electronic reprint
Journal ofApplied
Crystallography
ISSN 0021-8898

Deconvolution of instrumental aberrations for synchrotron powder
X-ray diffractometry
T. Ida, H. Hibino and H. Toraya

Copyright © International Union of Crystallography
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J. Appl. Cryst. (2003). 36, 181–187

Ida et al.  Deconvolution for synchrotron diffractometry

research papers
Journal ofApplied
Crystallography

Deconvolution of instrumental aberrations forsynchrotron powder X-ray diffractometry

ISSN 0021-8898

T. Ida,* H. Hibino and H. Toraya
Received 2 September 2002
Accepted 18 November 2002

# 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

A method to remove the effects of instrumental aberrations from the whole
powder diffraction pattern measured with a high-resolution synchrotron powder
diffractometer is presented. Two types of asymmetry in the peak pro®les causedby (i) the axial-divergence aberration of the diffractometer (diffractometer
aberration) and (ii) the aberration of the monochromator and focusing optics on
the beamline (beamline aberration) are both taken into account. The method isbased on the whole-pattern deconvolution by Fourier technique combined withthe abscissa-scale transformation appropriate for each instrumental aberration.
The experimental powder diffraction data of LaB6 (NIST SRM660) measured
on beamline BL-4B2 at the Photon Factory in Tsukuba have been analysed by
the method. The formula of the scale transformation for the diffractometer
aberration has a priori been derived from the instrumental function withgeometric parameters of the optics. The strongly deformed experimental peakpro®les at low diffraction angles have been transformed to sharp peak pro®leswith less asymmetry by the deconvolution of the diffractometer aberration. The
peak pro®les obtained by the deconvolution of the diffractometer aberration
were modelled by an asymmetric model pro®le function synthesized by the
convolution of the extended pseudo-Voigt function and an asymmetriccomponent function with an empirical asymmetry parameter, which werelinearly dependent on the diffraction angle. Fairly symmetric peak pro®les havebeen obtained by further deconvolution of the empirically determinedasymmetric component of the beamline aberration.

1. Introduction
Synchrotron radiation is a very powerful tool for the purposeof structure analysis by powder diffractometry, because thelow divergence of the source X-ray beam provides fairly highresolution in powder diffraction data. The sharp and simple
diffraction peak pro®les of a well crystallized sample
measured with a synchrotron light source enable precise
estimation of lattice constants (Hart et al., 1990) and accurate
structure determination (Estermann & Gramlich, 1993). The
sharp instrumental function also provides improved sensitivity
in size±strain estimation by line broadening analysis (Cox,1992).
It is strongly desired to establish the accurate formula of the
instrumental function for further improvement in the peakpro®le analysis. Previously, we have developed a mathematicalmodel for the instrumental function for a high-resolution
synchrotron powder diffractometer equipped with a crystal
analyser (Ida et al., 2001). However, it has been dif®cult toconstruct a model for the effects of the spectral distribution
and beam divergence of the source X-ray on the diffraction
peak pro®les, because those effects are not only affected by
J. Appl. Cryst. (2003). 36, 181±187

many parameters concerning the monochromator andfocusing optics on a beamline at a synchrotron facility, but arealso mutually correlated in a complex manner owing to thehigh coherence of the light source.
Recently, we have proposed a novel Fourier-based method
to eliminate the effects of instrumental aberrations from thewhole powder diffraction pattern measured with a conventional Bragg±Brentano powder X-ray diffractometer (Ida &Toraya, 2002). As the method is based on the analytical modelfor the effects of instrumental aberrations, the whole powder
diffraction pattern can immediately be deconvoluted without
the use of any standard materials.
In this study, we have applied the Fourier-based method toeliminate the effect of the axial-divergence aberration of ahigh-resolution synchrotron diffractometer (diffractometer
aberration) from the experimental diffraction data of LaB6
powder measured on beamline BL-4B2 at the Photon Factoryin Tsukuba, and investigated the effects of the beamline optics(beamline aberration) on the peak pro®les. We have also tried
to remove the residual asymmetric feature caused by thebeamline aberration, applying an empirical model for theasymmetry.
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2. Diffractometer aberration
In this section, the analytical formulae necessary for eliminating the diffractometer aberration from the entire diffraction data are described.

