electronic reprint
Journal ofApplied
Crystallography
ISSN 0021-8898

Peak profile function for synchrotron X-ray diffractometry
T. Ida, H. Hibino and H. Toraya

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J. Appl. Cryst. (2001). 34, 144–151

Ida, Hibino and Toraya  Peak profile functionresearch papers
Journal ofApplied
Crystallography

Peak profile function for synchrotron X-ray
diffractometry

ISSN 0021-8898

T. Ida,* H. Hibino and H. Toraya
Received 18 September 2000
Accepted 22 December 2000

# 2001 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 formula of the instrumental function for a high-resolution synchrotron X-ray
diffractometer, equipped with a ¯at crystal analyser and a set of Soller slits forlimiting the axial divergence of the diffracted beam, has been derived. The
formula incorporates the effects of (i) the axial divergence of the diffracted
beam limited by the Soller slits, (ii) the Bragg angle of the ¯at crystal analyser,and (iii) the tilt angle de®ned as the deviation of the normal direction of theanalyser face from the goniometer plane. The model pro®le function given by
the convolution of a Lorentzian function with the instrumental function has
been applied to ®t the experimental diffraction peak pro®les of standard Si
powder (NIST SRM640b) measured with a high-resolution synchrotron X-ray
diffractometer, MDS, on beamline BL4B2 at the Photon Factory in Tsukuba.
The convolution has been calculated by applying an ef®cient algorithm fornumerical integration. The pro®le function reproduces not only the experimental pro®les measured with a well aligned crystal analyser, but alsosigni®cantly distorted pro®les arising from misalignment of the analyser, with
Rp values within 1.4%, by varying only the instrumental parameter for the tilt
angle. It is suggested that further convolution with a Gaussian distribution ispractically not necessary for the model instrumental function to ®t the datacollected with MDS. More rapid computation can be achieved by applying ananalytical formula of the pro®le function, when the tilt angle of the crystal
analyser is within about 0.2 .

1. Introduction
By applying synchrotron X-ray radiation with negligible beam
divergence to powder diffractometry, most effects of opticalaberrations can be removed, while diffractometry withconventional X-ray sources is usually affected by the effects ofa ¯at specimen, sample transparency and displacement of thesample face from the goniometer axis. The effect of axial
divergence of the diffracted beam should not be neglected,
even in synchrotron X-ray diffractometry, although it is relatively easy to handle in this case compared with the conventional case, in which the axial divergence of both the incident
and the diffracted beam should be taken into account simultaneously (Ida, 1998).
Howard (1982) has shown that a model function forasymmetric neutron powder diffraction peaks is given by aconvolution of a symmetric function with an asymmetric
instrumental function. Laar & Yelon (1984) have proposed amodel instrumental function for axial divergence in neutron
diffractometry, incorporating the ®nite lengths of the sample
and detector slits along the axial direction. Finger et al. (1994)
have suggested that the model proposed by Laar & Yelon
might be applicable to the pro®les measured with a synchrotron X-ray diffractometer, but it seems that there remains

144

Ida, Hibino and Toraya  Peak profile function

ambiguity in the interpretation of the parameters of the model
function.
Hastings et al. (1984) have proposed a model pro®le function for a modern high-resolution synchrotron X-ray
diffractometer equipped with a crystal analyser on thediffracted-beam side, based on the assumption that theangular distribution of the diffracted beam along the axial
direction is expressed by a Gaussian function truncated within
the half-maximum region. The basic idea of the model seems
to be more appropriate than the treatment by Finger et al.
(1994). However, treatment of the axial divergence as thetruncated Gaussian function is questionable and the optimum
value of the axial divergence parameter to ®t the experimentalpro®le does not coincide with the angle de®ned as the ratio of
the spacing to the length of the foils of the Soller slits.
We have noticed that the experimental diffraction peakpro®les measured with the high-resolution synchrotron X-ray
diffractometer MDS (Toraya et al., 1996) on the beamline
BL4B2 at the Photon Factory in Tsukuba are considerably
deformed when the normal direction of the ¯at crystal
analyser is slightly (1 , for example) tilted out of the goniometer plane. Thus, a mathematical model explicitly including
the effect of the tilt angle of the analyser crystal is highly
desirable for high-precision analysis of experimental pro®les.

