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Journal ofApplied
Crystallography
ISSN 0021-8898

Deconvolution of the instrumental functions in powder X-ray
diffractometry
T. Ida and H. Toraya

Copyright © International Union of Crystallography
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J. Appl. Cryst. (2002). 35, 58–68

T. Ida and H. Toraya  Deconvolution of instrumental functions

research papers
Journal ofApplied
Crystallography

Deconvolution of the instrumental functions in
powder X-ray diffractometry

ISSN 0021-8898

T. Ida* and H. Toraya
Received 30 August 2001
Accepted 7 November 2001

# 2002 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

A novel method to deconvolute the instrumental aberration functions from theexperimental powder X-ray data has been developed. The method is based on
the combination of scale transformation, interpolation of data and fast Fouriertransformation. The effects of axial divergence, ¯at specimen, sample
transparency and spectroscopic pro®le of the source X-ray are eliminated fromthe entire observed diffraction pattern in three-step operations. The errors in
the deconvoluted data propagated from the statistical uncertainty in the source
data are approximated by the reciprocal of the square root of the correlation
between the reciprocal of the variance in the source data and the squared
instrumental function. The deconvolution of the instrumental aberration
functions enables automatic correction of peak shift and line broadening, andsupplies narrow and symmetric peak pro®les for a well crystallized sample,
which can be ®tted by a simple model function. It will be useful in preparatory
data processing for precise line pro®le analysis, accurate determination of lattice
parameters and whole pattern ®tting for crystal structure analysis.

1. Introduction
It has been well recognized that experimental X-ray diffraction pro®les are convolutions of the incident X-ray spectral
distribution, the various instrumental aberrations and the
intrinsic diffraction pattern of the powder or polycrystalline
sample (Ladell et al., 1959). A variety of methods have been
proposed for the elimination of instrumental broadening,
including that caused by the 1± 2 doublet (Rachinger, 1948;
Stokes, 1948; Keating, 1959; Gangulee, 1970; Ladell et al.,1975). However, none of these methods can be used to eliminate the instrumental aberrations from the entire observed
diffraction pattern in a one-step operation, except Keating's
method for elimination of the 2 peak, which is based on akind of series expansion. The other methods restricted thesolution of this problem to a small part of the pattern (a single
diffraction line). It is impossible to deconvolute all of the
instrumental effects from an entire diffraction pattern by astandard Fourier method, because the pro®le of the instrumental function for powder diffractometry is varied in acomplex manner depending on the diffraction angle 2. It
seems that no signi®cant improvement has been achieved in
Fourier-based methods for elimination of instrumental aberration from experimental peak pro®les, after the pioneering
work of Stokes (1948).
Recently, we have elucidated concrete mathematicalformulae for the instrumental aberration functions of widely
utilized powder diffractometers with Bragg±Brentano
geometry, and that the experimental diffraction peak pro®les
are well reproduced with the model pro®le function synthesized by multiple convolutions of a symmetric function with58

T. Ida and H. Toraya  Deconvolution of instrumental functions

those instrumental functions (Ida, 1998a,b; Ida & Kimura,
1999a,b). As the accurate mathematical formula of the
instrumental function is known, the intrinsic diffraction
pro®les of the sample should be extracted from the experimental pro®les by the deconvolution based on Fourier
methods.
In this paper, we present a novel method for eliminating theaberrations caused by the effects of axial divergence, ¯atspecimen, sample transparency and 1± 2 doublet of thesource X-ray. The method is based on the combination of scaletransformation, interpolation of data and fast Fourier transformation. The effects of instrumental aberrations can beremoved from the entire diffraction pattern in a three-step
operation. We also discuss how the statistical errors in source
data are propagated through the deconvolution process, andpropose an approximate indicator of the error in the deconvoluted data, which enables subsequent application of aweighted least-squares method to analyse the deconvoluteddata.

2. Theoretical framework
Although the pro®le of the total instrumental function for
powder diffractometry varies in a complex manner dependingon the diffraction angle 2, each component of the synthesized
instrumental function is not formidably complicated, as has
been shown in previous reports on this subject (Cheary &Coelho, 1992, 1998; Ida, 1998a; Ida & Kimura, 1999a,b). The
dependence of component instrumental pro®le on thediffraction angle can be removed by application of appro-

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J. Appl. Cryst. (2002). 35, 58±68

research papers
priate scale transformation to the variable, as shown in thefollowing sections.
2.1. Scale transformation

In this section, a general method to treat the instrumental
function, the pro®le of which varies depending on thediffraction angle 2, is described. We assume that the source
diffraction pattern is represented by the scattering intensity
data {Sm} for the diffraction angle f2m g (m = 0, . . . , M ÿ 1),
where M is the number of data points.
Provided that the change in the pro®le of the instrumental
function is speci®ed only by a line-width parameter 2
dependent on 2, the instrumental function ! 2 can be
written as

spaced intervals. In order to utilize fast Fourier transform,
they are interpolated to create evenly spaced data, the total
number of which should be an integer power of 2.
A correction should be applied to the error data fsm g,
because the sampling intervals are changed by interpolation.
When the local density of the data points is changed from n to
n0 by interpolation, the statistical properties of the data aremaintained by changing the error data from  to n0 =n1=2 .
Therefore, the error data fsm g are further modi®ed to fs0m g by
the following equation,

