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Journal ofApplied
Crystallography
ISSN 0021-8898

Editor: Gernot Kostorz

Symmetrization of diffraction peak profiles measured with ahigh-resolution synchrotron X-ray powder diffractometer
T. Ida and H. Hibino

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J. Appl. Cryst. (2006). 39, 90–100

T. Ida and H. Hibino ¯ Symmetrization of diffraction peak profilesresearch papers
Journal ofApplied
Crystallography
ISSN 0021-8898

Received 7 November 2005
Accepted 4 December 2005

Symmetrization of diffraction peak profiles
measured with a high-resolution synchrotron X-ray
powder diffractometer
T. Ida* and H. Hibino
Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi, 507-0071,
Japan. Correspondence e-mail: ida.takashi@nitech.ac.jp# 2006 International Union of Crystallography
Printed in Great Britain – all rights reserved

The asymmetry of diffraction peak profiles observed with a high-resolution
synchrotron powder X-ray diffractometer has been successfully removed by adouble deconvolution method. In the first step, the asymmetry caused by theaxial divergence aberration of the diffractometer is removed by a whole-patterndeconvolution method based on an a priori theoretical model for the aberration.
In the second step, the residual asymmetry, the origin of which can be ascribed tothe aberrations of the beamline optics, is also removed by a whole-patterndeconvolution method, based on an empirical model derived from the analysisof experimental diffraction peak profiles of a standard Si powder (NIST
SRM640b). The beamline aberration has been modelled by the convolution of apseudo-Voigt or Voigt function with an exponential distribution function. It has
been found that the angular dependence of the asymmetry parameter in the
exponential function is almost proportional to tan , which supports the idea thatthe residual asymmetry should be ascribed mainly to the intrinsic asymmetry inthe spectroscopic distribution of the source X-ray supplied by the beamline
optics of the synchrotron facility. Recently developed procedures of wholepattern deconvolution have been improved to treat the singularity of the
instrumental function in the measured angular range. Formulae for the wholepattern deconvolution based on the Williamson–Hall-type dependence of thewidth parameter of the instrumental function have also been developed. The
method was applied to the diffraction intensity data of a standard ZnO powder
sample (NIST SRM674) measured with a high-resolution powder diffractometeron beamline BL4B2 at the Photon Factory. The structure parameters of ZnO
were refined from the integrated peak intensities, which were extracted by anindividual profile fitting method applying symmetric profile models. The refinedstructure parameters coincide fairly well with those obtained from single-crystal
data.

1. Introduction
Previously, we have reported the exact theoreticalformulae of the axial-divergence aberration function for ahigh-resolution synchrotron X-ray powder diffractometerwith a crystal analyser (Ida et al., 2001), and also
proposed an analytical method based on a Fourier analysis
to remove the aberration from the observed data (Ida,
Hibino & Toraya, 2003). The peak profiles obtained by
the deconvolution of the axial-divergence aberration haveshown residual asymmetry, which could be ascribed to theaberration of the monochromator and focusing optics on
the beamline (beamline aberration). However, the originof the residual asymmetry was not clear, except that thedependence of the residual asymmetry on the diffraction
angles was found to be almost linear within a limited
angular range of 10  2  40 .

90

doi:10.1107/S0021889805040318

The application of the deconvolution method was restricted
to the low-angle data in our previous study, partly because theoriginal formula of the whole-pattern deconvolution withabscissa-scale transformation (Ida, Hibino & Toraya, 2003) for
the axial-divergence aberration with a crystal analyser with a
Bragg angle of A has a singularity at the diffraction angle 2= 90 + A, which means that an infinitely large number of
temporary data would be required near the singular point on
the numerical process. Furthermore, interpretation of the datameasured with a multiple-detector system (MDS) on beamline
BL4B2 at the Photon Factory was complicated because of thepossible differences in the instrumental functions of different
detectors.
Recently, we have developed an automatic method to adjust
the variation of the instrumental functions of multiple detectors (Ida, 2005), which greatly simplifies the interpretation of
the data measured with the MDS.

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J. Appl. Cryst. (2006). 39, 90–100

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In this study, a modified formula for the whole-patterndeconvolution, which can be applied to remove the axialdivergence aberration from all the diffraction intensity dataover the singular point, has been developed. Eleven diffraction peak profiles of a standard Si powder (NIST SRM674b) in
the range 24.0  2  148.4 , measured at a wavelength of  =1.306 Å, are analysed in detail. The formula for removing thebeamline aberration is modified according to the results of thedetailed analysis of the diffraction peak profiles of the standard sample. The improved deconvolution methods to remove
the axial-divergence and beamline aberrations are applied tothe diffraction intensity data of a standard ZnO powder (NIST
SRM674), measured in a multiple-detector mode, and thesymmetric features of the experimental peak profiles are
discussed.

2. Theory
2.1. Axial-divergence aberration function

The effect of axial divergence, limited by a set of Soller slitson the diffracted-beam side of the goniometer, is the only
significant aberration arising from the optics of a high-resolution synchrotron X-ray powder diffractometer (Ida et al.,2001). The instrumental function for the axial-divergenceaberration is given by8< jA j1 ½ð2=A Þ1=2  1!A ð2Þ ¼
for  1 < 2=A < 0;:0elsewhere;ð1Þ

A ¼ ð2 =2Þðcot 2 þ tan A Þ;