WA  with
 2 =2;2

A  cot 2  tan A ;3andfor any diffraction angles 2, where  is the full width at halfmaximum (FWHM) of the angular distribution of the axial
divergence limited by the Soller slits, and A is the Bragg
angle at the crystal analyser.
2.2. Scale transform of the instrumental function

Even though the width of the instrumental function!A 2 depends on the diffraction angle 2, an abscissaindependent formula can be derived by applying an appropriate scale transform (Ida & Toraya, 2002). Here we apply thefollowing scale-transform function GA 2,  GA 2 2 sin A ÿ cos A ln sin 2 tan A  cos 2 cos A ;4WA  wA  

A !A 2;6

the abscissa-independent formula of the instrumental function
is given by8< ÿ1  ÿ= ÿ1=2 ÿ 17
wA  
for ÿ <  < 0;:0 elsewhere:
Therefore, the intensity data multiplied by A accompaniedwith the transformed abscissa  are considered as a convolution with the abscissa-independent instrumental function
wA .
2.3. Fourier transform of the instrumental function

The Fourier transform of the instrumental function wA ,

182

Ida et al.  Deconvolution for synchrotron diffractometry8ÿ1=2 fC2 1=2  ÿ iS2 1=2 g
9where C(x) and S(x) are the Fresnel functions de®ned byS x 

Rx0sin t2 =2 dt

10

cos t2 =2 dt:

11

and
C x 

Rx
0

Reliable computer routines to evaluate Fresnel functions areavailable (Press et al., 1986).

3. Method of data processing
In this section, a series of actual procedures of the deconvolution are presented. Further details have been described
elsewhere (Ida & Toraya, 2002).
3.1. Scale transform of data

Since the polarization factor can be neglected in synchrotron diffractometry, only the corrections for the angularvelocity and powder diffraction intensity factors (Lipson &Langford, 1999) given byfcorr 2  1= sin  sin 2

12

are taken into account.
First, the following scale transforms for the diffraction angle
f2m g, intensity fSm g and error data fSm g (m = 0, . . ., M ÿ 1)
are applied:ÿ GA 2m ;

13

sm

ÿ Sm A 2m =fcorr 2m ;

14

sm

ÿ Sm A 2m =fcorr 2m ;

15

m

5

When we de®ne the instrumental function wA  for thetransformed abscissa  bywA x exp 2ix dx;

ÿ 2i ÿ1 1 ÿ exp ÿ2i ;in order to satisfy the following relation (Ida et al., 2001):
=2  1= cot 2  tan A   1= A :ÿ1

is solved directly from equation (7). The solution is given by

2.1. Instrumental function of the diffractometer

The axial-divergence effect limited by a set of Soller slits on
the diffracted beam side is the only signi®cant aberration of ahigh-resolution powder diffractometer (Ida et al., 2001). The
instrumental function for the axial-divergence aberration is
given by
8
ÿ1
ÿ1=2
<
ÿ 1
A   ÿ2=A
1
!A 2 
for ÿ A < 2 < 0;:0elsewhere;

R1

where fm g is the abscissa, fsm g the ordinate and fsm g theerror data.
3.2. Interpolation

N-point equidistant data are created by interpolating Mpoint non-equidistant data on the transformed scale. In orderto keep the statistical properties of the data unchanged by theinterpolation, the error data are modi®ed before the interpolation process by the following equation:
1=22m1 ÿ 2mÿ1 N
sm :
16
s0m ÿ
2 Mÿ1 ÿ 0  A 2m 
The equidistant data sets of abscissa fxn g, ordinate fyn g anderror fyn g (n = 0, . . ., N ÿ 1) are created by applying cubic
spline interpolation to the sets of fm g, fsm g and fs0m g.