1=2
9
s0m ÿ 2Mÿ1 ÿ 20 N 2m = Mÿ1 ÿ 0 M sm ;

2;

1

where is an arbitrary constant independent of 2.
We de®ne a function G(x) byR
G 2  d 2= 2;before the interpolation process.
Then, evenly spaced abscissa fxn g, ordinate fyn g, and errorfn g (n = 0, . . . , N ÿ 1) data are created from the source datafm g, fm g, fs0m g (m = 0, . . . , M ÿ 1), by a cubic spline interpolation (Press et al., 1986).

22.3. Deconvolution of instrumental function! 2; 2  1=

2w2=and change the abscissa from 2 to , by the equation  =G 2. Then, the following relation is derived,=2  1= 2;

3which means that the small width 2 on the 2 scalecorresponds to a constant step in  scale. By multiplying thescattering data by 2, the local integrated intensity is
maintained; that is, the diagram becomes equivalent to that
convoluted with the constant instrumental function:
ÿ1w = , on the transformed scale.
Here we take the angular velocity, polarization and powder
diffraction intensity factors given by (Lipson & Langford,
1999)
2

C 2  1  A cos 2= 1  A sin  sin 2

4

A  cos 2M ;5

andwhere 2M is the diffraction angle of the monochromator, tocorrect the scattering intensity. The overall scale transformfrom the source data, f2m g and fSm g, given bym ÿG 2m ;

6

m ÿSm 2m =C 2m ;7

andis applied before the following deconvolution process.
When the source data are collected by a counting method,
the intensity data are accompanied with the errors given bySm = S1=2
m . As a result of propagation of the error, the source
error fSm g is considered to be changed to fsm g as follows,sm ÿSm 2m =C 2m :

The Fourier transform W  of the instrumental function
w(x) is given by8

W  

R1
ÿ1

w x exp 2ix dx:

10Assuming that the instrumental function is expressed by adiscrete form fwn g (n = 0, . . . , N ÿ 1), the discrete Fouriertransform is formally given by
Wk 

Nÿ1
P
n0

wn exp 2ikn=Nk  ÿN=2; . . . ; N=2;

11

where the periodicity Wÿk = WNÿk is implicitly assumed.
The Fourier transform W  of the instrumental function
w(x) and the discrete Fourier coef®cients Wk satisfy thefollowing relation,
Wk  W k=Nk  ÿN=2; . . . ; N=2;

12

where  is the sampling interval of the data fyn g. Since aninstrumental function may have singularities, equation (12) iseasier to handle than equation (11) in numerical calculations,when the analytical formula of the Fourier transform isavailable.
The deconvoluted intensity data fzn g from the source data
fyn g are calculated by the inverse Fourier transformzn  1=N

Nÿ1
P
k0

Yk =Wk  exp ÿ2ikn=N;13

where fYk g is the discrete Fourier transform of fyn g.
2.4. Propagation of errors through deconvolution

The scale-transformed abscissa fm g, ordinate fm g anderror fsm g (m = 0, . . . , M ÿ 1) data are not sampled at evenly

In this section, we discuss how the statistical errors fn g
attached to the source data fyn g are propagated into thedeconvoluted data fzn g through the deconvolution process.
The convolution is generally expressed by

J. Appl. Cryst. (2002). 35, 58±68

T. Ida and H. Toraya  Deconvolution of instrumental functions

2.2. Interpolation of data

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yn 

Nÿ1
P
m0

znÿm wm ;where fwn g is the discrete form of the instrumental function.
We can de®ne the inverse instrumental function fwnÿ1 g bywnÿ1  1=N

Nÿ1
P
k0

1=Wk  exp ÿ2ikn=N;15

where fWk g is the discrete Fourier transform of the instrumental function. Then, fzn g is the convolution of fyn g andfwnÿ1 g; that is,
zn 

Nÿ1
P
m0

ynÿm wmÿ1 :16

The element of the covariance matrix Smn (m, n = 0, . . . , N ÿ1) for the deconvoluted data fzn g is immediately given by
Smn  zm ÿ hzm i zn ÿ hzn iNÿ1
P 2 ÿ1 ÿ1
l wmÿl wnÿl ;

l0

17

where hzi means the average of the variable z (see Appendix
A1). The above equation shows that the off-diagonal elementsof the covariance matrix Smn  are generally not equal to zero.
In principle, this problem can be exactly treated as follows.
In case that the off-diagonal elements have non-zero values,
the weighted least-squares ®tting method is identical to the
problem of minimization of the quantity
2 

Nÿ1
P Nÿ1
P
m0 n0zm ÿ f xm Tmn zn ÿ f xn ;

18

where f x is the model function, and Tmn is the element of theinverse covariance matrix (weight matrix). The weight-matrix
element Tmn is related to the source error fn g by the equation(see Appendix A2)
Tmn 