ð2Þ

with

where  is the full width at half-maximum (FWHM) of theangular distribution of the axially divergent beam restricted by
the Soller slits, and A is the Bragg angle at the crystal
analyser (Ida et al., 2001). Note that the instrumental function
approaches the Dirac delta function at the singular point 2s= /2 + A, where the characteristic width A of the aberration function approaches zero. The above formulae are valid
for both positive and negative value of A; that is, they cover
all the possible diffraction angles, 0 < 2 < , including therange /2 + A < 2 < , except the singular point at 2s.
Although the width of the instrumental function depends
on the diffraction angle 2, abscissa-independent formulae of
the instrumental function can be derived by applying thefollowing function for the scale transformation about theabscissa from 2 to  (Ida & Toraya, 2002): ð2= Þ cos A ;

¼  ð2 sin A  cos A ln j sin 2 tan A þ cos 2jÞ ð2=2 Þ cos A ;

ð4Þfor 2s < 2 < , both of which satisfy the following relation:
=2 ¼ 1=jA j:

ð5ÞThis means that the width and the shape of the aberration
function become constant on the transformed scale  on theabscissa.
The scale-transform function GA(2) diverges to 1 and1 at the singular point 2s for the ranges 2 < 2s and 2s< 2, respectively. The singularity or infinity can be avoided byseparating the data into three parts in numerical calculations,
namely the lower- and higher-angle regions, and the intermediate region around 2 ’ 2s. The method to treat theintermediate region is discussed in the following section indetail.
The abscissa-independent formula of the instrumental
function is then given by8< ðÞ1=2  1
ð6Þ
wA ðÞ ¼
for  1 <  < 0;:0elsewhere;for 2 < 2s, and8< ðÞ1=2  1
wA ðÞ ¼
for 0 <  < 1;:0
elsewhere;

ð7Þ

for 2 > 2s.
The Fourier transform of the abscissa-independent instrumental function wA(), defined byZ 1WA ðÞ ¼
wA ðÞ expð2iÞd;
1for the range 2 < 2s is given byWA ðÞ ¼ 1=2 ½Cð21=2 Þ  iSð21=2 Þ ð2iÞ1 ½1  expð2iÞ;ð8Þfor  > 0, andWA ðÞ ¼ jj1=2 ½Cð2jj1=2 Þ þ iSð2jj1=2 Þ ð2iÞ1 ½1  expð2iÞð9Þfor  < 0, where C(x) and S(x) are the Fresnel functions
defined byCðxÞ 

Rxcosðt2 =2Þ dt

ð10Þsinðt2 =2Þ dt:

ð11Þ

0

and

 ¼ GA ð2Þ
¼ ð2 sin A  cos A ln j sin 2 tan A þ cos 2jÞ
2

 ¼ GA ð2ÞSðxÞ 

Rx0

ð3Þ

for 0 < 2 < 2s, and

The Fourier formulae for 2 > 2s are simply given by the
Hermite conjugate of WA() for 2 < 2s.

J. Appl. Cryst. (2006). 39, 90–100

T. Ida and H. Hibino  Symmetrization of diffraction peak profiles

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When the angular dependence of  is assumed to be given

2.2. Treatment of the singularity in the axial-divergenceaberration function

by

The axial-divergence aberration function !A(2) has nonzero values only in the range |A| < 2 < 0 for 2 < 2s, and0 < 2 < |A| for 2 > 2s, as shown in equation (1). Then the
maximum width of the function !A(2) immediately corresponds to the valuejA j ¼ ð2 =2Þj cot 2 þ tan A j;
jA j ’  j2  2s j=2 cos A :2

ð12Þ

If the width of the instrumental function is much narrower
than the interval of the measurement step, the convolution ordeconvolution of the instrumental function will not affect theintensity profiles, practically. In the current method, the datain the range where the width of the instrumental function isnarrower than 1/10 of the measurement step are left
unchanged and simply combined with the deconvoluted datain the lower- and higher-angle ranges. Actually, the width of
the axial-divergence instrumental function was narrower than
0.0005 in the range 98.4 < 2 < 104.6 , for  = 1 and A =11.535 . The scheme for the segmentation, deconvolution andcombination of data is illustrated in Fig. 1.

ð14Þ

which is identical to the Williamson–Hall-type dependence on
the diffraction angle (Williamson & Hall, 1953), singularity
may appear at sin  = X/Y for the case Y/X < 1.
Formally, the singular point 2s is calculated by2s ¼ 2 arcsinðX =Y Þwhich can be approximated for a small deviation from 2s as2

 ¼ E ð2Þ ¼ X sec  þ Y tan ;

ð15Þfor Y/X < 1. The formulae for the scale transform, theabscissa-independent instrumental functions and the Fouriertransform are given byGE ð2Þ ¼ ð2=Y Þ ln jX =Y þ sin j;wE ðÞ ¼

expðxÞ
0

for x > 0,
elsewhere,

WE ðÞ ¼ 1=ð1  2iÞ

ð16Þ
ð17Þ
ð18Þ

for the case  > 0, andGE ð2Þ ¼ ð2=Y Þ ln jX =Y þ sin j;wE ðÞ ¼

for x < 0,elsewhere,expðxÞ
0

ð19Þ
ð20Þ

2.3. Asymmetry caused by the beamline aberration

The asymmetry caused by the beamline aberration ismodelled by the convolution with an exponential distributionfunction with the following form (Ida, Hibino & Toraya, 2003):
8 1< jj expð2=Þð13Þ!E ð2Þ ¼
for 2= > 0;:0elsewhere:
The parameter  can be treated as the measure of asymmetry.
The longer tail on the lower- or higher-angle side of a peakcorresponds to the negative or positive sign of , respectively.

Figure 1
Scheme for treating data near the singular point 2s of instrumental
functions in the whole-pattern deconvolution method. Three sets ofpartly overlapping segmented data are created from the initial data. The
lower- and higher-range data are treated by deconvolution and combinedwith the middle-range data, 2s  2s  2  2s + 2s, where thewidth of the instrumental function is narrower than 1/10 of themeasurement step interval. The data are combined again by calculating
weighted averages of data in the overlapped regions.