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J. Appl. Cryst. (2003). 36, 181±187research papers
3.3. Deconvolution by Fourier method

3.4. Error estimation by Fourier method

As fyn g is a set of equidistantly spaced data, the Fouriertransform fYk g,

The errors fzn g attached to the deconvoluted data fzn g aremodelled 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 (Ida & Toraya, 2002),
which are evaluated by the following equations:

ÿ1=2
Nÿ1
P ÿ2 2 
Vk Wk  exp ÿ2ikn=N;20zn  N ÿ1

Yk 

Nÿ1
P
0

yn exp 2ikn=Nk  ÿN=2; . . . ; N=2;17

can be rapidly evaluated by fast Fourier transform (FFT).
The discrete Fourier transform fWk g of the instrumental
function wA x is calculated by
Wk  WA k=Nxk  ÿN=2; . . . ; N=2;

k0

18

Vkÿ2 where x is the spacing of the data fyn g, that is, x = xn+1 ÿ xn.
Periodicities of Yk = YN+k and Wk = WN+k are assumed.
According to the convolution theorem, the deconvoluted
data fzn g are given byzn  N ÿ1

Nÿ1
P
k0

Yk =Wk  exp ÿ2ikn=N;

n0

Wk2 
wn  N ÿ119

which are also calculated by FFT method.

Nÿ1Pyn ÿ2 exp 2ikn=N;

Nÿ1
P
n0
Nÿ1
P
k0

21

w2n exp 2ikn=N;22

Wk exp ÿ2ikn=N;23

where Wk2  denotes the complex conjugate of Wk2.
3.5. Inverse scale transformation and interpolation

In order to make comparison of the deconvoluted andsource data easier, the deconvoluted data are mapped onto
the original scale again. Intensity ftm g and error ftm0 g data
mapped onto the sampling points fm g are created by cubic
spline interpolation from the deconvoluted intensity fzn g anderror fzn g data. The mapped error data ftm0 g are corrected
by

ÿ1=2
2m1 ÿ 2mÿ1 N
tm0
24
tm ÿ
2 Mÿ1 ÿ 0  A 2m 
to cancel the change in the statistical properties introduced by
the interpolation. The ®nal deconvoluted intensity fTm g anderror fTm g data for the diffraction angles f2m g are calculated by the following equations:Tm

ÿ tm fcorr 2m = A 2m 

25

Tm

ÿ tm fcorr 2m = A 2m :

26

and

4. Analysis of experimental data
4.1. Experimental

Figure 1(a) The experimental peak pro®le of LaB6 111 re¯ection measured with aÊ ). (b) The data given by the deconvolution
synchrotron X-ray ( = 0.707 Aof the instrumental function of the diffractometer (open circles) and the
optimized curve (solid line). (c) The errors in the deconvoluted data
estimated by Fourier method (broken line) and difference (solid line) inthe ®tting shown in (b). (d) The data given by the deconvolution of theasymmetry caused by beamline optics (open circles) and the optimized
curve (solid line). (e) The errors in the deconvoluted data (broken line)
and difference (solid line) for the ®tting shown in (d).
J. Appl. Cryst. (2003). 36, 181±187

The diffraction data of standard LaB6 powder [National
Institute of Standards & Technology, SRM660, a =Ê ] were collected with a high-resolution
4.15695 (6) A
synchrotron powder diffractometer, MDS (Toraya et al., 1996),
on beamline BL-4B2 at the Photon Factory (PF) in Tsukuba.
The PF storage ring stored 2.5 GeV electron beams and was
operated in multi-bunch mode. The beam radiated from abending magnet is used on beamline BL-4B2. The whiteradiation is monochromated with an Si(111) ¯at doublecrystal monochromator and focused with a cylindrical mirror(1000 mm in length, 62.5 mm in curvature) onto the specimen
Ida et al.  Deconvolution for synchrotron diffractometryelectronic reprint

183

research papers
centred at the diffractometer. Incident slits were inserted
between the focusing mirror and the specimen. The monochromator, focusing mirror, incident slits and specimen werelocated at distances of about 17, 18, 28 and 29 m from thebending magnet, respectively. The use of the cylindricalfocusing mirror and the rectangular cross section restricted bya couple of incident slits may cause asymmetric divergenceand spectral broadening of the incident beam.
The LaB6 powder was loaded into a ¯at aluminium sample
holder, which was mounted on an attachment, rotated at 1
revolution sÿ1 about the normal to the sample surface during
the measurements. The sample face was inclined by 4.885 tothe incident beam. The diffractometer was equipped with a setof Soller slits, with an axial-divergence angle  = 1 , and a
Ê ) adjusted at A =Ge(111) crystal analyser (d111 = 3.26638 AÊ
6.2 for the wavelength 0.707 A on the diffracted-beam side.
The incident X-ray beam was restricted to 2.5 mm in widthand 1 mm in height with a couple of incident slits.
The diffraction pattern was scanned over the angular range
9.0±37.8 (2), with a step length of 0.004 (2) and a counting
time of 4 s stepÿ1.
4.2. Deconvolution of the diffractometer aberration