Nÿ1
P
k0

kÿ2 wkÿm wkÿn :19

By applying (18) and (19), one can achieve a proper estimation of ®tting parameters.
However, usual library programs for least-squares ®tting
cannot deal with the weight matrix with non-zero off-diagonal
elements. Furthermore, huge memory space and efforts forcalculation will be needed if all the elements of the weightmatrix are explicitly taken into account in the case of a large
number of data points.
A usual ®tting program can still be applied, on theassumption that the diagonal elements of the weight matrix orthe covariance matrix are the indicators of the error in thedeconvoluted data.
When we de®ne the diagonal elements of the weight andand "2n , respectively, they arecovariance matrices as ÿ2nrelated to the error in the source data byÿ2n  Tnn 

Nÿ1
P
l0

lÿ2 w2lÿn

20

and

60

"2n  Snn 14

T. Ida and H. Toraya  Deconvolution of instrumental functions

Nÿ1
P
l0

h
i2
ÿ1
l2 wnÿl
;21

which mean that fÿ2
n g is the correlation of the reciprocal
variance fnÿ2 g with the squared instrumental function fw2n g,
and that f"2n g is the convolution of the variance fn2 g with thesquared inverse instrumental function f wnÿ1 2 g.
By applying the discrete correlation and convolution
theorems, the following convenient formulae are derived,
P ÿ1 h 2 i
1 Nÿ1

V
Wk exp ÿ2ikn=N;
22
ÿ2
n
N k0 k
and
"2n 

P
1 Nÿ1
V W ÿ2 exp ÿ2ikn=N;N k0 k k23where fVk g, fVkÿ1 g and fWkÿ2 g are the Fourier transforms ofthe variance fn2 g, reciprocal variance fnÿ2 g and the squared
inverse instrumental function f wnÿ1 2 g, respectively, andWk2  is the complex conjugate of the Fourier transform of
the squared instrumental function fw2n g.
The two indicators of the error in the deconvoluted data,{n} and {"n}, do not necessarily coincide. The difference
between the two error indicators will be discussed later.
2.5. Smoothing data

The deconvolution of the instrumental function from theexperimental data always exaggerates the noise in the source
data. Application of a slight smoothing ®lter is effective tosuppress the unwanted increase of the noisy structure. The
Gaussian ®ltering is implemented by the convolution with thefunctionwG x; G   1=1=2 G  exp ÿx2 = G2 ;24

the analytical Fourier transform of which is given by
WG  R1
ÿ1

wG x exp 2ix dx  exp ÿ2 2 G2 :

25

The minimal width of the Gaussian ®lter G to suppress theincrease of the noise on deconvolution is determined by
WG N=2  WO N=2 ;26

where f WO g is the Fourier transform of the target instrumental function, or more concretely,2
G  ÿln2

2
WO N=2 =22 N=2
;

27

where
N=2  1=2;28

in which  is the sampling interval and N=2 is equivalent to
the Nyquist critical frequency in time series analysis.
This smoothing process has a slight broadening effect, thewidth of which is comparable with the sampling interval of theinterpolated data, and narrower than the measurement step.
Even if a precise pro®le analysis is required after the deconvolution, this broadening effect can easily be taken into

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J. Appl. Cryst. (2002). 35, 58±68

research papers!A 2  !Aÿ 2  !A 2
R1
!Aÿ 2 ÿ y!A y dy:

account as a convolution with the Gaussian function with the
known width.

ÿ1

3. Instrumental functions
3.1. Axial divergence effect

The instrumental function for the axial divergence aberration is given by!A 2; 2  2=

2

Figs. 1(b) and 1(c) show how the function !A 2 isdecomposed to the two component functions !Aÿ 2 and!A 2.
By applying the scale transform given by the equations
2=x  exp t ÿ 1=t 2= 2 

 K0  t  1=t j2j= 2 ;

29

where t  tan , 2 ln 21=2 is the full width at halfmaximum (FWHM) of the axial divergence, and K0 x is themodi®ed Bessel function of the second kind (Ida, 1998a). The
pro®le of this function varies on the diffraction angle, in acomplex manner as shown in Fig. 1(a), and cannot satisfy therequirement given in equation (1); that is, the pro®le of thefunction !A 2; 2 cannot be determined by a single linewidth parameter.
However, this function is, fortunately, exactly equivalent tothe convolution of the following two functions:8< 2t=1=2 2  ÿ 2 =2t21=2 exp 2t2= 2 !Aÿ 2 
for 2  0:
0 elsewhere30

2

1 2  2= tan 

33

and
x  G1 2  ÿ ln cos 34

to the lower-angle component function !Aÿ 2, we obtain aformula of the instrumental function independent of thediffraction angle:

1=2
for x  035wAÿ x  exp x= A = A ÿx= A 0elsewhere
where
A 

2

=4:

36

The analytical formula of the Fourier transform of wAÿ x isgiven byWAÿ  and8< 2=1=2 t 2  t 2 =221=2 exp ÿ22=t!A 2 
for 2  0:
0 elsewhere