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J. Appl. Cryst. (2001). 34, 144±151

research papers
In this paper, we derive a geometrically correct formula ofthe instrumental function for a synchrotron X-ray diffractometer equipped with a ¯at crystal analyser and a set of Soller
slits for limiting axial divergence of the diffracted beam,
allowing a slight tilt angle of the analyser. The experimental
diffraction peak pro®les of standard Si powder (NIST
SRM640b) measured with MDS are analysed by a pro®le®tting method with the model pro®le function given by the
convolution of a Lorentzian function with the instrumental
function.
Figure 1

2. Geometric model
2.1. Geometry of diffractometric optics

The geometry of the powder diffractometer is shown
schematically in Fig. 1. The horizontal (axial) divergence ofthe beam diffracted from the specimen is restricted by a set ofSoller slits and the detector (scintillation counter) receives thebeam that satis®es the Bragg condition at the ¯at analyser
crystal. As shown in Fig. 1, the angles of the goniometer andthe analyser are respectively denoted as 2 and A. The
diffractometric measurement is conducted by varying 2,while A is ®xed after adjustment to the appropriate angle bya rocking curve measurement for the analyser crystal. Hereinafter, the optical geometry is treated analytically in a coordinate system taking the right direction as the x axis, theupward direction as the y and the forward direction as the zaxis, as shown in Fig. 1.
2.2. Tilt angle of the analyser

The diffraction peak pro®le is affected by the tilt angle Ade®ned by the angular deviation of the normal direction of theanalyser face from the goniometer (xy) plane, while thedeviation in A is expected to cause only constant shift of thepeak without deformation of the peak pro®le.
The normal vector of the analyser, a, is expressed by01
cos 2 ÿ A  ÿ sin 2 ÿ A  0
BCa  @ sin 2 ÿ A  cos 2 ÿ A  0 A
0

0

0
10

1

0
1
0
0
B
CB
C
 @ 0 cos A ÿ sin A A@ ÿ1 A
0 sin Acos A001
sin 2 ÿ A  cos ABC @ ÿ cos 2 ÿ A  cos A A;

1

Schematic illustration of the optical geometry of high-resolution
synchrotron X-ray diffractometry. 2 is the mechanical goniometer
angle and A is the analyser angle. The coordinate system chosen isshown in the upper left part.

cannot be neglected. When the beam diffracted from thespecimen has a Bragg angle of , the direction of the beam b isexpressed by
0
10
1
1
0
0
cos 2BCBCb  @ 0 cos ' ÿ sin ' A@ sin 2 A
0

Even if the axial divergence of the diffracted beam isrestricted by a set of Soller slits, the effect of ®nite divergence
J. Appl. Cryst. (2001). 34, 144±151

cos '
1

0
2

sin 2 sin 'where ' is an arbitrary parameter, when the axial divergence is
not restricted.
The angle of axial divergence is simply related to ' bytan

 sin 2 tan ':3

Note that we distinguish the diffraction angle 2 from thegoniometer angle 2 by different symbols.
2.4. Diffraction condition at the crystal analyser

The diffraction condition at the analyser crystal is that thebeam diffracted from the specimen should have an incident
angle of A with respect to the face of the analyser. We
assume that the area of the scintillator is wide enough todetect all beams diffracted from the analyser. Then the Bragg
condition at the analyser determines the relation between thediffracted beam vector b and the normal vector of theanalyser, a, as follows:b  a  cos =2  A   ÿ sin A :

ÿ sin A

2.3. Axial divergence and diffraction condition for a powder
samplesin 'cos 2BC
 @ sin 2 cos ' A;

1

which represents the rotation from the downward direction (0,ÿ1, 0) about the x axis by the angle A, followed by rotation
about the z axis by the angle 2 ÿ A.

04

2.5. Relation between the axial deviation and the apparent
diffraction angles

Here we derive the relation between the diffraction angle 2
and the goniometer angle 2 for the beam axially deviated by
the angle from the goniometer plane.
The de®nition of the tilt angle, equation (1), the diffraction
condition of the powder sample, equation (2), and theelectronic reprint

Ida, Hibino and Toraya  Peak profile function145research papers
diffraction condition of the analyser crystal, equation (4),immediately give the following exact relation:

The instrumental function for the axial divergence isexpressed by the following compact formulae:

sin 2 ÿ A  cos A cos 2

wH x ÿ cos 2 ÿ A  cos A sin 2 cos 'ÿ sin A sin 2 sin ' ÿ sin A :

