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Journal ofApplied
Crystallography
ISSN 0021-8898

Diffraction peak profiles from spherical crystallites with lognormal size
distribution
T. Ida, S. Shimazaki, H. Hibino and H. Toraya

Copyright © International Union of Crystallography
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J. Appl. Cryst. (2003). 36, 1107–1115

T. Ida et al.  Crystallite size distribution

research papers
Journal ofApplied
Crystallography

Diffraction peak profiles from spherical crystallites
with lognormal size distribution

ISSN 0021-8898

T. Ida,* S. Shimazaki, H. Hibino and H. Toraya
Received 24 December 2002
Accepted 27 May 2003

# 2003 International Union of Crystallography
Printed in Great Britain ± all rights reserved

Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi, Gifu 5070071, Japan. Correspondence e-mail: ida@crl.nitech.ac.jp

An ef®cient and accurate method to evaluate the theoretical diffraction peakpro®les from spherical crystallites with lognormal size distribution (SLN pro®le)is presented. Precise results can be obtained typically by an eight-term
numerical integral for any values of the parameters, by applying an appropriate
substitution of the variable to the integral formula. The calculated SLN pro®leshave been veri®ed by comparison with those calculated by inverse Fouriertransform from the exact analytical solution of the Fourier-transformed SLN
pro®le. It has been found that the shape of the SLN pro®le strongly depends onthe variance of size distribution. When the logarithmic standard deviation ! of
the size distribution is close to 0.76, the SLN pro®le becomes close to a
Lorentzian pro®le, and `super-Lorentzian' pro®les are predicted for larger
values of !, as has been concluded by Popa & Balzar [J. Appl. Cryst. (2002), 35,
338±346]. The intrinsic diffraction peak pro®les of an SiC powder sample
obtained by deconvolution of the instrumental function have certainly shown`super-Lorentzian' line pro®les, and they are well reproduced by the SLN pro®lefor the value ! = 0.93.

1. Introduction
Many important physical properties of solid materials dependon their microstructure, among which crystallite size andmicrostrain are the most important characteristics. In principle, these can be determined from the line pro®le analysis of
powder X-ray diffraction data.
Since experimental diffraction peak pro®les are broadenedby crystallite size, lattice defects and instrumental aberrations,
it is essential to separate these effects correctly. Three classicalmethods, namely (i) the Stokes (1948) method for eliminating
instrumental aberrations, and (ii) the Warren±Averbach(1950) and (iii) the Williamson±Hall (1952) methods forseparating size and strain effects, have been used during thepast ®ve decades.
The Stokes (1948) method is based on the deconvolution of
the instrumental function by a Fourier method. The experimental line pro®les of a standard sample, which should be
prepared by long-time annealing at suf®ciently hightemperatures, are used as instrumental functions in the Stokes
method.
The Warren±Averbach (1950) and Williamson±Hall (1952)
methods separate the size and strain effects by their different
dependence on the diffraction vector k. The former is basedon the Fourier coef®cients of the line pro®les, while the latteris based on the widths of the line pro®les.
In this paper, we focus our attention on the effects ofcrystallite size distribution on powder diffraction data. It is
known that the area-weighted average size hDiA is estimatedJ. Appl. Cryst. (2003). 36, 1107±1115from the initial slope of the Fourier transform in the Warren±
Averbach analysis, while the volume-weighted average size
hDiV is estimated by the Williamson±Hall method when it isapplied to the intrinsic integral breadths of the line pro®les(Guinier, 1963). It has been observed by many authors that thesize distribution of crystallites in a powder or polycrystalline
solid is lognormal (cf. Langford et al., 2000). By assuming thelognormal size distribution, the mean size and the variance,
which determine all the characteristics of the lognormal
distribution, can be estimated from the two different values of
hDiA and hDiV.
Recently, signi®cant improvements in diffraction line pro®leanalysis, which are particularly useful in the evaluation of
crystallite size distribution, have been achieved by Langford etal. (2000), UngaÂr et al. (2001) and Ida & Toraya (2002).
Langford et al. (2000) have shown that the size distribution
can be determined directly by a pro®le ®tting method,
applying the theoretical diffraction line pro®le from sphericalcrystallites with lognormal size distribution (hereafter abbreviated as the `SLN pro®le'). In principle, the SLN pro®le as anintegral formula can be evaluated numerically. The integral
breadth and the Fourier initial slope can simultaneously beevaluated for each diffraction peak by a simple pro®le ®tting
procedure applying the SLN pro®le function. When the size
effect is dominant in an observed line pro®le, the size distribution can be determined straightforwardly. However, theirevaluation and the application of the SLN pro®le have been
restricted to a small variance of the distribution (Popa &Balzar, 2002).