32



R1
ÿ1

wAÿ x exp 2ix dxnhÿ1=2 i1=2 hÿ1=242 2 2A  1 1 ÿi 42 2 2A  1
i1=2 o ÿÿ1=2
ÿ1:
2 42 2 2A  137

31
That is,Fig. 2 shows the axial divergence and its component functions
plotted on the ÿ ln cos  scale. It is found that the function!Aÿ 2 keeps the same pro®le on the current scale.
For the higher-angle component function !A 2,
application of the scale transform

Figure 1

The dependence of the axial divergence aberration functions on thediffraction angle. (a) The axial divergence aberration function !A 2,(b) the lower-angle component !Aÿ 2, and (c) the higher-angle
component !A 2 for the axial divergence angle A = 5 .
J. Appl. Cryst. (2002). 35, 58±68

Figure 2

The effect of the scale transform ÿ ln cos  on the axial divergenceaberration functions shown in Fig. 1.
T. Ida and H. Toraya  Deconvolution of instrumental functions

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2=x 

2 2  2 tan ;8< 1=2 F  ÿx= F ÿ1=2
wF x for ÿ F < x < 0:0 elsewhere

38

with
x  G2 2  ln sin ;
gives the angle-independent formula
1=2
wA x  exp ÿx= A = A x= A 039
for x  0
elsewhere40

The pro®le of the instrumental function on the transformed
scale is shown in Fig. 4(b).
The Fourier transform of the instrumental function is givenbyand the Fourier transform is given byWA  

R1
ÿ1

WF  wA x exp 2ix dxhÿ1=2 i1=2
1=242 2 2A  1 1  i 42 2 2A  1
i1=2 o ÿÿ1=2
ÿ1:
2 42 2 2A  1

Fig. 3 shows the axial divergence and its component functions
plotted on the ln sin  scale. The higher-angle component
function !A  is found to keep the same pro®le on thisscale.
Since equations (30) and (31) have an identical structure toequation (1), the axial divergence effect can be eliminated forthe entire diffraction pattern by two step deconvolutions of!Aÿ 2 and !A 2.
3.2. Flat-specimen effectThe ¯at-specimen aberration function on the 2 scale isgiven by (Ida & Kimura, 1999a)8< 1=2 F 1  ÿ2= F 1 ÿ1=2
42
!F 2 
for ÿ F 1 < 2 < 0;:0 elsewhere
where we de®ne F  2F =4 for the equatorial divergenceangle F . The de®nition of 1 is given in equation (33). The
dependence of the pro®le of !F 2 is shown in Fig. 4(a).
By applying the scale transform G1 2 = ÿ ln cos , theangle-independent instrumental function is derived as

R1
ÿ1

wF x exp 2ix dx 1=2  F 1=2 fC2  F 1=2  ÿ iS2  F 1=2 g;

nhÿ

41

43

44
where C x and S x are the Fresnel functions de®ned byRx45
S x  sin t2 =2 dt
0

andC x 

Rx0cos t2 =2 dt:

46

Reliable computer routines to evaluate the Fresnel functions
are available (Press et al., 1986).
3.3. Sample transparency effectIf the sample has such high transparency that the penetration depth of the source X-ray is comparable with or longer
than the thickness of the sample, the angular dependence ofthe transparency effect becomes complicated (Ida & Kimura,
1999b). In such cases, it may be better to conduct an analysis
applying the model pro®le function with convoluted formula,rather than deconvolution based on Fourier methods.
When the penetration depth is suf®ciently shorter than the
thickness of the sample, the deconvolution can be easily
achieved. In this case, the instrumental function on the 2
scale is given by

Figure 4
Figure 3

The effect of the scale transform ln sin  on the axial divergenceaberration functions shown in Fig. 1.

62

T. Ida and H. Toraya  Deconvolution of instrumental functions

The dependence of the ¯at-specimen aberration function on thediffraction angle. (a) The instrumental function !F 2 for theequatorial divergence angle of F = 2 on the 2 scale, and (b) thatplotted on the ÿ ln cos  scale.

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J. Appl. Cryst. (2002). 35, 58±68

research papers!T 2 n

1= T  exp 2= T 
0for 2 < 0elsewhere

where T  sin 2=2R for the linear absorption coef®cient and the goniometer radius R. The angular dependence of thepro®le is shown in Fig. 5(a).
By applying the scale transform de®ned as
3 2 sin 2=248

andG3 2  ÿ ln 1  cos 2= 1 ÿ cos 2;49

the instrumental function independent of 2 is given byn50
wT x  1= T  exp x= T  for x < 00elsewherewithT 

Rÿ1 :

51The pro®le on the transformed scale is shown in Fig. 5(b).
The Fourier transform of wT x is given by
WT  R1
ÿ1

wT x exp 2ix dx

 1 ÿ 2i T = 1  42 2 2T :

wi  i tan i ;