92

T. Ida and H. Hibino  Symmetrization of diffraction peak profiles

WE ðÞ ¼ 1=ð1 þ 2iÞð21Þ

for the case  < 0.
If necessary, the possible singularity would be treated by thesegmentation–deconvolution–combination scheme describedin the preceding section.
2.4. Removal of asymmetry

Assuming the Williamson–Hall-type dependence, theasymmetry caused by the beamline aberration would be
removed by deconvolution with an appropriate scale transformation as shown in x2.3. However, the deconvolution
generally causes an increase of noise to compensate for the
sharpened peak profiles in the deconvoluted data. Furthermore, there is no theoretical basis for the assumption that theexponential formula should be used as the model for theasymmetry caused by the beamline optics, while the formulafor the axial-divergence aberration has unambiguously been
derived from the geometry of the optics and the theory ofdiffraction.
In order to suppress the increase of noise, and also to
reduce changes in the characteristics of the peak profiles,
except the asymmetry, convolution with a symmetric function
wS() with the Fourier form
WS ðÞ ¼ jWE ðÞj
¼ ð1 þ 42 2 Þ1=2

ð22Þ

is simultaneously applied with the deconvolution of wE().
The symmetric function wS(), defined as the inverse Fouriertransform of WS(), is simply given byelectronic reprint

J. Appl. Cryst. (2006). 39, 90–100

research papers
wS ðÞ ¼ 1 K0 ðÞ;

ð23Þ

where K0(x) is the modified Bessel function of the secondkind.

3. Experimental
The powder diffraction data were collected with the multipledetector system on beamline BL4B2 at the Photon Factory in
Tsukuba (Toraya et al., 1996). The axial divergence of thediffracted beam was limited within  = 1 FWHM with a set ofSoller slits.
The Bragg angle of the Ge(111) crystal analyser for thecalibrated source X-ray wavelength of  = 1.306 Å was estimated at A = 11.535 .
Standard Si powder (NIST SRM640b), filled into the hollowof the flat-plate specimen holder of diameter 30 mm, was used
to evaluate the instrumental functions of the measurement
system, which may be affected by the spectroscopic intensity
distribution of the source X-ray. The specimen was rotatedabout its surface normal at one revolution s1 during themeasurement.
The diffraction intensity data of 11 reflection peaks from thestandard Si specimen were collected with a single-detector 2
scan mode, using one of the multiple detectors. The step
interval of the 2 scan was 0.005 . The measurement time ateach step was 1 s for the 111, 220, 311, 400 and 422 reflections,and 8 s for the 711/551, 642, 731/553 and 800 reflections.

The counting losses caused by the finite response time of thedetector system were corrected by an intermediately extended
dead-time model (Ida & Iwata, 2005).
A flat-plate specimen of standard ZnO powder (NIST
SRM674) was used as a test sample for the current method tosymmetrize the experimental diffraction peak profiles. The
diffraction data, ranging from 1.43 to 154.98 in 2, werecollected by a MDS measurement method. The measurement
time at each step was 4 s. Further details of the apparatus, themeasurement conditions for the ZnO specimen and themethod to connect the segmented intensity data from themultiple detectors, have been described elsewhere (Ida, 2005).

4. Analysis
4.1. Calibration of instrumental characteristics
4.1.1. Evaluation of asymmetry. In a first step, the aberrations caused by the divergence of the diffracted beam along
the axial direction were removed from the Si diffraction
intensity data by a deconvolution method recently developedby the authors (Ida, Hibino & Toraya, 2003) but with application of the modification described here in x2.2. The results ofthe deconvolution of axial divergence applied to the 111 and711/551 reflection data are shown as triangles in Figs. 2(b) and3(b).
After the removal of the axial-divergence aberrations, slightasymmetry of the diffraction peak profiles still remained,
similarly to our previous results (Ida, Hibino & Toraya, 2003).
The asymmetric deconvoluted profiles were fitted with anasymmetric function defined by the convolution of a pseudo-

Figure 2
The 111 peak profiles of standard Si (NIST SRM640b) powder. (a) Raw
intensity profile; (b) deconvoluted profile (triangles) and the fitted
asymmetrized pseudo-Voigt function; (c) difference plot of the fitting
shown in (b); (d) symmetrized profile (circles) and the fitted pseudo-Voigt
function; (e) difference plot of the fitting shown in (d). In the differenceplots, statistical uncertainties calculated based on counting statistics areshown as broken lines.
J. Appl. Cryst. (2006). 39, 90–100

Figure 3
The 711/551 peak profiles of standard Si (NIST SRM640b) powder. See
the caption of Fig. 2 for definitions.
T. Ida and H. Hibino  Symmetrization of diffraction peak profiles

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Table 1
Profile parameters of the deconvoluted Si standard diffraction peaks fitted with asymmetrized pseudo-Voigt function.
hkl

b

S

20 ( )S ( )

111
220
311
400
422
440
533
711/551642
731/553800

21 (7)
47 (4)
49 (5)
26 (3)
55 (4)
32.7 (9)27.3 (11)29.5 (8)38.5 (17)33.3 (8)21.1 (10)1459 (9)1080 (4)654 (4)186 (2)451 (2)142.9 (6)84.5 (7)144.8 (7)254.4 (16)191.9 (10)35.2 (13)24.0546 (3)39.7856 (2)47.0296 (5)57.5243 (10)72.2169 (4)85.7589 (3)104.1382 (5)118.4025 (5)128.3363 (6)134.9948 (6)148.388 (3)0.0208 (5)0.0266 (4)0.0307 (9)0.0395 (17)0.0429 (7)0.0518 (6)0.0701 (10)0.0877 (11)0.1044 (13)0.1215 (15)0.189 (7)0.829 (17)0.876 (14)0.882 (3)0.888 (4)0.813 (19)0.812 (14)0.77 (2)0.873 (16)0.826 (19)0.844 (17)0.88 (8) ( )

