e d b y H e p p a n d

B a e r l o c h e r ( 1 9 8 8 , p r e s e n t i s s u e p . 2 2 9 ) , i n w h i c h a n o p t i m u m f i t i s a c h i e v e d . L a n g f o r d

( 1 9 8 7 ) h a s r e v i e w e d p a t t e r n - d e c o m p o s i t i o n m e t h o d s a n d o t h e r p r o c e d u r e s f o r t o t a l

p a t t e r n f i t t i n g . T h e a i m o f p a t t e r n d e c o m p o s i t i o n i n t h i s c o n t e x t i s t o o b t a i n r e l i a b l e

e s t i m a t e s o f l i n e - p r o f i l e p a r a m e t e r s , p a r t i c u l a r l y t h e p o s i t i o n , h e i g h t , a r e a a n d b r e a d t h

( F W H M ) o f e a c h p e a k , f o r u s e i n a n a n a l y ' s i s o f s t r u c t u r a l i m p e r f e c t i o n s . T h e r e a r e

c u r r e n t l y t w o w a y s i n w h i c h t h i s c a n b e c a r r i e d o u t ; i n t h e f i r s t i t i s a s s u m e d t h a t t h e

l i n e p r o f i l e s a p p r o x i m a t e t o V o i g t c u r v e s , t h e c o n v o l u t i o n o f C a u c h y a n d G a u s s i a n

f u n c t i o n s ( L a n g f o r d 1 9 7 8 ) a n d i n t h e s e c o n d a p p r o a c h a d i r e c t a n a l y s i s o f t h e b r e a d t h

p a r a m e t e r s , g i v e n b y p a t t e r n d e c o m p o s i t i o n , i s c a r r i e d o u t ( K e i j s e r e t a l . 1 9 8 3 ) .

( a ) V o i g t A n a l y s i s 0 1 L i n e B r e a d t h s

A f t e r f i n d i n g t h e b r e a d t h s o f i n d i v i d u a l l i n e s , c o r r e c t i o n m u s t b e m a d e f o r

i n s t r u m e n t a l e f f e c t s , i f t h i s i s n o t i n c l u d e d i n t h e p a t t e r n d e c o m p o s i t i o n , a n d

t h e c o r r e c t e d b r e a d t h s a r e t h e n s e p a r a t e d i n t o o r d e r - i n d e p e n d e n t ( ' s i z e ' ) a n d o r d e r -

d e p e n d e n t ( ' s t r a i n ' ) c o m p o n e n t s . T h e r e a r e t h u s t w o s t a g e s o f d e c o n v o l u t i o n t o b e

c a r r i e d o u t . N o w i n s t r u m e n t a l p r o f i l e s h a v e C a u c h y - l i k e ( i n v e r s e - s q u a r e ) t a i l s , i f t h e

d a t a a r e o b t a i n e d w i t h a c o n v e n t i o n a l a n g l e - d i s p e r s i v e X - r a y d i f f r a c t o m e t e r , a n d t h e g

p r o f i l e s w i l l t e n d t o b e G a u s s i a n i f o b t a i n e d f r o m s y n c h r o t r o n o r c o n t i n u o u s - n e u t r o n

s o u r c e s . T h e t a i l s o f t h e l i n e p r o f i l e d u e t o s m a l l c r y s t a l l i t e s a r e d o m i n a t e d b y a n

i n v e r s e - s q u a r e t e r m , t h o u g h t h e p r e c i s e n a t u r e o f t h e p r o f i l e d e p e n d s o n t h e s h a p e o f

t h e c r y s t a l l i t e s a n d t h e d i s t r i b u t i o n o f s i z e , a n d i n g e n e r a l s t r a i n p r o f i l e s t e n d t o b e


Gaussian. There is thus both theoretical and experimental evidence to support the use of the Voigt function for the purpose of deconvolution in line-profile analysis. In the following it is assumed that the equivalent Voigt function has the same peak height, area and FWHM as the experimental line profile.

The Voigt function can be expressed as (Langford 1978, equation 22)

I(x) = Re[.8c/dO)(r){1T!x/Po+ikl1, (7)

where Pc and Po are the integral breadths of the Cauchy and Gaussian components, IdO) and 10(0) are their peak heights, k = PC/1Tl/2pO and (r){ zJ is the complex error function. If it is assumed that the experimental line profiles can be represented by a Voigt curve, then their constituent Cauchy and Gaussian components can be found and deconvoluted in the usual way (equations 16 and 17). In general the Voigt model fits experimental data well at intermediate and high angles and is usually adequate at low angles, provided that the lines are reasonably symmetrical (Langford 1987). [The error introduced by asymmetry has been considered by Keijser et al. (1982).]