47

56

where  is the fraction of the K 2 intensity, and i and i are
the wavelength and the full width at half-maximum, respectively, of the K i peak. Although many reports on theexperimental estimation of the spectroscopic pro®le of Cu K
radiation are available (Compton & Allison, 1935; Bearden &Shaw, 1935; HoÈlzer et al., 1997), the effective spectroscopic
parameters for a diffractometer may be different if a crystal
monochromator and/or pulse height analyser is used (Gangulee, 1970). For example, the parameters are approximately
Ê (Deslattes & Henins, 1973),
given as  = 1/3, 1 = 1.5405981 A
ÊÊ
Ê (Compton
2 = 1.54443 A, 1 = 0.00058 A, 2 = 0.00077 A& Allison, 1935).
For the purpose of elimination of the K 2 peak, theinstrumental function!X2 2  1 ÿ  2  fL 2; 22 ÿ 21 ; w2 ÿ w1 ;
57
where  x is the Dirac delta function, is used for the deconvolution. The total spectroscopic pro®le function is given by
the convolution of the single K 1 pro®le with !X2 2 as52

pX 2  fL 2; 21 ; w1   !X2 2:58Following Bragg's law  = 2d sin ,
3.4. Spectroscopic profile of Cu Ka radiation

The effect of the spectroscopic pro®le of the Cu K source
is approximated by the sum of two Lorentzian functions asfollows:
pX 2  1 ÿ fL 2; 21 ; w1   fL 2; 22 ; w2 ;53

fL x; xi ; wi   1=wi f1   x ÿ xi =wi 2 gÿ1 ;54

22  2 arcsin2 sin 21 =2=1 ;55

=2  d cos ;59=  1=2 tan :60which gives

Therefore, the scale transform required for the elimination of
the K 2 peak is G2 2 = ln sin , similarly to the higher-angle
component of the axial divergence aberration function.
Fig. 6 brie¯y illustrates the effect of the scale transform on
the spectroscopic pro®le function.
The instrumental function independent of the diffraction
angle is given by

Figure 5

The dependence of the sample-transparency aberration function on thediffraction angle. (a) The instrumental function !T 2 on the 2 scale,for the penetration depth of ÿ1 = 0.5 mm and the goniometer radius of R
= 185 mm, and (b) that plotted on the ÿ ln 1  cos 2= 1 ÿ cos 2scale.
J. Appl. Cryst. (2002). 35, 58±68

Figure 6

The dependence of the Cu K spectroscopic pro®le on the diffraction
angle. (a) The spectroscopic function pX 2 on the 2 scale, and (b) thatplotted on the ln sin  scale.
T. Ida and H. Toraya  Deconvolution of instrumental functions

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63

research papers
wX2 x  1 ÿ  x  fL x; x2 ; 2 ;61

x2  2 ÿ 1 =1

62

with
and
2 

2 ÿ 1 =21 ;63

tion. Although it is not dif®cult to deconvolute the effects of
the receiving slit and K 1 pro®le, they are left as a convolution
in the current analysis, because the deconvolution of the above
effects would result in very narrow deconvoluted pro®les forthe well crystallized Si sample, where severe loss of information may be caused by the ®nite interval of the data sampling.

the analytical Fourier transform of which is given by

4.1. Experimental

WX2   1 ÿ    exp ÿ2 2 cos 2x2   i sin 2x2 :
64

The standard Si powder (NIST SRM640b) was loaded into ahollow, of depth 0.549 (7) mm, of a glass holder with the ®lling
factor of 46.8 (6)%. The penetration depth of the Cu K X-ray
was estimated at 0.140 (2) mm, which is suf®ciently shorter
than the thickness of the sample.
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 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 by the symmetrically locatedsets of Soller slits. The pattern was scanned over the angularrange 20±155 (2), with a step length of 0.02 (2) and acounting time of 10 s stepÿ1.

3.5. The effects of the geometry of the beam source and thereceiving slit

An X-ray source with line focus is used for the Bragg±
Brentano diffractometer and the receiving slit is located at theother focal position. Since the beam source has ®nite widthalong the equatorial direction, it can slightly broaden the line
width according to the spatial intensity distribution of theradiation. The effects of the ®nite width of the beam source
and the receiving slit are both represented by instrumental
functions independent of the diffraction angle.
The instrumental function for the ®nite width of thereceiving slit is given by the following form with a rectangularshape:
n1= RS for ÿ RS =2 < x < RS =2
65
wRS x 
0elsewherewithRS  arctan WRS =R;66where WRS is the width of the receiving slit and R is thegoniometer radius. The Fourier transform of this function is
given by
WRS   exp i RS  ÿ 1=i RS :67The fundamental resolution of the diffractometer is usually
dominated by the ratio WRS =R, because the width of thereceiving slit is larger than the effective width of the beamsource. Practically, the fundamental resolution can bemodeled by the Gaussian function
wR x  1=1=2 R  expÿ x= R 2 ;

68

with the FWHM of 2 ln 21=2 R, which can be approximated by
RS given in equation (66).