Rp (%)

Rwp (%)

0.0038 (4)0.0067 (3)0.0081 (5)0.0091 (11)
0.0143 (4)0.0171 (3)0.0226 (6)0.0307 (6)0.0382 (7)0.0450 (8)0.060 (4)

2.12
1.74
2.90
3.95
2.26
1.54
1.85
1.76
1.84
1.94
4.89

4.26
3.26
4.37
6.13
3.79
2.62
2.43
2.55
2.57
2.72
6.01

Table 2
Profile parameters of the deconvoluted Si standard diffraction peaks fitted with the asymmetrized extended pseudo-Voigt function.
hkl

b

S

20 ( )L ( )

G

 ( )

Rp (%)

Rwp (%)

111
220
311
400
422
440
533
711/551642
731/553800

32 (7)
54 (4)
53 (6)
28 (3)
62 (4)
35.2 (11)29.5 (12)30.8 (9)41.9 (19)34.9 (9)21.3 (11)1453 (9)1075 (4)651 (5)185 (2)446 (3)141.0 (7)82.7 (7)143.3 (9)250.0 (19)189.3 (12)34.8 (15)24.0542 (4)39.7855 (3)47.0296 (5)57.5242 (10)72.2169 (4)85.7590 (3)104.1382 (5)118.4026 (5)128.3365 (6)134.9949 (6)148.388 (3)0.01633 (17)0.0225 (3)0.0261 (9)0.0336 (13)0.0323 (8)0.0385 (7)0.0475 (17)0.0720 (15)0.078 (2)0.095 (2)0.156 (18)0.0099 (6)0.0101 (9)0.0114 (17)0.15 (3)0.0207 (14)0.0257 (12)0.040 (2)0.036 (3)0.052 (3)0.056 (3)0.08 (2)0.0034 (4)0.0066 (3)0.0081 (6)0.0089 (11)
0.0143 (5)0.0172 (4)0.0226 (6)0.0308 (6)0.0384 (7)0.0451 (8)0.060 (4)

1.95
1.84
2.94
3.90
2.31
1.58
1.84
1.82
1.89
1.94
4.59

4.00
3.40
4.48
6.13
3.86
2.70
2.45
2.60
2.60
2.76
6.00

Voigt function with an asymmetric exponential function!E(2) given in x2.3. The pseudo-Voigt function is definedby
fpV ðx; S ; Þ ¼ ð1  ÞfG ½x; S =2ðln 2Þ1=2  þ fL ðx; S =2Þ;ð24Þparameter for the pseudo-Voigt function , and the asymmetry
parameter  in the asymmetric component function, aretreated as adjustable profile parameters to be optimized.
Figs. 2(b), 2(c), 3(b) and 3(c) show the results of fitting anddifference plots. The optimized profile parameters for the 11
reflection of Si are listed in Table 1. The R factors defined bywhere fG ðx; G Þ is the Gaussian function given byfG ðx; G Þ ¼ 1=2 G1 expðx2 =G2 Þ;

ð25Þ

Rp ¼

i

and fL ðx; L Þ is the Lorentzian function given byfL ðx; L Þ ¼ 1 L =ðL2 þ x2 Þ:

P

ð26Þ

jyi ðobs:Þ  yi ðcalc:Þj=Pijyi ðobs:Þj

ð27Þ

and

The constant background b, integrated intensity S, peak
position 20, FWHM of the symmetric component S, shape

Figure 5

Figure 4

The angular dependence of the asymmetry parameter  for themeasurement of Si powder (triangles). The optimized Williamson–Halltype dependence is shown as a solid curve.

94

T. Ida and H. Hibino  Symmetrization of diffraction peak profiles

The angular dependence of the Lorentzian width L (triangles) and theGaussian width G (crosses) of the Voigtian profile assigned to thesymmetric part of the deconvoluted diffraction peak profile of Si powder,evaluated by profile fitting with asymmetrized Voigt profiles to the data
after deconvolution of axial divergence aberration. The optimized
dependences obtained by a non-linear fitting procedure are shown as
solid and broken curves, respectively.

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Table 3
Profile parameters for the symmetrized Si diffraction peaks individually fitted with a pseudo-Voigt function.
hkl

b

S

20 ( )S ( )

111
220
311
400
422
440
533
711/551642
731/553800

15 (6)
31 (4)
37 (5)
24 (4)
40 (3)
26.8 (9)22.7 (10)23.4 (8)23 (2)
26.1 (8)18.4 (11)1463 (8)1089 (4)662 (4)188 (2)462 (2)147.5 (6)88.4 (7)151.8 (8)272.1 (19)202.8 (10)39.5 (15)24.05478 (7)39.78555 (6)47.02953 (13)57.5252 (2)72.21623 (12)85.75891 (9)104.13891 (15)118.40265 (16)128.3361 (2)134.9944 (2)148.3912 (9)0.0232 (2)0.0317 (2)0.0369 (5)0.0460 (8)0.0537 (4)0.0646 (3)0.0861 (5)0.1118 (6)0.1337 (8)0.1564 (8)0.239 (4)Rp (%)

Rwp (%)

0.770 (13)0.80 (9)0.805 (17)0.83 (3)0.763 (13)0.781 (9)0.780 (15)0.850 (10)0.846 (13)0.842 (10)0.94 (5)