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T. Ida et al.  Crystallite size distribution1107

research papersUngaÂr et al. (2001) have shown that the exact solution of theFourier transform of the SLN pro®le is available in closed
form. They have applied the formula to ®t the Fourier-transformed experimental peak pro®les of ®ne silicon nitride
powder. Both the Fourier initial slopes and the integral
breadths have been evaluated by ®tting to the Fourier-transformed line pro®les, similarly to the method of Langford et al.
(2000). It has been reported that the size distribution determined by the ®tting method is in good agreement with theresults of transmission electron microscopy (TEM).
We have recently developed a new technique to eliminate
the instrumental aberrations from the experimental diffraction data (Ida & Toraya, 2002). Since the method is based on
analytical expressions for the instrumental aberrations (Ida,
1998a; Ida & Kimura, 1999a,b), no measurement of standard
samples is required, and the whole diffraction pattern cansimultaneously be treated no matter how complicated thepattern may be. Estimation of the errors in the deconvoluted
data have also been enabled. The method would be particularly useful in the estimation of size distribution by ®tting the
SLN pro®le to experimental peak pro®les of relatively large
crystallites, where the effects of instrumental aberrations are
not negligible.
In this paper, we present an ef®cient algorithm to evaluate
theoretical SLN pro®les in high precision for any values of thelogarithmic standard deviation !. The calculated line pro®lesare veri®ed by comparison with the inverse Fourier transformof the exact analytical solution of the Fourier-transformed line
pro®le given by UngaÂr et al. (2001). The possibility of aLorentzian or a `super-Lorentzian' size effect for lognormal
distribution is theoretically examined. Although Langford etal. (2000) have concluded that a Lorentzian or `superLorentzian' size effect might be attributed to a non-unimodal
size distribution, it seems that their results also suggest thatthe line pro®les of lognormal size distribution can be
Lorentzian or become `super-Lorentzian' in shape for large
values of the logarithmic standard deviation !. The intrinsic
diffraction peak pro®les of an SiC powder sample, obtained by
the whole-pattern deconvolution method, are analysed byapplying the SLN pro®le model. The results are comparedwith the experimental results of laser diffraction, scanningelectron microscopy (SEM), and the speci®c surface area
measured by the Brunauer±Emmett±Teller (BET) method.

2

which givesfS k; D RDÿD

AS L; D exp ÿ2ikL dL

 3Dsÿ2 1 ÿ 2sÿ1 sin s  4sÿ2 sin2 s=2:3

It should be noted that the following relations,
IS k; D  D3 =6fS k; D4

fS k; D ! 3D=4 k ! 0;5

andare readily derived from the above equations.
It is assumed that the crystallite size obeys lognormal
distribution with the median m and logarithmic standard
deviation !, the density function of which is given by
fLN D; m; !  D! 21=2 ÿ1 expfÿln D=m2 =2!2 g;

6

and the cumulative distribution function is
FLN D; m; !  1=2erfcÿ ln D=m=21=2 !;7

where the function erfc(x) is the complementary error function de®ned by
erfc x  2=1=2 

R1xexp ÿt2  dt:

8

The properties of the lognormal distribution have been brie¯yreviewed by Langford et al. (2000). The jth moment of the
distribution is given by
hDj i R1
0

Dj fLN D; m; ! dD

 mj exp j2 !2 =2;9from which the arithmetic mean ishDi  m exp 0:5!2 ;10

 2  hDi2 exp !2  ÿ 1;11

the variance isthe area-weighted mean is
hDiA  hD3 i=hD2 i  m exp 2:5!2 ;

12

and the volume-weighted mean is
hDiV  hD4 i=hD3 i  m exp 3:5!2 :

2. Theory13

The diffraction peak intensity pro®le from lognormally
distributed spherical particles is given by

2.1. Line profiles

According to Langford & Wilson (1978), the diffraction
peak intensity pro®le for the diffraction vector k = 2(sin  ÿ
sin Bragg)/ from a spherical crystallite is given by4

ÿ2

IS k; D  D =2s 1 ÿ 2s

ÿ1

sin s  4s

ÿ2

2

sin s=2;1

where D is the diameter and s = 2kD. The normalized
formula for the peak pro®le function fS(k; D) is derived fromthe normalized Fourier formula for a spherical crystallite:1108

AS L; D  1 ÿ 3jLj=2D  jLj3 =2D3 ;T. Ida et al.  Crystallite size distribution

ISLN k; m; ! 