The deconvolution by the method described in xx2 and 3
was applied to the experimental powder diffraction data. The

Figure 2(a) The experimental peak pro®le of LaB6 220 re¯ection measured with aÊ ). See the caption of Fig. 1 for (b), (c), (d)synchrotron X-ray ( = 0.707 A
and (e).

184

Ida et al.  Deconvolution for synchrotron diffractometry

number of sampling points of the interpolated data was N =32768. Figs. 1±3(a) show the experimental peak pro®les of 100,
220 and 321 re¯ections, and Figs. 1±3(b) show the dataobtained by the deconvolution of the diffractometer aberration as open circles. Note that the whole diffraction pattern
was simultaneously treated by the method, though only
restricted regions are shown in Figs. 1±3.
As shown in Figs. 1(a) and 1(b), the heavily distorted peakpro®les at low diffraction angles are transformed to sharp andalmost symmetric pro®les by the deconvolution of the
instrumental function of the diffractometer, which indicatesthat the peak pro®les at low diffraction angles are dominatedby the axial-divergence aberration of the diffractometer.
In contrast, the peak pro®les at higher diffraction angles
show signi®cant asymmetry, having longer tails on the higherangle side of the peak. The higher-angle peak pro®les are only
slightly changed by the deconvolution of the diffractometer
aberration, as can be seen in Figs. 2(a), 2(b), 3(a) and 3(b). The
residual asymmetry in the deconvoluted pro®les is naturally
ascribable to the beamline aberration.
4.3. Evaluation of the beamline aberration

It is dif®cult to deduce a mathematical model from thegeometry of the beamline optics for the following reasons: (i)Figure 3(a) The experimental peak pro®le of LaB6 321 re¯ection measured with aÊ ). See the caption of Fig. 1 for (b), (c), (d)synchrotron X-ray ( = 0.707 A
and (e).

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J. Appl. Cryst. (2003). 36, 181±187research papers
the effects of the monochromator and focusing mirror on thediffraction peak pro®les are affected by many geometricalparameters of the optics; (ii) the spectral distribution anddivergence of the source X-ray beam are heavily correlated;(iii) the wavelength, focusing condition and cross section of
the X-ray beam are frequently changed; and (iv) the characteristics of the X-ray beam are slightly varied on each
injection of the electrons to the storage ring.
In this section, we analyse the asymmetry caused by thebeamline aberration in the experimental peak pro®lesextracted by the deconvolution of the diffractometer aberration. The dependence of the extracted asymmetry on thediffraction angle is estimated by a curve-®tting method
applying an asymmetric model peak pro®le function. We use
the model function synthesized by the convolution of theextended pseudo-Voigt function (Ida et al., 2000) with thefollowing exponential function as an asymmetric component:( ÿ1
E exp ÿ2= E 27
!E 2 
for 0 < 2;0elsewhere:
The convoluted model pro®le has been calculated by applying
an ef®cient algorithm for the numerical integral (Ida, 1998; Ida& Kimura, 1999). The extended pseudo-Voigt functions can beconsidered to be virtually identical to the Voigt function
de®ned as the convolution of the Lorentzian and Gaussianfunctions (Ida et al., 2000). Only the parameter E in theasymmetric component function speci®es the asymmetry of
the pro®le caused by the beamline aberration.
The model pro®le function includes the peak position 20 ,integrated intensity I, Lorentzian FWHM ÿL , Gaussian
FWHM ÿG , asymmetry parameter E , and constant background b as adjustable parameters. The parameters are optimized by a least-squares method for the data obtained by thedeconvolution of the diffractometer aberration. The peakpro®le analysis has been conducted for all the diffraction
peaks in the measured range, from 100 to 321 re¯ections.
Figs. 1±3(b) show the optimized asymmetric model peakpro®les for 100, 220 and 321 re¯ections as solid curves. The