4. Analysis of experimental data
In this section, the K 2 lines and the aberrations caused by theaxial divergence, ¯at specimen and sample transparency are
removed from the whole experimental diffraction pattern of Si
standard powder by the deconvolution technique. As a resultof the deconvolution, the peak pro®les in the lower-angle
region will be dominated by the fundamental resolution or thereceiving slit width, while those in the higher-angle region will
be dominated by the spectroscopic pro®le of the K 1 radia-

64

T. Ida and H. Toraya  Deconvolution of instrumental functions

4.2. Calculation

All calculations were conducted with commercial softwarefor data analysis (WaveMetrics, IgorPro) on a personal
computer (Apple, iBook, PowerPC G3 500 MHz). The built-in
cubic spline and fast Fourier transform functions were utilized.
The data were deconvoluted by the following three steps.
(i) The elimination of K 2 and the higher-angle componentof the axial divergence aberration on the ln sin  scale.
(ii) The elimination of the lower-angle component of theaxial divergence aberration and ¯at-specimen aberration on
the ÿ ln cos  scale.
(iii) The elimination of the aberration caused by the sample
transparency on the ÿ ln 1  cos 2= 1 ÿ cos 2 scale.
The total number of source data points was 6751, while alarger number of interpolated data were temporarily createdfor each deconvolution process; 65 536, 65 536 and 16 384 data,
including 190-, 216- and 13-point zero-padding, were created
for the ®rst, second and third steps, respectively.
On the elimination of the K 2 peak, the following parameters were assumed: the intensity ratio,  = 1/3; the difference of the wavelength between K 2 and K 1, 2 ÿ 1 =1 =0.002480; the slight broadening of the K 2 peak, 2 ÿ
1 =21 = 0.00006.
The FWHM values of the Gaussian ®lter for noise
suppression were 3.88  10ÿ5 on the ln sin  scale at the step(i), 4.16  10ÿ5 on the ÿ ln cos  scale at step (ii), and 5.17 
10ÿ4 on the ÿ ln 1  cos 2= 1 ÿ cos 2 scale at step (iii).
On the 2 scale, the broadening in the Gaussian FWHM isgiven byelectronic reprintÿ2G  ÿ21 tan2   ÿ22 = tan2   ÿ23 sin2 2;69

J. Appl. Cryst. (2002). 35, 58±68

research papers
diffraction pattern have become
narrow and symmetrical singlepeak pro®les.
4.4. Peak profile analysis

The diffraction pro®le dataobtained by the deconvolution
are analysed by a pro®le ®tting
and compared with the results of®tting to the raw data. We also
compare the least-squares results
applying the two different indicators of error, the square root of
the reciprocal of the diagonal
elements of the weight matrix,{n}, and the square root of the
diagonal elements of the covariance matrix, {"n}.
The raw data are ®tted byapplying the model function
synthesized by the convolutionof the pseudo-Voigt function(Thompson et al., 1987) with the
instrumental function (Ida &Figure 7
Kimura, 1999b), where the
The changes in the diffraction pro®les at each deconvolution step. (a) The raw data of the diffraction pro®lesof Si (NIST SRM640b), (b) the results of the deconvolution of K 2 and the higher-angle component of theGaussian FWHM is ®xed ataxial divergence effect, (c) the results of the deconvolution of the lower-angle component of the axial
0.003438 , which is the value
divergence effect and the ¯at-specimen effect, and (d) the results of the deconvolution of the samplecalculated by equation (66). The
transparency effect.
deconvoluted pro®les are simply®tted by the pseudo-Voigt function, where the Gaussianwith
FWHM is corrected by equation (69). In both of the ®tting
ÿ1  0:00444 ; ÿ2  0:00476 ; ÿ3  0:01481 ;analyses, the location of the K 1 peak 21, the integrated
intensity I, the Lorentzian width ÿL, and the constant backfor each value of 2.
ground b are treated as variable ®tting parameters. Even
The fast Fourier algorithm enables rapid calculation forthough the convoluted model pro®le function is evaluated byprocessing the large number of data. The deconvolution andan ef®cient algorithm (Ida & Kimura, 1999b), the ®tting of thethe evaluation of the errors have been completed in 20 s by apseudo-Voigt function to the deconvoluted data is much moreprogram written in the macro language IgorPro.
rapid in computation and easier to implement.
The results of the ®tting are shown in Figs. 8, 9 and 10 for
4.3. Results of deconvolution
the 111, 422 and 533 peaks, respectively. The deconvoluted
Fig. 7 shows how the peak pro®les of 111, 422, 533 re¯ecpeak pro®les are very well ®tted by the symmetric pseudotions are modi®ed by each step of the deconvolution.
Voigt function. The ®tting curves optimized for the two
At the ®rst step (Fig. 7b), where K 2 and the higher-angle
different indicators of error are almost identical. The differpart of the axial divergence effect are deconvoluted, it is foundence between the deconvoluted data and the ®tting curve liesthat not only the K 2 peaks for all re¯ections, but also theroughly within {n}, and no signi®cant systematic deviation ishigher-angle tail of the 533 re¯ection is signi®cantly removed
found. The indicator denoted by {"n} tends to overestimate theby the deconvolution. At the second step (Fig. 7c), where theerrors in the deconvoluted data, as is clearly seen in thelower-angle part of the axial divergence effect and the ¯atdifference plots for the 422 and 533 re¯ections.
specimen effect are deconvoluted, the lower-angle tail of the
The pro®le parameters optimized by the ®tting method are
111 re¯ection is removed, and also the peak shift is autolisted in Tables 1±3. The values obtained by the pro®le ®tting
matically corrected. The effect of the third step (Fig. 7d),to the deconvoluted data are close to those obtained by thewhere the sample transparency effect is deconvoluted, is notdirect analysis of the raw data. The slight difference can bevery signi®cant, but it seems that the lower angle tail of the 422
caused by the assumed spectroscopic parameters of the sourcere¯ection is reduced and the slight peak shift is corrected. As a
X-ray, which are not optimized for the current diffractometer.
result, all the peak pro®les obtained by the deconvolution of
The standard errors of the ®tting parameters, which are thethe instrumental functions from the experimental Si powder
square root of the diagonal elements of the covariance matrix
J. Appl. Cryst. (2002). 35, 58±68