R1
0

IS k; DfLN D; m; ! dD;

14

while the normalized formula is given by
fSLN k; m; ! 

R1
0

fS k; DfLN D; m exp 3!2 ; ! dD:

15

From equations (5) and (9), the peak top value offSLN(k; m, !) is given byelectronic reprint

J. Appl. Cryst. (2003). 36, 1107±1115

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fSLN k; m; ! ! 3m=4 exp 3:5!2 

k ! 0;and the integral breadth isSLN 

2
4=3hDiÿ1
V  4=3m exp ÿ3:5! :  FLN D; m; !:

16

17

Alternatively, Popa & Balzar (2002) have proposed thenumerical formulae
fSLN k; m; ! ' m exp 3:5!2 

2.2. Fourier-transformed line profile

Although it is dif®cult to solve the integral in equation (15)analytically, the exact solution of the Fourier transform
ASLN L; m; ! 

R1
ÿ1

fSLN k; m; ! exp 2ikL dk

18

is available (UngaÂr et al., 2001). The solution is
ASLN L; m; !
 1=2erfcfln jLj=m ÿ 3!2 =21=2 !gÿ 3jLj=4m exp ÿ2:5!2 erfcfln jLj=m ÿ 2!2 =21=2 !g jLj3 =4m3  exp ÿ4:5!2 erfcfln jLj=m=21=2 !g:19

The initial slope of the Fourier transform is then given by@ASLN =@L !  3=2m exp ÿ2:5!2   3=2hDiÿ1
A

L ! 0

20

Fig. 1 shows the Fourier transforms ASLN(L; m, !) calculated by equation (19) for the common value of hDiV = 1 onvariation of the logarithmic standard deviation !.

22

N
P
j1

wj fs k; Dj =Dj

23

and
Dj  m exp 4!2  21=2 !yj ;24

where {yj} and {wj} are the abscissas and weights for the Nterm Gauss±Hermite quadrature. The above formula gives
Pthe
exact peak value for a normalized set of weights {wj}, i.e. j wj
= 1. Popa & Balzar (2002) have treated the pro®les calculated
by a 16-term Gauss±Hermite quadrature on equation (23) as
the `exact' pro®le for the range 0  !  1.39. However,application of such formulae given by equations (23) and (24)is only valid for small values of !, which correspond to a slightmodi®cation from fS(k; m). For large !, the calculated pro®leswould be considerably deviated from the exact solution andthe accuracy would only be improved slowly by increasing thenumber of terms N. Fig. 2 show how the results calculated byequations (23) and (24) change on the increasing N, for thecase ! = (ln7)1/2. The SLN pro®le calculated by a 65536-point
inverse Fourier transform of the exact Fourier-transformedpro®le given by equation (19) with a step size of 1/409.6 on the
L scale is also shown as markers in Fig. 2.

2.3. Evaluation of the line profile

Even though the SLN prolile is clearly de®ned by equation(15) in integral form, it is not easy to evaluate the integral
numerically, especially for large values of !. Langford et al.
(2000) have presented the formulafSLN k; m; ! 

R1
0

 R1

IS k; D d0

D3 =6 d;

21

where

Figure 2
Figure 1

The Fourier transforms of the SLN pro®les for the median m =exp(ÿ3.5!2) (hDiV = 1) and the logarithmic standard deviation ! = 0, 0.5,
1, 1.5.
J. Appl. Cryst. (2003). 36, 1107±1115

Evaluation of the SLN pro®le for m = exp(ÿ3.5!2) (hDiV = 1) and ! =(ln7)1/2 = 1.39496 on increasing the number of terms N, applying
equations (23) and (24) proposed by Popa & Balzar (2002). The open
diamonds mark the values obtained by the inverse Fourier transform of
the exact Fourier-transformed pro®le.