Before embarking on a Voigt analysis of the data, the Voigt parameter , the ratio of the FWHM to the integral breadth, is obtained for each line in the g and h patterns. (If Ka radiation is used, then the Ka2 component must be removed before obtaining the breadths.) The Voigtian can only be considered as a suitable model if

I 2/1T (=06366) .;; .;; 2(ln 2/1T)~ (=0.9394). (8) (Cauchy limit) (Gaussian limit)

Values of slightly less than 2/1T can occur with high-resolution instrumental functions, which are usually assumed to be Cauchy. Data with > 09394 are suspect.

Cauchy and Gaussian Components: The Cauchy and Gaussian integral breadths are obtained from the experimental values of P and for the g and h profiles. Fractional components Pc/P can be found by interpolation from Table 1 or graphically from Fig. 2. Alternatively, approximate values of the constituent breadths, accurate to within 1 %, are more conveniently given by (Keijser et al. 1982, equations 3-5)

Pc = P(2.0207-0.4803-1.77562), (9)

Po = P{0.64420+ 1.4187(-2/1T)! -2.2043+ 1.87062J . (10)

Voigt Parameters from Pc and Po: If Pc and Po are known, the corresponding Voigt function, aside from a scale factor which depends on the peak heights of the component functions, can be obtained from (7). The integral breadth is given by (Langford 1978, equation 25)

P = Po exp(-~)/{l-erf(k)J (11)

1 8 0


T a b l e 1 .

F r a c t i o n a l C a u c h y a n d G a u s s i a n c o m p o n e n t s o f t h e i n t e g r a l b r e a d t h o f a V o i g t

f u n c t i o n f o r v a l u e s o f t h e V o i g t p a r a m e t e r ~ i n t h e r a n g e 2 / 1 r . ; ; ~.;; 2 ( 1 n 2 / 1 r ) 1 ! 2