T. Ida and H. Toraya  Deconvolution of instrumental functions

electronic reprint65

research papers
for the parameters, are shown in parentheses in the tables. In
the case of f"ÿ2
n g-weight ®tting to the deconvoluted data, theestimated error values are several to ten times larger thanthose estimated by the ®tting to the raw data, which corresponds to the overestimation of the error in the indicator {"n}.
The errors estimated by the fÿ2
n g-weight ®tting are similar to
or at most three times larger than those estimated from the®tting to the raw data. Therefore, it is concluded that thesquare root of the reciprocal diagonal elements of the weightmatrix (i.e. the inverse matrix of the covariance matrix), {n}, ismore appropriate than the square root of the diagonal
elements of the covariance matrix, {"n}, as the indicator of theerror in the deconvoluted data.
Tables 1±3 also list the R factors for pro®le ®tting, Rwp and
Rp, de®ned byP
2 1=2i wi Y 2i obs ÿ Y 2i calc 
;
70
Rwp 
P
2
i wi Y 2i obs
and

P
Rp 

Figure 8

i jY 2i obs ÿ Y 2i calc jP

i Y 2i obs;

71The results of pro®le ®tting to the Si 111 re¯ection. (a) The raw data
(open circles) and the ®tting curve (solid line), (b) the difference plot(solid line) of the ®tting to the raw data and the standard deviation(broken lines), (c) the deconvoluted data (open circles) and the ®tting
curve (solid line) optimized for the weight fÿ2
n g, (d) the error indicator{n} (broken lines) and the difference (solid line) in the fÿ2
n g-weight®tting, and (e) the error indicator {"n} (broken lines) and the difference(solid line) in the f"ÿ2
n g-weight ®tting.

where Y 2i obs is the observed pro®le intensity, Y 2i calc the
calculated intensity, and wi the weight for each datum. The
values listed in Tables 1±3 clearly show larger values of the R
factors for the ®tting to the deconvoluted data. However, it
seems that this is not a serious problem for the purpose ofevaluating the pro®le parameters. Lower resolution and/or

Figure 9

Figure 10

The results of pro®le ®tting to the Si 422 re¯ection. See Fig. 8 forde®nitions.

66

T. Ida and H. Toraya  Deconvolution of instrumental functions

The results of pro®le ®tting to the Si 533 re¯ection. See Fig. 8 forde®nitions.

electronic reprint

J. Appl. Cryst. (2002). 35, 58±68

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Table 1Pro®le parameters for Si 111 re¯ection.
21 is the position of the K 1 peak, I is the integrated intensity, ÿL is the
FWHM of the Lorentzian broadening, and b is the constant background.
Deconvoluted data21 ( )IÿL ( )b
Rwp (%)
Rp (%)

Raw data

fÿ2
n g

f"ÿ2
n g

28.47323 (9)10280 (15)0.0098 (2)89 (5)
3.2
2.2

28.47354 (5)9847 (23)0.0069 (2)92 (17)4.4
6.9

28.4735 (4)10313 (168)0.009 (2)65 (45)8.9
6.8

Table 2Pro®le parameters for Si 422 re¯ection.
See Table 1 for de®nitions.

squares ®tting becomes much faster in terms of computation
time and easier to implement. It is also expected that theeffects of ®nite size, dislocations and stacking faults in the
crystallites are sensitively detected, because the intrinsic peakpro®les caused by crystallographic imperfection can bereasonably extracted by the deconvolution method.