optimized pro®les are very close to the data obtained by thedeconvolution of the diffractometer aberrations (opencircles). The differences are comparable with the errors estimated by the Fourier method, and no systematic behaviours
are found in the difference, as shown in Figs. 1±3(c). We canconclude that the model applied here has been validated
within the experimental errors, even though the geometricdescription of the beamline optics has not yet been established.
The dependence of the optimized asymmetry parameter Eon the diffraction angle is shown in Fig. 4. The optimized
linear dependence is given byE  a  b 2;The dependence of the asymmetry parameter ( E ) for the beamline
aberration upon the diffraction angle (open circles). Error bars indicate
1 the standard uncertainty. The optimized linear dependence is shownby the thin line.
J. Appl. Cryst. (2003). 36, 181±187b  0:000248 9:
28

The linear dependence on the diffraction angle 2, rather thantan  or 1= cos , suggests that the asymmetry is caused by the
instrumental aberrations of the beamline optics, which areconsidered to be mutually correlated effects of the spectral
distribution and divergence of the source X-ray beam.
4.4. Elimination of experimentally evaluated asymmetry

Assuming the linear dependence of the asymmetry parameter E on the diffraction angle, the residual asymmetry canalso be eliminated by the deconvolution based on the Fourier
and scale-transform methods from the whole diffraction
pattern.
We use the scale-transform function
GE 2  1=b ln j2  a=bj

29in order to satisfy the relationdGE 2=d 2  1=a  b 2  1= E 2:The transformed instrumental function is given by
exp ÿx for x > 0,wE x 0elsewhere,

30

31

and the Fourier transform is
WE  Figure 4a  ÿ0:00093 19 ;R1
ÿ1

wE x exp 2ix dx  1= 1 ÿ 2i:32

The actual procedures for the deconvolution of the beamline
aberration are similar to those described in x3, except thatGA 2, A 2 and WA  are substituted by GE 2, E 2
and WE .
The results of the deconvolution are shown as open circles
in Figs. 1±3(d). The sampling-point number of the interpolated
data was N = 16384. It has been found that sharpened andfairly symmetrized pro®les are obtained by the deconvolution
for all the diffraction peaks.
Signi®cant peak shifts by the deconvolution are also
observed in Figs. 1±3. This indicates that the experimental
peak positions are shifted by the instrumental aberrations, andthe deconvolution automatically corrects the systematic
errors, which enables accurate determination of the intrinsic
peak positions (Ida & Kimura, 1999).
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Table 1

Optimized parameters for the symmetrized diffraction peak pro®les.
hkl

20 ( )IÿL ( )

ÿG ( )b

RwP (%)

100
110
111
200
210
211
220
300/221
310
311
222
320
321

9.7615 (1)13.8185 (1)16.9439 (2)19.5888 (2)21.9257 (2)24.0505 (2)27.8421 (4)29.5678 (2)31.2061 (3)32.7703 (4)34.2707 (11)43.5200 (9)37.1116 (4)316 (9)698 (14)357 (11)202 (8)469 (13)263 (10)112 (7)331 (12)235 (10)156 (8)27 (4)
39 (5)
201 (10)0.0051 (6)0.0055 (7)0.0056 (10)0.0060 (14)0.0058 (11)0.0075 (16)0.0059 (29)0.0066 (18)0.0080 (18)0.0056 (31)0.0059 (107)0.0083 (75)0.0098 (26)0.0049 (5)0.0061 (5)0.0071 (7)0.0066 (10)0.0078 (7)0.0082 (11)0.0093 (17)0.0100 (11)0.0088 (13)0.0110 (17)0.0127 (57)0.0122 (49)0.0112 (19)40 (17)74 (28)41 (21)41 (18)42 (26)37 (21)18 (15)33 (25)35 (22)18 (19)32 (14)23 (14)27 (23)