~

f 3 c 1 f 3

f 3 0 1 f 3

~

f 3 c 1 f 3

f 3 0 1 f 3

0 6 4 6 4

0 9 6 9 8 0 1 4 0 3 0 6 7 0 9

0 8 9 7 5

0 2 6 6 5

0 6 4 6 9

0 9 6 8 5 0 1 4 3 8

0 6 7 4 0

0 8 8 8 8 0 2 7 8 6

0 6 4 7 4

0 9 6 6 7 0 1 4 7 4

0 6 7 7 4

0 8 7 8 9 0 2 9 1 7

0 6 4 8 0

0 9 6 5 4

0 1 5 1 3

0 6 8 1 2

0 8 6 7 8 0 3 0 6 0

0 6 4 8 6

0 9 6 3 4 0 1 5 5 3 0 6 8 5 4

0 8 5 5 0

0 3 2 1 6

0 6 4 9 2

0 9 6 1 2

0 1 5 9 5

0 6 9 0 3

0 8 4 0 5 0 3 3 8 7

0 6 4 9 9

0 9 5 9 3 0 1 6 4 0 0 6 9 6 1

0 8 2 4 0 0 3 5 7 6

0 6 5 0 7

0 9 5 6 8 0 1 6 8 7

0 7 0 2 6 0 8 0 5 0 0 3 7 8 5

0 6 5 1 6 0 9 5 4 4

0 1 7 3 7

0 7 0 9 9 0 7 8 3 2

0 4 0 1 7

0 6 5 2 5

0 9 5 1 8

0 1 7 9 0

0 7 1 8 4 0 7 5 7 9

0 4 2 7 6

0 6 5 3 5 0 9 4 8 9

0 1 8 4 6 0 7 2 8 2

0 7 2 8 2 0 4 5 6 5

0 6 5 4 6

0 9 4 5 4 0 1 9 0 5 0 7 3 9 7 0 6 9 3 5

0 4 8 9 1

0 6 5 5 7

0 9 4 2 3 0 1 9 6 9 0 7 5 3 0

0 6 5 2 5 0 5 2 5 9

0 6 5 7 0

0 9 3 8 3

0 2 0 3 6 0 7 6 8 1

0 6 0 3 8 0 5 6 7 8

0 6 5 8 5 0 9 3 4 1

0 2 1 0 8 0 7 8 6 6 0 5 4 5 6 0 6 1 5 7

0 6 6 0 0

0 9 2 9 5 0 2 1 8 5 0 8 0 7 9 0 4 7 5 6 0 6 7 0 8

0 6 6 1 7 0 9 2 4 2

0 2 2 6 7 0 8 3 2 6 0 3 9 0 6

0 7 3 4 6

0 6 6 3 6

0 9 1 8 7

0 2 3 5 6 0 8 6 2 8 0 2 8 6 8

0 8 0 9 0

0 6 6 5 8

0 9 1 2 3

0 2 4 5 1

0 8 9 7 7

0 1 5 8 9 0 8 9 6 5

0 6 6 8 2

0 9 0 5 4

0 2 5 5 4 0 9 3 9 5 0 0 0 0 0 1 0 0 0 0

o r b y t h e a p p r o x i m a t i o n o f K e i j s e r e t a l . ( e q u a t i o n 6 ) :

1 3 = 1 3

0

f { - ! ' 7 T h + ! ( ' 7 T k 2 + 4 ) t - 0 . 2 3 4 k e x p ( - 2 . 1 7 6 k ) ] . ( 1 2 )

T h e F W H M 2 w , g i v e n b y ( L a n g f o r d 1 9 7 8 , e q u a t i o n 2 7 )

R e { w ( ' 7 T t w / l 3

o

+ i k ) ] = 1 3

0

/ 2 1 3 ,

( 1 3 )

c a n b e o b t a i n e d f r o m t h e a p p r o x i m a t i o n f o r 4 > d e v i s e d b y A h t e e e t a l . ( 1 9 8 4 ,

e q u a t i o n 7 ) :

4 > = 2 ( l n 2 I ' 7 T ) t ( 1 + 0 9 0 3 9 6 4 5 k + O 7 6 9 9 5 4 8 k

2

) / ( 1 + 1 3 4 6 2 1 6 k

+ 1 . t 3 6 1 9 5 k

2

) . ( 1 4 )

C o r r e c t i o n f o r I n s t r u m e n t a l B r o a d e n i n g : F r o m ( 3 ) , i f f ( x ) , g ( x ) a n d h ( x ) a r e

a s s u m e d t o b e V o i g t i a n , t h e n

h = h e * h o = f e * f o * g e * g o

( 1 5 )

T h e C a u c h y a n d G a u s s i a n c o m p o n e n t s o f t h e i n t e g r a l b r e a d t h s o f g ( x ) a n d h ( x ) a r e

o b t a i n e d f r o m 1 3 ! 1 ' 4 > g ' 1 3 h ' 4 > h b y m e a n s o f T a b l e 1 , F i g . 2 o r e q u a t i o n s ( 9 ) a n d ( t o ) ,

a n d t h e c o r r e s p o n d i n g b r e a d t h s o f f ( x ) a r e g i v e n b y

1 3 I e = 1 3 h C - 1 3 g C ,

1 3 } 0 = 1 3 k - I 3 ! o .

( 1 6 )

( 1 7 )

Profile Analysis for Microcrystalline Properties

/3GI/3

"r! t:: "

~

"'f ~ 6 1"9-

1 8 2

J . I . L a n g f o r d e t o L

b y t h e a p p l i c a t i o n o f ( 1 ) a n d ( 2 ) , i s c a r r i e d o u t i n t h r e e s t a g e s : ( a ) W i l l i a m s o n - H a l l

p l o t ; ( b ) m u l t i p l e - l i n e a n a l y s i s ; a n d ( c ) s i n g l e - l i n e a n a l y s i s .

( a ) W i l l i a m s o n - H a l l p l o t : T h e g r a p h o f / 3

1

c o s O v e r s u s s i n O ( b r e a d t h s i n u n i t s o f

2 0 ) o r / 3

1

v e r s u s 1 / d ( b r e a d t h s i n r e c i p r o c a l u n i t s ) g i v e s a u s e f u l v i s u a l i n d i c a t i o n

o f t h e n a t u r e o f a n y i m p e r f e c t i o n s p r e s e n t i n t h e s a m p l e , s i n c e t h e s l o p e d e p e n d s

o n s t r a i n a n d t h e i n t e r c e p t v a r i e s a s t h e r e c i p r o c a l o f t h e s i z e o f t h e c r y s t a l l i t e s o r

d o m a i n s . [ I t s h o u l d b e n o t e d t h a t W i l l i a m s o n a n d H a l l ( 1 9 5 3 ) a l s o u s e d a p l o t o f