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T. Ida et al.  Crystallite size distribution1109

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Table 1

The abscissas xj and weights wj for eight-term (N = 8) Gauss±Legendre
quadrature.
For j  5, use the following relations: xN+1ÿj = 1 ÿ xj; wN+1ÿj = wj.

xj
wj

j=1

j=2

j=3

j=4

0.01985507
0.05061427

0.10166676
0.11119052

0.23723380
0.15685332

0.40828268
0.18134189

The results given in Fig. 2 by the method proposed by Popa
& Balzar (2002) are found to be considerably deviated fromthe pro®le calcuclated by the inverse Fourier transform of theexact analytical solution. The maximum deviation of theresults of the 16-term numerical integral exceeds 1.08% relative to the peak height, the amount of which cannot be
neglected in the application to pro®le ®tting analysis.
As one of the authors has already reported (Ida & Kimura,
1999a,b), convolution of peak pro®le functions can be properly evaluated by applying formulae derived from an appropriate substitution of the variable. The method is also
applicable to the evaluation of the SLN function fSLN(k; m, !)with only slight modi®cation, which gives the following
formulae:fSLN k; m; ! ' H0

N
P
j1

wj gj =G0j ;D0  m exp 4!2 ;

25
26

Figure 3

Evaluation of the SLN pro®le for m = exp(ÿ3.5!2) (hDiV = 1) and ! =(ln7)1/2 = 1.39496 on increasing the number of terms N, applying
equations (30) and (31). The open diamonds mark the values obtained by
the inverse Fourier transform of the exact Fourier-transformed pro®le.

1110

T. Ida et al.  Crystallite size distribution

G0  kÿ1 arctan 3kD0 =2;27

H0  2ÿ1 exp ÿ0:5!2 G0 ;28

j  xj G0 ;29

D0j  4=3k tan kj ;30

Dj  m exp4!2 ÿ 21=2 ! erfcÿ1 D0j =D0 ;

31

gj  fS k; Dj =Dj ;32

G0j  3=41  3kD0j =42 ÿ1 ;33

j  1; . . . ; N
where {xj} and {wj} are the abscissas and weights of the Gauss±
Legendre quadrature (Press et al., 1986). The details of thederivation are described in Appendix A. A numerical routine
for evaluating the inverse complementary error functionerfcÿ1(x) is given in Appendix B.
Fig. 3 shows the results of the calculation by the formulae
given by equations (25)±(33) for ! = (ln7)1/2 = 1.39496, on
increasing the number of terms N up to 16. Interestingly, eventhe two-term numerical integral (N = 2) gives values considerably close to those calculated by the inverse Fourier transform of ASLN(L; m, !), and the numerical integral rapidly
approaches the expected value on increasing N. The deviationof our eight-term numerical integral is within 0.32% relative tothe peak height.
Table 1 lists the abscissas {xj} and weights {wj} for the eightterm (N = 8) Gauss±Legendre integral, created by a routinegauleg() given by Press et al. (1986). The SLN pro®les for !
= 0, 0.5, 1, 1.5, calculated by the eight-term numerical integrals,
are shown in Fig. 4. All of the calculated pro®les coincided
well with those derived from the inverse Fourier transform of
ASLN(L; m, !) calculated by equation (19).

Figure 4

The SLN pro®les for the median m = exp(ÿ3.5!2) (hDiV = 1) and thelogarithmic standard deviation ! = 0, 0.5, 1, 1.5. Markers denote thevalues obtained by the inverse Fourier transform of the Fouriertransformed pro®le.

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2.4. Comparison with the Lorentzian profile

Although Langford et al. (2000) have suggested that theLorentzian or `super-Lorentzian' size effect might be attributed to a non-unimodal size distribution, it seems that bothshapes can be rather naturally explained by the unimodal
lognormal size distribution with large logarithmic standard
deviation !, as has been concluded by Popa & Balzar (2002).
The Lorentzian pro®le with the full width at half-maximum(FWHM) of 2 is given by
fL k;    ÿ1 1  k= 2 ÿ1 ;34

and the Fourier transform is
AL L;   exp ÿ2 jLj:35

The integral breadth is then  and the initial slope of theFourier coef®cient is |(@AL/@L)0| = 2 . Those properties areexactly the same as the SLN pro®le with m = (3/4)6/ and ! =[ln(16/9)]1/2 = 0.758528.
Fig. 5 shows both the Lorentzian pro®le and the SLN pro®lewith ! = [ln(16/9)]1/2. The close resemblance clearly indicatesthat the line pro®les of crystallites with lognormal size distribution can certainly become a `Lorentzian-like' pro®le for a
realistic value of ! ' 0.76, even though it would be intermediate between Lorentzian and Gaussian for a narrower
distribution of size (Langford et al., 2000). Furthermore,`super-Lorentzian' (Wertheim et al., 1974) line pro®les, whichhave sometimes been reported (e.g. PleÂvert & LoueÈr, 1990),
are also likely to be observed in the case of a broader size
distribution.