2.18
1.60
2.72
3.52
2.32
1.52
1.76
1.67
1.97
1.78
3.84

3.86
2.96
3.84
4.78
3.46
2.42
2.45
2.34
3.21
2.47
4.94

Table 4
Profile parameters for the symmetrized Si diffraction peaks individually fitted with the Voigt function.
hkl

b

S

20 ( )L ( )

G ( )

Rp (%)

Rwp (%)

111
220
311
400
422
440
533
711/551642
731/553800

31 (5)
45 (4)
47 (6)
27 (3)
50 (4)
30.0 (13)24.2 (12)24.8 (10)25 (2)
27.4 (10)18.1 (10)1454 (7)1080 (4)655 (5)186 (2)454 (3)144.8 (8)86.8 (8)150.0 (10)269 (2)200.1 (13)39.7 (14)24.05477 (6)39.78555 (7)47.02951 (14)57.5252 (2)72.21622 (14)85.75891 (12)104.13889 (16)118.40266 (19)128.3361 (3)134.9944 (3)148.3912 (9)0.0165 (2)0.0234 (3)0.0274 (6)0.0350 (8)0.0365 (8)0.0451 (8)0.0599 (15)0.0881 (15)0.105 (2)0.122 (2)0.221 (14)0.0122 (4)0.0159 (5)0.0184 (10)0.0221 (17)0.0302 (11)0.0351 (11)0.0471 (17)0.051 (2)0.061 (3)0.073 (3)0.06 (2)

1.93
1.79
2.87
3.43
2.40
1.78
1.86
1.95
2.23
1.95
3.83

3.37
3.00
3.91
4.66
3.48
2.57
2.55
2.51
3.35
2.62
4.95

Rwp ¼


P
i





1=2
wi jyi ðobs:Þ  yi ðcalc:Þj2
P
i

1=2
wi y2i ðobs:Þ

;

ð28Þ

where wi is the weight for the least-squares fitting, are also
listed in the table.
The optimized value of asymmetry parameter  for eachreflection is plotted versus the diffraction angle in Fig. 4. The
dependence of the asymmetry parameter  is fitted by theWilliamson–Hall-type dependence given by equation (14).
The optimized coefficients of the secant and tangent terms, Xand Y, areobtained from the profile fitting using the pseudo-Voigt
function.
Fig. 5 shows the angular dependence of the Lorentzian andGaussian FWHM values of the extended-pseudo-Voigt
approximation to the Voigt profile assigned to the symmetricpart of the deconvoluted profile. The angular dependences of
the Lorentzian and Gaussian widths are assumed to be
expressed by the following equations:L ¼ LX sec  þ LY tan ;ð29Þ

2G ¼ 2GX sec2  þ 2GY tan2 :

ð30Þ

and

The optimized values of the coefficients, LX, LY, GX andGY areX ¼ 0:0000 ð3Þ ;Y ¼ 0:0184 ð5Þ :LX ¼ 0:0105 ð3Þ ;LY ¼ 0:0281 ð7Þ ;The apparent asymmetry has almost a purely tangential
dependence on 2, which supports the idea that it is caused by
the spectroscopic distribution of the source X-ray supplied by
the beamline optics.
The deconvoluted profiles were also analysed by fitting witha function calculated as the convolution of the extendedpseudo-Voigt function (Ida et al., 2000), which is practicallyequivalent to the exact Voigt profile, with the asymmetriccomponent function !E(2) as the profile model. The optimized profile parameters are listed in Table 2. The values of
the asymmetry parameter  coincide reasonably with those

The coefficients of the secant terms, LX and GX, should be
unaffected during the removal of the asymmetric component,
because the asymmetry has purely tangential dependence. If
the secant terms of the Lorentzian and Gaussian components
are assigned to isotropic size broadening of spherical crystallites, the area- and volume-weighted average diameters areformally estimated at

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GX ¼ 0:0077 ð4Þ ;GY ¼ 0:0235 ð5Þ :electronic reprint95

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Table 5

Profile parameters for the symmetrized Si diffraction peaks fitted with the Voigt function with LX = 0.0082 , LY = 0.0398 , GX = 0.0105 and GY =0.0311 .
hkl

b

S

20 ( )L ( )

G ( )

Rp (%)

Rwp (%)

111
220
311
400
422
440
533
711/551642
731/553800

25 (7)
49 (4)
57 (6)
37 (3)
40 (3)
25.7 (7)20.8 (5)28.2 (5)27.2 (11)28.9 (5)21.6 (3)1457 (11)1078 (4)648 (6)179 (3)461 (2)147.5 (5)89.0 (3)146.6 (5)266.0 (11)198.1 (6)34.8 (3)24.05473 (11)39.78556 (7)47.02963 (18)57.5255 (4)72.21628 (13)85.75888 (12)104.13884 (16)118.4027 (2)128.3359 (3)134.9943 (3)148.3913 (9)0.016850.023100.026240.031170.039140.048100.06434
0.08269
0.10091
0.11735
0.17046

0.01260
0.01583
0.017710.02083
0.026120.032210.04337
0.05601
0.06855
0.07984
0.11629

2.75
1.90
3.63
5.90
2.57
1.82
1.95
2.09
2.47
2.10
3.96

3.77
3.07
4.73
7.51
3.75
2.78
2.70
2.67
3.44
2.68
5.18

hDiA ¼ 340 ð9Þ nm;
hDiV ¼ 479 ð17Þ nm:(see Appendix A).
4.1.2. Validation of the evaluated asymmetry. The Si
diffraction peak profiles obtained by the deconvolution of theaxial-divergence aberration from the observed profiles are
symmetrized by the deconvolution of the exponential asymmetry followed by the convolution of the corresponding
Bessel function, as described in x2.4. It should be noted that
this process might not be necessary for analysis of the standard
data for characterization of the instrument, but is useful for
testing the validity of the analytical method. The angulardependence given by equation (14), and the values X =0.0000 and Y = 0.0184 are applied.
The symmetrized diffraction peak profiles of Si 111 and 711/
551 reflections are shown as circles in Figs. 2(d) and 3(d). The
symmetrized profiles were fitted with the Voigt function,
applying the numerical method proposed by Poppe & Wijers(1990), which is more accurate but takes more computation
time than the extended-pseudo-Voigt approximation. The
application of the more accurate formula is possible because
there is no need for the calculation of the convolution with anasymmetric function for the analysis of the symmetrized
profiles.