/ 3 } v e r s u s 1 / d

2

. ] I t i s p o s s i b l e t o t e l l a t a g l a n c e i f s t r a i n i s s i g n i f i c a n t , w h e t h e r t h e

c r y s t a l l i t e s h a p e i s a n i s o t r o p i c , o r t h e n a t u r e o f a n y m i s t a k e s p r e s e n t ( L a n g f o r d 1 9 6 8 ;

L a n g f o r d e t a l . 1 9 8 6 ) . H o w e v e r , s i n c e C a u c h y f u n c t i o n s a r e i m p l i e d f o r b o t h s i z e

a n d s t r a i n c o n t r i b u t i o n s , t h e p l o t s h o u l d n o t b e u s e d q u a n t i t a t i v e l y .

/ 3 s

( 1 ) S i z e

< :

\ ' ! . " ) / 3 s c

/ 3 j C ( 8 )

/ 3 D C

~

\'l") / 3 S G

/ 3 j O

( 9 )

/ 3 D G

/ 3 D

( 2 ) S t r a i n

F i g . 4 . P r o c e d u r e f o r m u l t i p l e - l i n e s i z e - s t r a i n a n a l y s i s b y t h e

V o i g t m e t h o d .

( b ) M u l t i p l e - l i n e a n a l y s i s : I f r e l i a b l e d a t a a r e a v a i l a b l e f o r t w o o r m o r e o r d e r s o f

a r e f l e c t i o n a n d t h e o r d e r - i n d e p e n d e n t ( ' s i z e ' ) a n d o r d e r - i n d e p e n d e n t ( ' s t r a i n ' ) p r o f i l e s

a r e a s s u m e d t o b e V o i g t i a n , t h e n t h e C a u c h y c o n t r i b u t i o n s / 3 s c a n d / 3 D C a r e o b t a i n e d

f r o m

/ 3 l c c o s O = / 3

s c

+ / 3 D C s i n O ( / 3 / c i n 2 0 )

( 1 8 a )

o r

/ 3 l c = / 3

s c

+ / 3

I x

/ d

( / 3 I C i n r e c i p r o c a l u n i t s ) ,

( 1 8 b )

a n d t h e G a u s s i a n c o n t r i b u t i o n s / 3 S G a n d / 3 D O f r o m

/ 3 } G c o s

2

0 = /3~G + / 3 k s i n

2

0 ( / 3 I G i n 2 0 )

( 1 9 a )

o r

/ 3 } G = /3~G + / 3 k / d

2

( / 3 I G i n r e c i p r o c a l u n i t s ) .

( 1 9 b )

T h e c o n t r i b u t i o n s / 3

s c

a n d / 3

S G

a r e c o m b i n e d b y m e a n s o f ( 1 1 ) o r ( 1 2 ) t o g i v e

/ 3 s ' t h e o r d e r - i n d e p e n d e n t p a r t o f / 3 1 ' a n d t h i s c a n b e i n t e r p r e t e d i n t e r m s o f

c r y s t a l l i t e / d o m a i n s i z e b y m e a n s o f ( 1 ) . S i m i l a r l y , / 3 D C a n d / 3 D O a r e c o m b i n e d t o

g i v e / 3 D ' t h e o r d e r - d e p e n d e n t p a r t o f / 3 1 ' a n d t h e c o r r e s p o n d i n g e s t i m a t e o f s t r a i n i s

g i v e n b y ( 2 ) . T h e p r o c e d u r e i s s u m m a r i s e d i n F i g . 4 .


(c) Single-line analysis: When a multiple-line analysis cannot be applied, because higher orders of a reflection are absent or the breadths are inaccurate, size and strain can be separated for a single line if it is assumed that the size profile is Cauchy (f3 D = f3 fd and that the strain distribution is pure Gaussian ({3 D = f3 fG)' Estimates of size and strain are then given directly by (1) and (2).