Fig. 7 shows the scanning electron micrograph (SEM) of thepowder. Although it is dif®cult to estimate the size distribution
from the SEM photograph, the apparent particle size is foundto be widely distributed, ranging from 50 to 1000 nm.
The speci®c surface of the powder, measured by the methodof Brunauer±Emmett±Teller (BET), was 22.93 m2 gÿ1, fromwhich the area-weighted average size was estimated as hDiA =80.9 nm.
The powder diffraction pattern was measured with aconventional powder X-ray diffractometer (Rigaku, RAD2C)
with a Cu K radiation tube operated as a line focus at 40 kV
and 30 mA, and with a curved graphite monochromator on thediffracted-beam side. The radius of the goniometer circle was
R = 250 mm; a receiving sit of 0.15 mm width, and 1 open
divergence and scattering slits were used. The axial divergence
was limited to A = 2.5 in FWHM by symmetrically locatedsets of Soller slits. The SiC powder was loaded into the hollow[0.553 (7) mm depth] of a glass holder with the ®lling factor of
35.8 (4). The penetration depth of the Cu K X-ray was estimated at 0.199 (2) mm, which was suf®ciently shorter than the
thickness of the sample. The pattern was scanned over the
angular range 33±125 (2), with a step length of 0.05 (2) anda counting time of 40 s per step.

3. Application to an SiC powder sample
3.1. Experimental

A commercial SiC powder sample (JFCC, RP-2) was used
for the experiments. The cumulative size distribution
measured with a laser diffraction instrument (Malvern,
MasterSizer) and the optimized lognormal distribution are
shown in Fig. 6. The optimized values of the parameters forthe distribution are hDiV ' 600 nm and ! = 0.97 (3).

Figure 5Comparison between the Lorentzian (dotted) and SLN (solid) line
pro®les.
J. Appl. Cryst. (2003). 36, 1107±1115

Figure 6

Particle size distribution of SiC powder measured with a laser diffraction
instrument. Open circles denote the observed values and the line shows
the optimized lognormal distribution.

Figure 7

SEM micrograph of an SiC powder sample.

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Figure 8

The 111 diffraction peak pro®les of an SiC powder sample: (a) raw
experimental data, (b) deconvoluted data (thin line) and the optimized
SLN pro®le (thick line), (c) the difference plot (solid line) and theestimated error (broken lines).

Figure 9

The 331 and 420 diffraction peak pro®les of an SiC powder sample: (a)raw experimental data, (b) deconvoluted data (thin line) and the
optimized SLN pro®le (thick line), (c) the difference plot (solid line) andthe estimated error (broken lines).

3.2. Elimination of instrumental aberrations

The K 2 sub-peak and instrumental aberrations caused by
the axial divergence, ¯at-specimen and sample transparency
were removed from the experimental data by the wholepattern deconvolution method recently developed by theauthors (Ida & Toraya, 2002). The effects of the ®nite widthsof the X-ray source and receiving slit were also deconvoluted,
assuming the effective width of the X-ray source to be 0.1 mm.
The smoothing ®lter was not applied for simplicity in theestimation of the pro®le parameters. The errors in thedeconvoluted data were evaluated by the correlation of thereciprocal variance in the source data and the squared
instrumental function.
Figs. 8(a) and 9(a) show the raw diffraction data around the111, 331 and 420 re¯ections of the SiC powder, while Figs. 8(b)and 9(b) show the deconvoluted pro®les. The deconvoluted
pro®les have symmetric shape, sharpened peak-tops and alsosigni®cantly long tails, which are the characteristic features ofa so-called `super-Lorentzian' pro®le (Wertheim et al., 1974).