The profile parameters optimized by the individual Voigtian
profile fittings are listed in Table 4.
Even though the uncertainties of the FWHM values of theLorentzian and Gaussian components, L and G, optimized
for the symmetrized profiles, are not significantly different
from those obtained in x4.1.1, the effect of mutual correlation
between L and G appears more pronounced here, as can beseen in Table 4 and the plot in Fig. 5. It suggests the degradation of the deconvoluted data, which may be caused by theneglect of the off-diagonal elements of the error matrix (Ida &Toraya, 2002).
Therefore, an iterative profile fitting procedure was
performed, applying the constraints given by equations (29)and (30). The optimized values of the coefficients, LX, LY,
GX and GY areLX ¼ 0:0082 ð4Þ ;LY ¼ 0:0398 ð8Þ ;GX ¼ 0:0105 ð6Þ ;GY ¼ 0:0311 ð7Þ :Figure 6

The angular dependence of the Lorentzian width L (triangles) and theGaussian width G (crosses) of the Voigtian profile fitted to thesymmetrized peak profile of Si powder, evaluated by profile fitting with
the Voigt profile to the symmetrized data. The optimized dependences
evaluated by a constrained profile fitting procedure are shown as solidand broken curves, respectively.

96

T. Ida and H. Hibino  Symmetrization of diffraction peak profiles

Figure 7
Validation of the calibrated wavelength and systematic errors. The
nominal wavelength hkl (crosses) and the values calculated by the
optimized model for systematic errors (solid line) are shown in the upper
part. The lower part shows the difference plot (solid line) with the errors(broken lines) evaluated from the results of the profile fitting.

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Figure 8

Figure 10

The 100 peak profiles of ZnO (NIST SRM674) powder. See the caption ofFig. 2 for definitions.

The 216 peak profiles of ZnO powder. See the caption of Fig. 2 fordefinitions.

The above values of the coefficients for the tangent terms, LYand GY, should be affected by the convolution with the Bessel
function for smoothing, and be reasonably larger than thevalues estimated in the preceding section. The values of thesecant-term coefficients, LX and LY, are again attributed tothe size broadening of spherical crystallites, and the area- andvolume-weighted averages are estimated at
hDiA ¼ 440 ð20Þ nm;
hDiV ¼ 490 ð20Þ nm:
The dependences calculated by the above parameters are
drawn as lines in Fig. 6. The profile parameters optimized withthe constraints are listed in Table 5. The changes in the optimized profile parameters or R values on application of theconstraints are not significant, except the increase in the Rvalues of the 400 reflection of Si, which can be caused by the
intrinsically broader line width of the 400 reflection comparedwith the value calculated on the assumption of isotropic
broadening.
4.1.3. Calibration of wavelength and systematic errors of
the goniometer. Assuming the certified lattice constant of thestandard Si powder (NIST SRM640b), a = 5.430940 (34) Å,
the peak wavelengths for the symmetrized peak profiles areevaluated by a least-squares method. The optimized peak
positions obtained by the fitting with the pseudo-Voigt function to the symmetrized profiles, listed in Table 3, are used toevaluate the wavelength.
The following model for systematic errors is assumed:2hkl ¼ 2hkl  20  21 cosð2hkl  22 Þ;

ð31Þ

where 2hkl is the true diffraction angle related to the wavelength  by the Bragg equation,
Figure 9 ¼ 2aðh2 þ k2 þ l2 Þ1=2 sin hkl ;The 205 peak profiles of ZnO powder. See the caption of Fig. 2 fordefinitions.
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ð32Þ

97

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Table 6Structure parameters of ZnO.
This work

Abrahams & Bernstein (1969)

Albertsson & Abrahams (1989)

Specimen
SourceNo. of reflections

PowderSynchrotron X-ray
42

Single crystal
Laboratory X-ray
141

Single crystal
Neutron394

z(O)U(Zn) (Å2)
U(O) (Å2)
U11(Zn) (Å2)
U33(Zn) (Å2)
U11(O) (Å2)
U33(O) (Å2)

0.3827 (8)0.3825 (14)0.0080 (3)0.0086 (9)0.3819 (1)R (%)
Rw (%)

0.85
1.57

0.0079 (2)0.0080 (4)0.0065 (11)0.0095 (21)0.0073 (4)0.0094 (4)0.0056 (4)0.0064 (4)
3.78
5.12

2hkl is the observed peak position, and 20, 21, 22 areparameters for modelling mechanical systematic errors of thegoniometer. The second term in equation (31) models aconstant offset error, and the third term models a deviation
with the periodicity of 360 , which may be assigned to a gear
eccentricity error (Toraya et al., 1993; Hart et al., 1990).
The optimized peak wavelength, offset and eccentricity
errors are estimated at
 ¼ 1:306348 ð12Þ Å;20 ¼ 0:006 ð2Þ ;21 ¼ 0:0122 ð6Þ ;22 ¼ 107 ð8Þ :