5

We de®ne the deviation of the apparent diffraction angle 2
from the true diffraction angle 2 as   2 ÿ 2, and derive
the second-order Taylor expansion of  by ' and A:
 




@
@
1 @2 ' '2
 ' 0 @' 0@A 0 A 2 @'2 0
 2 


@
1 @2 

' 
2 ;
6
@'@A 0 A 2 @2A 0 A



R1
ÿ1

R1
ÿ1

R1
ÿ1

 x ÿ fSoller  d

 x ÿ Au2 ÿ B0 u ÿ C0  1 ÿ juj dux ÿ A u  B2 ÿ C 1 ÿ juj du;

14

where
B  B0 =2A;15C  C0 ÿ B02 =4A;

16

where the subscript 0 indicates the value for ' = 0 and A = 0.
It is not dif®cult to derive the following relations from equation (5):
  

@
@

 0;
7
0 
@' 0
@A 0

A  ÿ 2H =2 cot 2  tan A ;

17

B0  H A sec A ;18C0  ÿ 2A =2 tan A ;

19

 2 
@
cos 2 ÿ 2A  sin 2
ÿ
;
@'2 0
cos Aand (x) is the Dirac delta function. As discussed later, thestatistical properties of the instrumental function are directly
derived from the above compact formulae.



@2 
@'@A



8

sin 2
;cos A

9

 2 
@
 tan A :@2A 0100Therefore,  is naturally approximated by'2 cos 2 ÿ 2A  sin 2 'A sin 2 2A tan A
:


'ÿ
2
2 cos Acos A11
Since ' ' /sin 2, as derived from equation (3),  is related tothe axial deviation angle by
'ÿ

2

2

cot 2  tan A  A2ÿ A tan A :cos A
2

12

The ®rst term of the above equation is identical to the formula
reported by Hastings et al. (1984).

3. Instrumental function for axial divergence3.1. Transmission function of the Soller slits

The transmission function of the Soller slits is given by thefollowing triangular form:
1=H  1 ÿ j j=H  for ÿH < < H ,fSoller  
0elsewhere ,13
where H is the axial divergence angle de®ned as the ratio of
the spacing to the length of the foils of the Soller slits.

146

Ida, Hibino and Toraya  Peak profile function

3.2. Concrete formulae of the instrumental function

In this section, the concrete formulae of wH(x), as thesolution of equation (14), are shown for the case A  0 and B 0. The solution for A < 0 or B < 0 is given by
wH x; A; B; C  wH ÿx; ÿA; B; ÿC20

wH x; A; B; C  wH x; A; ÿB; C:

21

and
In the case A = 0 , 2 = A + /2, the instrumental function
wH(x) is given by
8
1=jB0 j1 ÿ j x ÿ C0 =B0 j
><
for C0 ÿ jB0 j < x < C0  jB0 j;
22
wH x >:0 elsewhere:
When 0 < A, the concrete formulae of wH(x) are:8
 1 ÿ B=A x ÿ C=Aÿ1=2>>>>
for C < x  C  AB2 ;
>
>
>
>
>
>
>
>
>
1=Af x ÿ C=Aÿ1=2 ÿ 1g><
for C  AB2 < x  C  A 1 ÿ B2 ;wH x 
>
>
>
ÿ1=2
>
>
ÿ 1g
>
> 1=2Af 1  B x ÿ C=A
2
>
>
<
x

C  A 1  B2 ;
for
C

A
1
ÿ
B
>
>
>
>
>
:
0 elsewhere;23
for 0  B < 1/2;

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J. Appl. Cryst. (2001). 34, 144±151

research papers
8 1 ÿ B=A x ÿ C=Aÿ1=2>>>>
for C < x  C  A 1 ÿ B2 ;
>
>
>
>
>
>
>
>
>
1=2Af 1 ÿ B x ÿ C=Aÿ1=2  1g><
for C  A 1 ÿ B2 < x  C  AB2 ;wH x >>>>>
1=2Af 1  B x ÿ C=Aÿ1=2 ÿ 1g>>>>
for C  AB2 < x  C  A 1  B2 ;
>
>
>
>
>
:
0 elsewhere;24

for 1/2  B < 1; andwH x 8>
1=2Afÿ B ÿ 1 x ÿ C=Aÿ1=2  1g>>>>
for C  A B ÿ 12 < x  C  AB2 ;
>
>
>
<
>
>
>
>
>
>
>
>
:

ÿ1=2

ÿ 1g
1=2Af B  1 x ÿ C=A
for C  AB2 < x  C  A B  12 ;0

Figure 325

elsewhere;for 1  B.
Figs. 2 and 3 show the variation of the shape of the instrumental function wH(x) for some realistic values of diffraction
angle 2, analyser angle A, axial divergence angle H andanalyser tilt angle A.
It is found that not only the shift and broadening, but also
the asymmetry of the peak is signi®cantly affected by the ®nite
values of the tilt angle A in the case of low diffraction angle.
The peak-top position may shift to the higher angle as a resultof the effect of the tilt angle, as shown in the pro®les in Fig. 2,
while the centroid always shifts to the lower angle, as will be
discussed in the following section.
When the diffraction angle is near (90 + A), the shift andbroadening of the peak become the main effects of ®nite A,
as shown in Fig. 3.

Pro®les of the instrumental function wH(x) for the case 2 = 80 . See Fig. 2
for de®nitions.

When the crystal analyser is precisely adjusted and theeffect of the tilt angle A is negligible, the instrumental
function has the following simple formula:
ÿ1=2
ÿ 1 for 0 < x=A  1,
wH x  1=jAj x=A0elsewhere.
26

3.3. Statistical properties of the instrumental function

The mean, xH, and the variance, H2 , of the instrumental
function wH(x), de®ned by
xH 

R1

x wH x dx;

27

x ÿ xH 2 wH x dx;

28

ÿ1

and
H2 

R1
ÿ1

respectively, are easily calculated from equation (14) rather
than from the results presented in the previous section.
The statistical mean of the instrumental function is derived
as follows:
xH 



R1
ÿ1

R1
ÿ1

R1
ÿ1

x

R1
ÿ1

 x ÿ Au2 ÿ B0 u ÿ C0  1 ÿ juj du dx

1 ÿ jujR1
ÿ1x x ÿ Au2 ÿ B0 u ÿ C0  dx du1 ÿ juj Au2  B0 u  C0  du

 A=6  C0
 ÿ 2H =12 cot 2  tan A  ÿ 2A =2 tan A :29Figure 2Pro®les of the instrumental function wH(x) for the case 2 = 20 . A is theanalyser angle, H is the axial divergence angle and A is the tilt angle of
the analyser.
J. Appl. Cryst. (2001). 34, 144±151

The above equation shows that the average shift of the peakconsists of the terms proportional to the squared axial divergence 2H and the squared tilt angle 2A, the latter of which iselectronic reprint

Ida, Hibino and Toraya  Peak profile function147research papers
independent of the diffraction angle 2 and always causes ashift to the lower angle.
The variance is similarly given byH2  7A2 =180  B02 =6
 74H =720 cot 2  tan A 2  2H 2A =6 sec2 A :30The broadening arising from the tilt angle A, which is givenby the second term of the above equation, is also found to be
independent of 2.

qi x 4.1. Symmetric profile function

We assume that the experimental diffraction peak pro®lesare modelled by the convolution of a symmetric pro®lefunction with the instrumental function. The effect of ®nite
size of crystallites on the diffraction peak broadening is
empirically known to have a Lorentzian distribution (Keijser
et al., 1982), while the basic resolution function of the optics is
expected to be approximated by a Gaussian function (Rietveld, 1969). Thus it seems that the Voigt pro®le function
de®ned as the convolution of Lorentzian and Gaussian functions is the most natural description of the symmetric featureof the pro®le function. However, our preliminary investigations on the experimental peak pro®les of standard Si powder(NIST SRM 640b) measured with MDS have shown that theobserved peak pro®les are well reproduced by a convolutionof a Lorentzian function,
31

with the instrumental function wH(x); the Gaussian distribution is practically negligible. Neglecting the Gaussian distribution, a numerically precise evaluation of the convolution
can easily be achieved as described in the following section.

i1
Rai
aiÿ1

qi x;
33
fL x ÿ ywH y dy;

where a0  a and aM  b. The numerical evaluation of each
integral qi(x) is ef®ciently conducted by applying the following
formulae (Ida & Kimura, 1999):qi x '

i ÿ

iÿ1 

N
wj wH yj 
FL x ÿ y 0  P;
x ÿ y0 j1 wH WHÿ1 j 

34

where FL(x) is the primitive function of fL(x), i.e.
WH(x) is the primitive function of wH(x),F x ÿ y0  
x ÿ WHÿ1 j  ;yj  x ÿ FLÿ1 L
x ÿ y0