Errors in the Voigt Analysis: It should be remembered that in general the errors in any estimate of size and strain from a Voigt analysis of line breadths will be large. This is partly due to counting statistics (Langford 1980; Keijser et al. 1982). It is usually impracticable to spend as long counting at each step for the total pattern as it is for individual lines with the Fourier and variance methods, and random errors will in general be greater. Errors due to residual sample broadening of the 9 pattern can be avoided by careful preparation of the standard. Within the limitations of the assumptions made in the Voigt approach, truncation of lines is not a serious problem, since areas are calculated from analytical functions and, provided that about 8 to 10 points are recorded across the range of the half-width 2 w for each line, errors due to sampling interval should be inappreciable. Systematic errors will be introduced by the use of approximations (9), (10) and (12) and, for Ka radiation, by the removal of the Ka2 component, if this is done analytically (Keijser et al. 1982). Inadequacy of the Voigt model gives rise to a systematic error which cannot readily be quantified. While this will affect the absolute accuracy of any derived parameters, experience has shown that self-consistent relative values are given by the method. The use of the single-line approach rather than a multiple-line analysis can introduce a further systematic error, as is shown below.

(b) Direct Approach to the Analysis of Line Breadths We assume a shape function h(h1' ~, ... ) for the measured profiles of the specimen

investigated and a shape function g(g1' rh, ... ) for the instrumental profile, where h1' ~, ... and g1' rh' .,. are parameters. In this approach there are no mathematical limitations concerning hand g; e.g. asymmetric functions are allowed. Within the scheme of the shape functions adopted, a mathematical relation between the parameters h1'~' ... and g1' rh' ... and the area-weighted crystallite size D and the r.m.s. strain < e2)l!2 is obtained by the following procedure:

(i) determine analytically the Fourier coefficients H(n, h1' ~, ... ) and G(n, g1' rh, ... ) of hand 9 normalised so that H( n = 0) = G( n = 0) = 1, n being the harmonic number;

(ii) obtain the Fourier coefficient F of the pure structurally broadened profile in terms of n, h1' ~, ... , g1' rh' ... by application of Stokes' procedure (1948),

F(n, h1' ~, ... , {h, rh' ... ) = " +i Bf = H(n, h1' ~, ... ) . G(n, g1' rh, ... ) , (20)

(iii) express in terms of h1' ~, ... and g1' rh' ...

D = _ d { I dd~ I )-1 , n_O

(21)

I d2Af I dn2 n_O I d2As I dn2 -4172 P< e2), (22) n_O

1 8 4


w h e r e A S i s t h e F o u r i e r c o e f f i c i e n t o f t h e s i z e - b r o a d e n e d p r o f i l e , d i s t h e i n t e r p l a n a r

s p a c i n g a n d I i s t h e o r d e r o f t h e r e f l e c t i o n .

A c c o r d i n g t o ( 2 1 ) , D c a n b e d e t e r m i n e d f r o m a s i n g l e l i n e , w h e r e a s , f r o m ( 2 2 ) ,

( t ? ) 1 1 2 [ a n d d

2

A S ( n ) / d n 2 ] c a n b e d e t e r m i n e d f r o m t w o o r d e r s o f a r e f l e c t i o n . B e c a u s e

t h e p r o f i l e f i t s a r e d i f f e r e n t f o r d i f f e r e n t o r d e r s , t h e r e l i a b i l i t y o f a m u l t i p l e - l i n e m e t h o d

o n t h e b a s i s o f ( 2 1 ) a n d ( 2 2 ) i s v e r y l i m i t e d a n d r e s u l t s m a y b e m i s l e a d i n g . H o w e v e r ,

i f t h e s i z e e s t i m a t e s , f r o m s e p a r a t e o r d e r s , a c c o r d i n g t o ( 2 1 ) , c o i n c i d e , a r e l i a b l e s t r a i n

v a l u e m i g h t b e o b t a i n e d .

I n v i e w o f t h e a b o v e a s i n g l e - l i n e a n a l y s i s w i l l b e a p p r o p r i a t e i n m a n y c a s e s a n d i s

u n a v o i d a b l e i f m U l t i p l e o r d e r s a r e a b s e n t . I n t h e s i n g l e - l i n e a p p r o a c h a n a d d i t i o n a l

a s s u m p t i o n i s r e q u i r e d a b o u t t h e t e r m I d

2

A S / d r ? I n . . . . . 0 i n ( 2 2 ) ( f o r d e t a i l s s e e K e i j s e r

e t a l . 1 9 8 3 ) . S o m e p r o f i l e - s h a p e f u n c t i o n s a r e u n s u i t a b l e f o r u s e w i t h t h i s d i r e c t

a p p r o a c h , a s t h e i r m a t h e m a t i c a l p r o p e r t i e s a r e i n c o m p a t i b l e w i t h ( 2 1 ) a n d / o r ( 2 2 ) .