3.3. Profile fitting

The deconvoluted pro®les were individually ®tted with the
SLN peak pro®le function fSLN(k; m, !), calculated by theef®cient algorithm based on the eight-term numerical integral.
A combination of two pro®le functions was used for two pairsof re¯ections, (311 and 222) and (331 and 422). A quadratic
background was assumed for each analysed data range. Both

1112

T. Ida et al.  Crystallite size distribution

Figure 10

The optimized values of the logarithmic standard deviation ! of the SLN
pro®les for 111±422 re¯ections.

the median diameter m and the logarithmic standard deviation! were treated as adjustable parameters.
The results of pro®le ®tting for the 111, 331 and 420
re¯ections are shown in Figs. 8(b), 8(c), 9(b) and 9(c). As seenin Figs. 8(b) and 9(b), the SLN pro®les ®t the deconvoluted
data very well. No signi®cant systematic deviation is found in
the difference plots in Figs. 8(c) and 9(c).
3.4. Line profile analysis

The optimized values of the logarithmic standard deviation! for the 111±422 re¯ections are shown in Fig. 10. The

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weighted average has been estimated at h!i = 0.93 (3), which
is close to the value of 0.97 (3) evaluated by the laser
diffraction method. The observed `super-Lorentzian' pro®les
are naturally ascribed to the broad size distribution.
= (4/The integral breadth calculated by
SLN3m)exp(ÿ3.5!2) for each re¯ection is plotted versus dhkl in
Fig. 11, which is identical to the Williamson±Hall plot
(Williamson & Hall, 1953). The optimized linear dependence
is given by
SLN  0:024

4 nmÿ1  0:0040 6 dhkl :The volume-weighted average size of crystallites is formallyestimated at 50±70 nm, which appears to be considerably
smaller than the particle size observed in laser diffraction and
BET experiments. It has been suggested that the coherently
diffracting domain of the SiC sample is smaller than theapparent particle size. It should be noted that the line widths
might be affected by the line broadening caused by stacking
faults, which may arise in the SiC crystallites (Paterson, 1952).

4. Conclusion
We have developed an ef®cient and accurate algorithm toevaluate the theoretical diffraction peak pro®les from spherical crystallites with lognormal size distribution. `Lorentzianlike' or `super-Lorentzian' line pro®les are naturally predicted
for broad size distributions, as has been concluded by Popa &Balzar (2002). The properties of the size distribution can be
estimated by a simple curve ®tting method applied to theintrinsic diffraction peak pro®les from small crystallites, which
can be extracted from the experimental data by the wholepattern deconvolution method recently developed by theauthors (Ida & Toraya, 2002).

f xg x dx

36

is obtained by the substitution of the variable x by , whichapproximates the primitive function of the integrand, i.e.
R  H x ' f xg x dx:
37
The integral is exactly evaluated by the following formula:
S

HR b
H a

ff H ÿ1 gH ÿ1 =H 0 H ÿ1 g d:

G x '

Rg x dx:40

Let us examine the following formula for H(x):H x  GAF x  B=A;

41

42

Then, H(x) will become an approximation for the primitive of[f(x)g(x)] if [AF(x) + B] ' x is satis®ed. Since the value of[f(x)g(x)] is signi®cant at x = x1, x2, the following set of linear
algebraic equations is assumed:

AF x1   B  x1 ;43AF x2   B  x2 ;the solutions of which are given byA  x2 ÿ x1 =F x2  ÿ F x1 44B  x1 F x2  ÿ x2 F x1 =F x2  ÿ F x1 :

45

and

38

When the integrand in equation (38) gradually varies with the
integration variable , the Gauss±Legendre quadrature, or any
other numerical integral, becomes valid for the precise
evaluation of the integral. Even when the function H(x) is only
J. Appl. Cryst. (2003). 36, 1107±1115

and' f xgAF x  B:An ef®cient algorithm for numerical evaluation of theintegral with the formulaaa rough approximation for the primitive function of the integrand in equation (35), it can considerably improve the ef®ciency of the numerical calculation.
We assume that the functions f(x) and g(x) have peaks
located at x1 and x2, respectively, and approximate formulae ofthe primitive functions are given by F(x) and G(x), i.e.
RF x ' f x dx
39

H 0 x  F 0 xG0 AF x  BA1. General formula

Rb

The integral breadth SLN plotted versus the reciprocal interplanar
distance dhkl (Williamson±Hall plot). The dotted line denotes the
optimized linear dependence.

where A and B are arbitrary constants. The derivative of H(x)is given by

APPENDIX A
Derivation of efficient algorithm for the numericalintegralS

Figure 11

The ef®ciency of the numerical integral is expected to be
improved by the substitution of the variable  = G[AF(x) + B]/
A, with the constants A, B given by equations (44) and (45),because it roughly satis®es the relation given by equation (37).