W
i
H xÿ
j 


x ÿ y0FL x ÿ a i  ;FL x ÿ y 0 

iÿ1 

i ÿ

iÿ1 xj ;

35

36
37
38

xj and wj are the abscissa and weights of an N-term Gauss±
Legendre integral (Press et al., 1986), and y0 is chosen as themost signi®cant point for the instrumental function, i.e. thepoint where wH(x) has the maximum value in each divided
integral region. A 16-term (N = 16) Gauss±Legendre integral
gives almost exact results for any combination of realistic
values of the parameters.
4.2.2. Profile function for negligible tilt angle. The convolution of a Lorentzian function with the formula of the
instrumental function for negligible tilt angle, equation (26),
has an analytical solution expressed by a combination ofelementary functions as follows:
p x; L ; A  1= L fLH x= L ; A= L ;39fLH u; v  1=2wp ln v  p  w= v ÿ p  w 1=wqf=2 ÿ arctan w ÿ v=qg

4.2. Calculation of convolution

ÿ 1=vf=2 ÿ arctan 1  u2 =v ÿ ug
for v > 0;404.2.1. Convolution of a Lorentzian function with thegeneral formula of instrumental function. The pro®le function

de®ned as the convolution of a Lorentzian function fL(x) with
the instrumental function wH(x) is given byp x  fL x wH xRb fL x ÿ ywH y dy;a32

where a and b are respectively the lower and upper limits ofthe variable having non-zero values of the instrumental
function wH(x).
As the instrumental function is given as a piecewise
analytical function, the integral range is divided into two or
three regions according to the cases presented in x3.2. The
total convolution is expressed by

148

M
P

FL x  1= arctan x= L ;

4. Model profile function as a convolution

fL x  1= L 1  x= L 2 ÿ1 ;p x 

Ida, Hibino and Toraya  Peak profile function

with
fLH ÿu; ÿv  fLH u; v;41

where w  (u2 + 1)1/2, p  [2v(w + u)]1/2, q  [2v(w ÿ u)]1/2. As
the above formulae include only three square-roots, one
logarithm and two arctangents (apart from arithmetic operations), they will be computed more rapidly and safely than the
numerical evaluation described in the previous section.

5. Analysis of experimental profiles
5.1. Experimental

The experimental diffraction peak pro®les of standard Si
powder (NIST SRM 640b) were collected with a high-reso-

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research papers
lution synchrotron powder X-ray diffractometer, MDS(Toraya et al., 1996), on beamline BL4B2 at the Photon
Factory in Tsukuba. The Si powder was loaded into a cylindrical aluminium sample holder, which was mounted on an
attachment rotated, at about one revolution sÿ1, about thenormal to the sample surface during measurements. The
diffractometer is equipped with a set of Soller slits, with anaxial divergence angle H = 1 , and a Ge(111) crystal analyserÊ ) adjusted at A = 6.2 for the wavelength(d111 = 3.26638 AÊ
0.707 A on the diffracted-beam side. The incident X-ray beam
was restricted to 2.5 mm in width and 1 mm in height with acouple of slits on the incident-beam side.
The full width at half-maximum (FWHM) value of therocking curve for the Ge(111) analyser, measured by scanning2 around zero with an attenuated direct beam, was estimatedto be about 0.005 , which is close to the value reported by
Hastings et al. (1984).

andpi x  fL x ÿ 2i ; wi =2

Y xcalc 

and

P
Rp 

2
P
j0

bj x ÿ 21  j 

3
P
i1

Ii pi x4243The experimental pro®les observed in two different
measurement runs have been analysed. The ®rst measurement
was conducted with an accidentally tilted analyser, theapparent tilt angle of which was 1.0 (1) , while the second wasmeasured after the tilt angle of the analyser had been carefully
adjusted to 0.0 (1) .
The results of the pro®le ®ttings are shown in Figs. 4 and 5,
and listed in the second and third columns of Table 1. 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 
;
44
Rwp 
P
2
i wi Y 2i obs