APPENDIX A

A1. Covariance matrix after deconvolution

The deconvolution of the instrumental function is equivalent to the convolution of the `inverse instrumental function'.
When the source data, deconvoluted data and inverse
instrumental function are expressed by fyn g, fzn g and fwnÿ1 g,
respectively, this relation is given by

Deconvoluted data21 ( )IÿL ( )b
Rwp (%)
Rp (%)

Raw data

fÿ2
n g

f"ÿ2
n g

88.0481 (3)1742 (8)0.0271 (7)92 (3)
8.1
5.5

88.0496 (6)1712 (21)0.023 (2)116 (12)8.4
8.7

88.049 (2)1733 (67)0.024 (5)113 (30)
11.3
9.1

zn 

m0

ÿ1
ym wnÿm
:72

The average of zn, hzn i, is related to the average of yn by
Nÿ1
 Nÿ1
P
P
ÿ1
ÿ1
ym wnÿm 
hym iwnÿm
:73
hzn i m0

m0Therefore, the covariance matrix Smn  of the deconvoluted
data is given by

Table 3Smn  zm ÿ hzm i zn ÿ hzn i
Nÿ1

Nÿ1
P
ÿ1 P
ÿ1
yk ÿ hyk iwmÿkyl ÿ hyl iwnÿl

Pro®le parameters for Si 533 re¯ection.
See Table 1 for de®nitions.

k0

Deconvoluted data21 ( )IÿL ( )b
Rwp (%)
Rp (%)Nÿ1
P

Raw data

fÿ2
n g

f"ÿ2
n g

136.8993 (8)786 (8)0.050 (3)
67 (4)
6.4
5.2

136.900 (2)797 (18)0.049 (6)57 (10)12.4
13.7

136.899 (6)807 (53)
0.05 (2)54 (25)14.9
14.0

higher background in data tend to give smaller R factors
despite the clear degradation in the informational quality. It issuggested that R factors for pro®le ®tting are not always good
indicators of either the quality of the source data or thevalidity of the theoretical models.

5. Conclusions



l0

Nÿ1
P
P Nÿ1
k0 l0ÿ1
ÿ1
yk ÿ hyk i yl ÿ hyl i wmÿk
wnÿl
:

74
When the covariance matrix of the source data is diagonal andthe diagonal element is given by fn2 g,
 2
k for k  l75
yk ÿ hyk i yl ÿ hyl i 
0for k 6 l
Equations (74) and (75) give the matrix elements of thecovariance matrix of the deconvoluted data as
Smn 

Nÿ1
P
k0

ÿ1
ÿ1
k2 wmÿk
wnÿk
:

76

A2. Least-squares fitting to the deconvoluted data

The effects of dominant aberrations in powder diffractometerswith conventional geometry are effectively deconvoluted fromthe experimental diffraction patterns, by utilizing a fast
Fourier method accompanied by scale-transform and interpolation of data. The errors in the deconvoluted data areapproximated by the square root of the reciprocal diagonal
elements of the weight matrix, which are estimated as thecorrelation of the reciprocal variance of the source data withthe squared instrumental function. As the peak pro®les in thedeconvoluted data are modelled by symmetrical functions forwell crystallized samples, the data analysis based on least-

where fn2 g is the variance and g(x) is the model function.
When the data fyn g are the convolution of fzn g and the
instrumental function fwn g, and the model g(x) is also the
convolution of a function f(x) and fwn g, they are related by thefollowing equations:

J. Appl. Cryst. (2002). 35, 58±68

T. Ida and H. Toraya  Deconvolution of instrumental functions

The weighted least-squares ®tting is identical to the
problem of minimizing the following quantity:
2 electronic reprint

Nÿ1
P
k0

kÿ2 yk ÿ g xk 2 ;

77

67

research papers
yk 

Nÿ1
P
n0

zn wkÿn

78

andg xk  

Nÿ1
P
n0

f xn wkÿn :

79


2
Nÿ1
P ÿ2 Nÿ1P
2
kzn ÿ f xn wkÿn
 



k0

n0

Nÿ1
P Nÿ1
P

k0

kÿ2

Nÿ1
P Nÿ1
P
m0 n0m0 n0zm ÿ f xm Tmn zn ÿ f xn ;

80

where Tmn is the weight matrix given by
Nÿ1
P
k0

kÿ2 wkÿm wkÿn :81

Let us calculate the matrix elements of the product Umn  ofthe weight matrix Tmn  and the covariance matrix Smn . The
matrix element of Umn  is given by
Umn 



Nÿ1
P
j0

Smj Tjn

Nÿ1
P Nÿ1
P
j0 k0Nÿ1
P Nÿ1
Pk0 llÿ1
ÿ1
k2 wmÿk
wjÿk

Nÿ1
P
l0

ÿ1
k2 lÿ2 wmÿkwlÿnlÿ2 wlÿj wlÿn

Nÿ1
P
j0

ÿ1
wjÿk
wlÿj :

82

When we apply the following relation to equation (82),

68



1
0

for
for

n0
n 6 0

83


Umn 

1
0

for
for

mn
m 6 n84

is derived; that is, Umn  is the unit matrix and Tmn  is theinverse matrix of Smn .

Referenceszm ÿ f xm zn ÿ f xn wkÿm wkÿn

Tmn 

k0

ÿ1
wk wnÿk
which is immediately derived from the de®nition of wnÿ1, thefollowing result,

Therefore, the objective function for the weighted leastsquares method is written as

Nÿ1
P

Nÿ1
P

T. Ida and H. Toraya  Deconvolution of instrumental functions

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