2.31
3.67

0 ð2Þ ¼ ð= sin Þ sin½ þ 20 =2
þ ð21 =2Þ cosð2  22 Þ;ð34Þ

are compared in Fig. 7. It has been confirmed that goodcoincidence is achieved by the above model for mechanicalsystematic errors.
When the strain broadening is assumed to be negligible inthe standard Si powder, the tangent terms in equation (14) are
assigned to the spectroscopic distribution of the source X-ray,while the secant terms in equations (29) and (30) areconnected with the size broadening of the Si crystallites.
4.2. Analysis of ZnO powder diffraction data

and the curve calculated by

4.2.1. Deconvolution of axial-divergence aberration. The
effect of the axial-divergence aberration has been removed
from the powder diffraction intensity data of ZnO by amodified algorithm for the whole-pattern deconvolution
method, as described in x2.2.
The singular point of the aberration function is located at2s = 90 + A = 101.54 . By applying equation (12) for thetolerable width of the aberration function, 0.0005 , which is 1/

Figure 11

Figure 12

The Lorentzian (triangles) and Gaussian (crosses) FWHM values of theZnO diffraction peaks. The optimized and difference curves are shown in
the upper and lower panels, respectively.

The structure factor |F| of ZnO powder. The experimental values areshown as crosses and the calculated values are drawn as a line in theupper part. The lower part shows the difference.

In order to examine the appropriateness of the assumed
systematic errors, the nominal wavelength hkl calculated fromeach peak position 2hkl byhkl ¼ 2dhkl sin hkl ;

98

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10 of the measurement step, the data ranging from 98.39 to104.69 in 2 are assumed to be unaffected by the axialdivergence aberration. The lower-angle data below 100.49
and the higher-angle data above 102.59 were separately
treated by the original whole-pattern deconvolution method(Ida, Hibino & Toraya, 2003), and combined again with theoriginal data in the range 98.39–104.69 .
Figs. 8–10 show the change in the peak profiles of the 100,
205 and 216 reflections of ZnO, through the deconvolution
and symmetrization processes.
The effects of the deconvolution of the axial-divergenceaberration are only significant in the low-angle range, as canbe seen for the 100 reflection in Fig. 8. The asymmetry of theinitial data is reduced and the peak position is shifted to thehigher-angle side after the deconvolution of the axial-divergence effect.
The peak profile of the 205 reflection of ZnO located near
the singular point 2s is not changed in the deconvolution
process. The higher-angle tail of the 216 reflection, locatedabove 2s, is slightly reduced and the peak is shifted to thelower-angle side after the deconvolution of the axial-divergence aberration.
4.2.2. Symmetrization of the peak profiles. The asymmetry
modelled by the convolution with an exponential distributionfunction was also removed by the whole-pattern deconvolution method.
As can be seen in Figs. 8–10, the asymmetry of thediffraction peak profiles of the ZnO powder has certainly been
removed by the symmetrization process. Shifts of the apparent
peak positions through the symmetrization process are also
observed. The peak positions of the symmetrized peak profiles
naturally correspond to the hypothetical symmetric spectral
distribution of the source X-ray, the peak position of which islocated at  = 1.306348 (12) Å. Note that the value of the peakwavelength was determined from the symmetrized peak
profiles of the standard Si powder obtained by the same
method.
The isotropic broadening model given by equations (29)and (30) is applied to fit the line-width parameters L and G
evaluated by the individual peak profile fittings with the Voigt
function to the symmetrized ZnO peak profiles. The results of
the least-squares fitting are shown in Fig. 11. The optimized
parameters are

symmetrized diffraction peaks. The formula for the systematic
errors given by equation (31) is again assumed, fixing the
wavelength at  = 1.306348 Å. The lattice parameters a and c,
the constant shift 20, and parameters for the eccentricity
errors 21 and 22, are treated as variable parameters to fit
the observed peak positions. The optimized values of theparameters are
a ¼ 3:249847 ð18Þ Å;c ¼ 5:20650 ð3Þ Å;20 ¼ 0:0036 ð9Þ ;21 ¼ 0:0077 ð3Þ ;22 ¼ 88 ð5Þ :The evaluated lattice constants coincide well with the literature values, a = 3.24990 (4) Å and c = 5.20660 (6) Å at 300 K,
estimated by the Bond method (Albertsson & Abrahams,
1989).
4.2.4. Structure refinement. The structure parameters of
ZnO are refined to fit the integrated peak intensities evaluatedby application of the individual profile fitting method applying
the Voigt function as the symmetric profile model. The
enhancement factor for asymmetric reflection measurement(Toraya et al., 1993), and a modified formula for March–
Dollase preferred-orientation correction for asymmetricreflection (see Appendix B) have been applied. The linear
absorption coefficients were calculated by using X0h, developed by Stepanov (2005).
The optimized structure parameters are listed in Table 6.
All the parameters coincide well with the results obtained bysingle-crystal methods. The March–Dollase preferred-orientation parameter has been estimated at r = 1.055 (4), whichmeans that the c axis of the crystallites has a slight tendency toorient parallel to the specimen face, which is reasonablebecause it is known that crystallites of ZnO are usually elongated along the c axis.
The observed and calculated values of the structure factors
are shown in Fig. 12. The slight systematic deviation may be
attributed to the change in electron charge density from theneutral atoms assumed for the applied atomic scatteringfactors.