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T. Ida et al.  Crystallite size distribution1113

research papers
andA2. Evaluation of synthesized peak profile

The formula of the SLN peak pro®le given by equation (15)can be rewritten as
R1
fSLN k; m; !  m exp 3:5!2  fS k; x=xÿ xj ;

6146The general formula, described in the preceding section, isstraightforwardly applied to the above equation for evaluating
the SLN peak pro®le.
The function fLN[x; mexp(4!2); !] has a maximum at x1 =m exp 4!2 , and the exact primitive function is assigned toF(x) byF x  1=2erfcfÿln x=m ÿ 4!2 =21=2 !g:

where {wj} are the appropriate weights for the abscissas {xj}( j = 1, . . . , N) ranging from 0 to 1.

0

 fLN x; m exp 4!2 ; ! dx:

j 

APPENDIX B
Numerical routine for the inverse error function
A concise approximation for the inverse complementary error
function erfcÿ1 x, written in C, is presented in Fig. 12.

47

The function [fS(k; x)/x] has a maximum at x2 = 0, and can beroughly approximated by the Lorentzian function with thesame area and integral breadth,G0 x  3=41  3kx=42 ÿ1 ;48

the primitive function of which is given by
G x  kÿ1 arctan 3kD=4:49

Then, the constants A and B are given byA  2m exp 4!2 50

B  0:

51

andThe substitution of the variable  GAF x

52

gives the following formula for the SLN pro®le:R
fSLN k; m; !  1=2 exp ÿ0:5!2  g x=G0 y d;

53

where
 GAF 0  0;54

 GAF 1  G2m exp 4!2 ;55

y  Gÿ1 ;

56

ÿ1

57

xF

y=A:

Note that the inverse functions of F(x) and G(x),
F

ÿ1

2

1=2

z  m exp4! ÿ 2 ! erfc

ÿ1

2z

58

and
Gÿ1   4=3k tan k;59

are both available.
Finally, equations (25)±(33) are derived by applying thefollowing general quadrature formulae:R1114h  d '

ÿ 

N
P
j1

wj h j 

T. Ida et al.  Crystallite size distribution

60

Figure 12

Concise approximation for the inverse complementary error function.

The authors are grateful for the ®nancial support of Nippon
Sheet Glass Foundation for Materials Science and Engineering.

References
Guinier, A. (1963). X-ray Diffraction. San Francisco: Freeman.
Ida, T. (1998a). Rev. Sci. Instrum. 69, 2268±2272.
Ida, T. (1998b). Rev. Sci. Instrum. 69, 3837±3839.
Ida, T. & Kimura, K. (1999a). J. Appl. Cryst. 32, 634±640.
Ida, T. & Kimura, K. (1999b). J. Appl. Cryst. 32, 982±991.
Ida, T. & Toraya, H. (2002). J. Appl. Cryst. 35, 58±68.
Langford, J. I., LoueÈr, D. & Scardi, P. (2000). J. Appl. Cryst. 33, 964±974.

electronic reprint

J. Appl. Cryst. (2003). 36, 1107±1115

research papers
Langford, J. I. & Wilson, A. J. C. (1978). J. Appl. Cryst. 11, 102±113.
Paterson, M. S. (1952). J. Appl. Phys. 23, 805±811.
PleÂvert, J. & LoueÈr, D. (1990). J. Chim. Phys. 87, 1427±1440.
Popa, N. C. & Balzar, D. (2002). J. Appl. Cryst. 35, 338±346.
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T.
(1986). Numerical Recipes. Cambridge University Press.
Stokes, A. R. (1948). Proc. Phys. Soc. London, 61, 382±393.

J. Appl. Cryst. (2003). 36, 1107±1115UngaÂr, T., Gubicza, J., RibaÂrik, G. & BorbeÂly, A. (2001). J. Appl.
Cryst. 34, 298±310.
Warren, B. E. & Averbach, B. L. (1950). J. Appl. Phys. 21, 595±
599.
Wertheim, G. K., Butler, M. A., West, K. W. & Buchanan, D. N. E.
(1974). Rev. Sci. Instrum. 11, 1369±1371.
Williamson, G. K. & Hall, W. H. (1953). Acta Metall. 1, 22±31.

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T. Ida et al.  Crystallite size distribution

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