4.0
4.1
3.5
5.1
5.0
8.4
5.7
7.2
6.1
7.8
14.0
15.6
9.0

4.5. Analysis of symmetrized peak profiles

The symmetrized diffraction peak pro®les obtained by the
double deconvolutions have been analysed by a curve-®tting
method with the extended pseudo-Voigt function (Ida et al.,2000) as a pro®le model. The model pro®le function includes
the peak position 20 , integrated intensity I, Lorentzian
FWHM ÿL , Gaussian FWHM ÿG , and constant background b
as adjustable ®tting parameters.
The optimized values of the pro®le parameters are listed inTable 1. All the optimized values seem to be reasonable exceptthat considerably larger errors are estimated as compared withtypical results of pro®le analysis for raw diffraction data.
The reliability factor RwP , de®ned by the following equation, is also listed in Table 1.
P
RwP 

2 1=22m Tm ÿ T 2m calc  = Tm ;P 2
2m Tm = Tm 

33

where fTm g and fTm g are the deconvoluted intensity dataand error data estimated by the Fourier method, respectively,
and T 2m calc is the intensity value calculated with the optimized symmetric model function.

Figure 5

The dependence of the Lorentzian width (ÿL ) on the diffraction angle.
Error bars indicate 1 the standard uncertainty. The error bar for the 222
re¯ection (34.27 ) is off the scale.

186

Ida et al.  Deconvolution for synchrotron diffractometry

The optimized model pro®les for 100, 220 and 321 re¯ections are shown as solid curves in Figs. 1±3(d). The errors
estimated by the Fourier method in the symmetrized pro®les
are shown as broken lines, and the difference plots for thesymmetric model are drawn as solid lines in Figs. 1±3(e). The
differences are comparable with the estimated errors, andshow no systematic behaviour.
The dependences of the optimized Lorentzian FWHM ÿL
and Gaussian FWHM ÿG , obtained by the pro®le-®tting
method, on the diffraction angle are shown in Figs. 5 and 6.
The values and dependences of those parameters are ascribable to (i) the intrinsic size and strain broadening and (ii) the®nite resolution of the beamline optics.
Although it has been suggested that SRM660 LaB6 shows
small strain broadenings (Rasberry, 1989), quantitativedescriptions of the broadenings have not been reported. The
plots in Figs. 4 and 5 indicate that ®nite values of about 0.002±
0.005 remain at the diffraction angle 0 for both Lorentzianand Gaussian FWHM values. As the wavelength of the X-ray
Ê , those angular widths correspond to the coherenceis 0.707 A
length of about 1 mm, which is smaller than the typical particle
size of 10 mm of the LaB6 (SRM660) powder (Rasberry, 1989).
The symmetric component of the experimental pro®le should

Figure 6

The dependence of the Gaussian width (ÿG ) on the diffraction angle.
Error bars indicate 1 the standard uncertainty.

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J. Appl. Cryst. (2003). 36, 181±187research papers
be attributed not only to the intrinsic broadening of thesample, but also to the ®nite resolution of the beamline optics.
The symmetric instrumental broadenings could be experimentally determined for any given arrangements of thebeamline optics, if a standard material with well de®ned
intrinsic peak pro®les were available. Highly sensitive peakpro®le analysis will be achieved by correction or further
deconvolution of the symmetric instrumental broadenings.

5. Conclusion
We have developed a deconvolution method to remove thediffractometer aberration and the asymmetry caused by thebeamline optics from the whole powder diffraction patternmeasured with a high-resolution synchrotron diffractometerequipped with a crystal analyser. The fairly symmetric peakpro®les obtained by the deconvolution are well approximatedby the Voigt pro®le, and the errors in the deconvoluted data
are also reasonably estimated by a Fourier-based method. The
deconvolution method provides automatic correction of thesystematic errors in the peak position to improve the accuracy
in the lattice-parameter determination. As the elimination of
the asymmetry by the deconvolution certi®es the validity ofusing symmetric model pro®le functions, the ef®ciency of
structure analysis based on the Rietveld method will be

J. Appl. Cryst. (2003). 36, 181±187

enhanced. Highly sensitive peak pro®le analysis will immediately be achieved, if a well de®ned standard material for the
intrinsic peak pro®les is available.

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