G a u s s i a n a n d g e n e r a l l y P e a r s o n V I I f u n c t i o n s d o n o t c o n f o r m t o ( 2 1 ) b e c a u s e t h e

d e r i v a t i v e o f t h e F o u r i e r t r a n s f o r m f o r n - + 0 i s a l w a y s z e r o , i m p l y i n g t h a t D i s

i n f i n i t e . T h e u s e o f t h i s a p p r o a c h w i t h V o i g t a n d p s e u d o - V o i g t s h a p e f u n c t i o n s h a s

b e e n d e s c r i b e d b y K e i j s e r e t a l . ( 1 9 8 3 ) .

s

4

X

' ' ' '

2

, , .

8 =

7~7

4 0 0

0 <

' - '

~ 2 0 0

o

0 . 4

0 . 6

. . -

. . - . . .

,

. ,

~

~~

0 . 8

l i d ( X - I )

,

~

,

1 D

( a )

. . . .

( b )

. . . .

1 2

F i g . 5 . E s t i m a t e s f o r Z n o o f ( a ) t h e s t r a i n a n d ( b ) t h e s i z e a g a i n s t 1 1 d

o b t a i n e d b y t h e s i n g l e - l i n e m e t h o d ( f r o m S o n n e v e l d e t a l . 1 9 8 6 ) .

4 . E x a m p l e o f L i n e P r o f i l e A n a l y s i s f r o m T o t a l P a t t e r n F i t t i n g

A s a n e x a m p l e o f e s t i m a t i n g c r y s t a l l i t e s i z e a n d s t r a i n b y t o t a l p a t t e r n f i t t i n g , d a t a

o b t a i n e d f r o m a s a m p l e o f Z n O ( Z n O - B o f L a n g f o r d e t a l . 1 9 8 6 ) w i l l b e c o n s i d e r e d .

T h e P e a r s o n V I I f u n c t i o n , w i t h a l l o w a n c e f o r a s y m m e t r y , w a s u s e d t o m o d e l t h e

i n d i v i d u a l p e a k s o f t h e p a t t e r n . T h e r e s u l t s o f a p p l y i n g a s i n g l e - l i n e V o i g t a n a l y s i s a r e

s h o w n i n F i g . 5 ( s e e a l s o T a b l e 1 o f S o n n e v e l d e t a l . 1 9 8 6 ) . F o r r e f l e c t i o n s o f w h i c h

m U l t i p l e o r d e r s a r e p r e s e n t , s i n g l e - l i n e a n d m u l t i p l e - l i n e a n a l y s e s m a y b e c o m p a r e d

i i i T a b l e 2 a n d F i g . 6 . C l e a r l y t h e l a r g e s t s i z e a n d s m a l l e s t s t r a i n v a l u e s a r e o b s e r v e d

f o r t h e ( 0 0 2 ) a n d ( 0 0 4 ) r e f l e c t i o n s [ s e e b e l o w f o r ( 0 0 6 ) ] . F o r t h e o t h e r r e f l e c t i o n s n o

Profile Analysis for Microcrystalline Properties

Table 2. Comparison of size and strain values for ZnO obtained from single-line and multiple-line analyses (Sonneveld et al. 1986)

hkl Size D (A.) Strain ex 103 Multiple Single Multiple Single

(100) 392 38 (hOO) (200) 318 352 29 27

(300) 186 19 (101) 327 33

(hOh) 305 32 (202) 238 25 (002) 484 23

(001) 417 20 (004) 391 17

(a)

_(0) (b)

.., r:-~ 8 3 _____ X 001 hOh ,. '" -~

I 0' 10

lid (,&.-1) Fig. 6. Comparison between (a) size and (b) strain for ZnO estimated by single-line (solid) and multiple-line (dashed) methods.

185

distinct anisotropy with respect to size and strain is apparent (Fig. 5). These results are similar to those obtained for a ZnO powder which only exhibited size broadening (ZnO-A of Langford et af. 1986), where the more accurate Fourier and variance methods were applied.