LX ¼ 0:0372 ð6Þ;
LY ¼ 0:0127 ð10Þ;

5. Conclusion

GX ¼ 0:0076 ð18Þ;
GY ¼ 0:054 ð2Þ:
Even though the main features of the observed dependence
are apparently reproduced by the above parameters, thenegative sign of LY indicates that the isotropic broadening
modelled by the combination of secant and tangent dependences is not reasonable for the angular dependence of theobserved profiles.
4.2.3. Evaluation of lattice constants. The lattice constantsof the ZnO sample are evaluated from the positions of the

A convenient method to symmetrize the diffraction peak
profiles measured with a high-resolution powder diffractometer has been developed. Application of the symmetrization procedure has following advantages. (i) The integrated
intensities of the peaks can be extracted by a simple curve
fitting method with a symmetric profile function if the intrinsic
diffraction peak profiles of the sample are symmetric. (ii) The
observed peak position unambiguously corresponds to thecalibrated peak wavelength, while it can be affected bybroadening effects in the asymmetric case. (iii) Intrinsic
asymmetry in diffraction peak profiles, which may be caused

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by stacking faults for example, can properly be extracted fromthe experimental data.

APPENDIX B
Modification of the March–Dollase correction forasymmetric reflection
The pole-figure profile of the March–Dollase model (Dollase,1986) is given by

APPENDIX A
Evaluation of average crystallite size from Voigtian
profile parameters
The normalized Lorentzian and Gaussian functions are givenbyfL ðx; L Þ ¼ ð1=L Þ½1 þ ðx=L Þ2 1

ð35Þ

fG ðx; G Þ ¼ ð1=1=2 G Þ exp½x2 =G2 ;

ð36Þ

and

the Fourier transforms of which are respectively given byFL ð; L Þ ¼ expð2jjL Þ

ð37Þ

ð44Þ

Since the scattering vector is parallel to the face normal
direction in flat-plate symmetric reflection measurements, thepreferred-orientation factor for a given reciprocal-lattice
vector at the angle  from the axis of symmetry is equivalent
to the pole-figure profile.
However, the direction of the scattering vector is generally
deviated by   |i  | from the face normal in asymmetricreflection measurements, where i is the incident glancing
angle and  is the Bragg angle. Therefore, the preferredorientation factor should be evaluated byPð; Þ ¼

R2

p½arccosðsin  sin  cos ’ þ cos  cos Þ d’

0andFG ð; G Þ ¼ expð2 2 G2 Þ:ð38Þ

The Fourier initial slope of the Voigtian profile AVoigt
specified by the widths of the Lorentzian and Gaussiancomponents,  L and  G, is immediately given by
AVoigt ¼ 2L :ð39Þ

The integral breadth BVoight of the Voigtian profile is exactly
given byB1
Voigt ¼R1
1FL ð; L ÞFG ð; G Þ d

¼ ð1=1=2 G Þ expðL2 =G2 Þ erfcðL =G Þ;

ð40Þ

where erfc(x) is the complementary error function defined byerfcðxÞ ¼ ð2=1=2 Þ

R1expðt2 Þ dt:

ð41Þ

x

When the crystallites have spherical shape, the areaweighted average diameter hDiA  hD3 i=hD2 i is related to the
Fourier initial slope of the peak profile on the d* scale by
hDiA ¼ ð3=2ÞA1
Voigt ;

ð42Þ

and
the
volume-weighted
averagediameterhDiV  hD4 i=hD3 i is related to the integral breadth as follows(Ida, Shimazaki et al., 2003):
hDiV ¼ ð4=3ÞB1
Voigt :

100pðÞ ¼ ð2Þ1 ðr2 cos2  þ r1 sin2 Þ3=2 :ð43Þ

T. Ida and H. Hibino  Symmetrization of diffraction peak profilesR¼ 1 ½r1 þ ðr2  r1 Þðsin  sin  cos ’0þ cos  cos Þ2 3=2 d’:ð45Þ

The present study was financially supported by Nippon
Sheet Glass Foundation for Materials Science and Engineering. The authors are indebted to the staff of the Photon
Factory for their aid and for making the facilities available.

References
Albertsson, J. & Abrahams, S. C. (1989). Acta Cryst. B45, 34–40.
Abrahams, S. C. & Bernstein, J. L. (1969). Acta Cryst B25, 1233–1236.
Dollase, W. A. (1986). J. Appl. Cryst. 19, 267–272.
Hart, M., Cernik, R. J., Parrish, W. & Toraya, H. (1990). J. Appl. Cryst.
23, 286–291.
Ida, T. (2005). J. Appl. Cryst. 38, 795–803.
Ida, T., Ando, M. & Toraya, H. (2000). J. Appl. Cryst. 33, 1311–1316.
Ida, T., Hibino, H. & Toraya, H. (2001). J. Appl. Cryst. 34, 144–151.
Ida, T., Hibino, H. & Toraya, H. (2003). J. Appl. Cryst. 36, 181–187.
Ida, T. & Iwata, Y. (2005). J. Appl. Cryst. 38, 426–432.
Ida, T., Shimazaki, S., Hibino, H. & Toraya, H. (2003). J. Appl. Cryst.
36, 1107–1115.
Ida, T. & Toraya, H. (2002). J. Appl. Cryst. 35, 58–68.
Poppe, G. P. M. & Wijers, C. M. J. (1990). ACM Trans. Math. Software,
16, 38–46.
Stepanov, S. A. (2005). X-ray Dynamical Diffraction Data on the Web,http://sergey.gmca.aps.anl.gov/x0h.html.
Toraya, H., Hibino, H. & Ohsumi, K. (1996). J. Synchrotron Rad. 3,
75–83.
Toraya, H., Huang, T. C. & Wu, Y. (1993). J. Appl. Cryst. 26, 774–777.
Williamson, G. K. & Hall, W. H. (1953). Acta Metall. 1, 22–31.

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