As regards the accuracy of the results obtained, the quality of fit in this example for the Pearson VII function, can be assessed from the factors Rp and Rwp (Young et al. 1982). From the values of these factors for the ZnO sample (Langford et af. 1986, Table 3), it is evident that the accuracy is lowest at high angles, where the intensity of lines tends to be low and, in this case, there is more peak overlap. This suggests that, for size-strain analysis in conjunction with pattern fitting, in many cases size and strain data can be obtained by performing a single-line analysis on lines considered to be fitted best. This is corroborated by the fact that, in this example, a combination of (002), (004) and (006) in a multiple-line analysis gives physically unrealistic results. [However, it should be noted that the (006) has a very low intensity.]

1 8 6

J . I . L a n g f o r d e t a L

I n t h e p a r t i c u l a r c a s e o f s i z e a n d s t r a i n b r o a d e n i n g b e i n g V o i g t i a n i n c h a r a c t e r ,

a n d i f ( 1 ) a n d ( 2 ) h o l d f o r t h e i r C a u c h y a n d G a u s s i a n c o m p o n e n t s , t h e n t h e f o l l o w i n g

t r e n d s a r e t o b e e x p e c t e d i f a s i n g l e - l i n e a n a l y s i s i s a p p l i e d t o d i f f e r e n t o r d e r s o f a

r e f l e c t i o n :

( a ) S i n c e i t i s a s s u m e d t h a t f J s = f J f e ' a G a u s s i a n o r d e r - i n d e p e n d e n t c o m p o n e n t

i s i g n o r e d a n d a C a u c h y o r d e r - d e p e n d e n t p a r t i s i n c l u d e d . T h e e s t i m a t e o f f J s t h u s

i n c r e a s e s w i t h 1 / d ( o r a n g l e ) a n d t h e e s t i m a t e d s i z e d e c r e a s e s ( T a b l e 2 ) .

( b ) B y a s s u m i n g t h a t f J D = f J f G ' a C a u c h y o r d e r - d e p e n d e n t c o m p o n e n t i s i g n o r e d

a n d a n o r d e r - i n d e p e n d e n t G a u s s i a n c o n t r i b u t i o n i s i n c l u d e d . T h e e s t i m a t e o f f J D '

a n d h e n c e t h e e s t i m a t e d s t r a i n , t h e n d e c r e a s e s w i t h 1 / d ( T a b l e 2 ) .

T h e e f f e c t o f t h e s e t e n d e n c i e s i s s t r i k i n g l y a p p a r e n t f r o m F i g . 6 a n d i n t h i s p a r t i c u l a r

e x a m p l e t h e s i z e a n d s t r a i n v a l u e s g i v e n b y t h e t w o m e t h o d s d i f f e r b y u p t o 3 0 % .

T h e t r e n d i s a l s o e v i d e n t f o r b o t h s i z e a n d s t r a i n v a l u e s i n F i g . 5 . H o w e v e r , t h e s i g n

a n d m a g n i t u d e o f s y s t e m a t i c e r r o r s i n t r o d u c e d b y a p p l y i n g t h e s i n g l e - l i n e a p p r o a c h

c a n n o t r e a d i l y b e a s s e s s e d ; f o r s i z e e s t i m a t e d f r o m t h e C a u c h y c o m p o n e n t o f t h e t o t a l

p r o f i l e , a n u n k n o w n s t r a i n c o n t r i b u t i o n i s i n c l u d e d , b u t t h e u n k n o w n s i z e c o n t r i b u t i o n

t o t h e G a u s s i a n c o m p o n e n t i s i g n o r e d .

F u r t h e r , i n a m u l t i p l e - l i n e a n a l y s i s , t h e e r r o r i n v a l u e s f o r s i z e a n d s t r a i n i s i n

g e n e r a l g o v e r n e d b y t h a t o r d e r o f a r e f l e c t i o n f o r w h i c h t h e q u a l i t y o f t h e b r e a d t h

p a r a m e t e r s i s l o w e s t . A l s o , n o i n f o r m a t i o n i s n o r m a l l y a v a i l a b l e a b o u t t h e p r e c i s e

s h a p e o f t h e s i z e a n d s t r a i n c o m p o n e n t p r o f i l e s , i . e . w h e t h e r o r n o t t h e V o i g t i a n i s

i n f a c t a c l o s e a p p r o x i m a t i o n . N o g e n e r a l s t a t e m e n t c a n t h e r e f o r e b e m a d e ' r e g a r d i n g

t h e a c c u r a c y o f s i z e - s t r a i n d a t a o b t a i n e d f r o m a s i n g l e - l i n e a n a l y s i s , c o m p a r e d w i t h

t h a t f r o m a m u l t i p l e - l i n e a p p r o a c h b a s e d o n a s s u m e d p r o f i l e s h a p e s .

R e f e r e n c e s

A h t e e , M . , U n o n i u s , L . , N u r m e l a , M . , a n d S u o r t t i , P . ( 1 9 8 4 ) . J . A p p l . C r y s t . 1 7 , 3 5 2 - 7 .

A v e r b a c h , B . L . , a n d W a r r e n , B . E . ( 1 9 4 9 ) . J . A p p l . P h y s . 2 0 , 8 8 5 - 6 .

D e l h e z , R . , K e i j s e r , T h . H . d e , a n d M i t t e m e i j e r , E . J . ( 1 9 8 0 ) . I n ' A c c u r a c y i n P o w d e r

D i f f r a c t i o n ' , N B S S p e c . P u b . N o . 5 6 7 ( E d s S . B l o c k a n d C . R . H u b b a r d ) , p p . 2 1 3 - 5 3 ( N B S :

W a s h i n g t o n , D C ) .

D e l h e z , R . , K e i j s e r , T h . H . d e , a n d M i t t e m e i j e r , E . J . ( 1 9 8 2 ) . F r e s e n i u s Z . A n a L C h e r n . 3 1 2 , 1 - 1 6 .

D e l h e z , R . , K e i j s e r , T h . H . d e , M i t t e m e i j e r , E . J . , a n d L a n g f o r d , J . I . ( 1 9 8 6 ) . J . A p p l . C r y s t . 1 9 ,

4 5 9 - 6 6 .

D e l h e z , R . , K e i j s e r , T h . H . d e , M i t t e m e i j e r , E . J . , a n d L a n g f o r d , J . I . ( 1 9 8 8 ) . A u s t . J . P h y s . 4 1 ,

2 1 3 - 2 7 .

G u i l l a t t , I . F . , a n d B r e t t , N . H . ( 1 9 7 0 ) . P h i / o s . M a g . 2 1 , 6 7 1 - 8 0 .

G u i l l a t t , I . F . , a n d B r e t t , N . H . ( 1 9 7 1 ) . P h i / o s . M a g . 2 2 , 6 4 7 - 5 3 .

H a l d e r , N . C . , a n d W a g n e r , C . N . J . ( 1 9 6 6 ) . A d v . X - r a y A n a l . 9 , 9 1 - 1 0 2 .

H a l l , W . H . ( 1 9 4 9 ) . P r o c . P h y s . S o c . L o n d o n A 6 2 , 7 4 1 - 3 .

H e p p , A . , a n d B a e r l o c h e r , C h . ( 1 9 8 8 ) . A u s t . J . P h y s . 4 1 , 2 2 9 - 3 6 .

H u a n g , T . C . ( 1 9 8 8 ) . A u s t . J . P h y s . 4 1 , 2 0 1 - 1 2 .

K e i j s e r , T h . H . d e , L a n g f o r d , J . I . , M i t t e m e i j e r , E . J . , a n d V o g e l s , A . B . P . ( 1 9 8 2 ) . J . A p p l .

C r y s t . 1 5 , 3 0 8 - 1 4 .

K e i j s e r , T h . H . d e , M i t t e m e i j e r , E . J . , a n d R o z e n d 8 l 1 1 , H . C . F . ( 1 9 8 3 ) . J . A p p l . C r y s t . 1 6 , 3 0 9 - 1 6 .

K l u g , H . P . , a n d A l e x a n d e r , L . R . ( 1 9 7 4 ) . ' X - r a y D i f f r a c t i o n P r o c e d u r e s f o r P o l y c r y s t a l l i n e a n d

A m o r p h o u s M a t e r i a l s ' , 2 n d e d n ( W i l e y : N e w Y o r k ) .

L a n g f o r d , J . I . ( 1 9 6 5 ) . N a t u r e 2 0 7 , 9 6 6 - 7 .

L a n g f o r d , J . I . ( 1 9 6 8 ) . J . A p p l . C r y s t . 1 , 1 3 1 - 8 .

L a n g f o r d , J . I . ( 1 9 7 8 ) . J . A p p l . C r y s t . 1 1 , 1 0 - 1 4 .


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Manuscript received 7 Ianuary, accepted 15 February 1988

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