5.2. Analysis

The observed 111, 220 and 311 re¯ection peaks of Si havebeen ®tted simultaneously by a least-squares method with thepro®le function de®ned by the convolution of a Lorentzianfunction with the instrumental function, which is evaluated by
a 16-term numerical integral for each divided region. The
analyser angle and the axial divergence angle were ®xed at A= 6.2 and H = 1 .
The position 2i, the integrated intensity Ii and theLorentzian FWHM value wi of each peak (i = 1, 2,3 ), the tilt
angle of the analyser A, and the background parameters b0,
b1 and b2, are treated as independent variable parameters inthe ®tting.
The overall pro®les are given by

wH x; 2i ; A ; H ; A :

i jY 2i obs ÿ Y 2i calc jP

i Y 2i obs;

45

respectively, are also listed in Table 1.
It should be emphasized that the variations of the threeexperimental peak pro®les, strongly dependent on thediffraction angle 2, are all well reproduced by the model
function, applying the common values of the instrumental
parameters H and A, which supports the validity of thecurrent model for the instrumental function. The excellently
small R values of Rwp = 2.0±2.1% and Rp = 1.3±1.4%, and thenearly statistical behaviour of the deviation shown in the
difference plot, mean that almost all the information in theexperimental pro®les has already been extracted by the pro®lefunction. It is suggested that further convolution with aGaussian distribution is practically not necessary for thecurrent model instrumental function to ®t the experimental
data collected with MDS.

Figure 4

Ê X-rays and a Ge(111) analyser
Synchrotron diffraction data for 111, 220 and 311 re¯ections from Si (NIST SRM 640b) collected with 0.707 A
accidentally tilted by an angle of about 1 . Open circles in the upper part are the experimental points, the solid line is the least-squares ®t and the dottedline is the extracted Lorentzian component. The lower part shows the difference plot (solid line) with the standard uncertainties (dashed line) of the data,
drawn on a magni®ed scale.
J. Appl. Cryst. (2001). 34, 144±151

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Ida, Hibino and Toraya  Peak profile function149research papers

Figure 5

Ê X-rays and a well aligned Ge(111)Synchrotron diffraction data for 111, 220 and 311 re¯ections from Si (NIST SRM 640b) collected with 0.707 A
analyser. See Fig. 4 for symbolism.

The signi®cant distortion of the experimental pro®les,showing broadening and enhanced asymmetry, for the tilted
analyser are well simulated by varying only the instrumental
parameter A in the model pro®le function. The difference inthe values of A, estimated as 1.435 (7) and 0.23 (2) , coincides with the apparent tilt angle of the analyser, within
experimental uncertainty.
From equation (30), the line broadening effect for thevalues A = 6.2 , H = 1 and A = 0.23 , is roughly evaluatedin terms of Gaussian FWHM value as2 ln 2=31=2 H A sec A ' 0:004 ;46

which is close to the FWHM value of the rocking curve, 0.005 .
However, it is dif®cult to express clearly how the width of therocking curve, which is affected by spectral width and divergence of the source X-ray, as well as mosaicity and strain in theanalyser crystal (Hastings et al., 1984), propagates into theTable 1Pro®le parameters estimated by a least-squares method.
2hkl, Ihkl and whkl are respectively the position, the integrated intensity andthe Lorentzian FWHM value of each peak. A is the tilt angle of the analyser.
bj (j = 0, 1, 2) are the background parameters.

2111
I111
w111
2220
I220
w220
2311
I311
w311
A
b0
b1
b2
Rwp (%)
Rp (%)

150

Well aligned analyser

Misaligned analyser
Numerical model

Numerical model

Analytical model

12.94010 (6)421 (2)0.0128 (2)21.20680 (6)807 (3)0.0157 (2)24.92175 (4)1618(4)0.0177 (2)1.435 (7)22 (4)
ÿ18 (3)2.9 (3)2.11
1.34

12.94375 (4)561 (2)0.01272(9)21.21339 (4)795 (3)0.01565 (9)24.92893 (4)1657(4)0.01791 (8)0.23 (2)
21 (4)
ÿ14 (3)2.8 (3)2.04
1.39

12.94386 (4)561 (2)0.01281 (9)21.21351 (3)797 (3)0.01587 (8)24.92904 (4)1661(4)0.01817 (6)0 (®xed)20 (4)
ÿ13 (3)2.7 (3)
2.16
1.49

Ida, Hibino and Toraya  Peak profile function

powder diffraction peak pro®le. It may contribute not only tothe parameter A but also to the Lorentzian width wj in thecurrent model pro®le function.
The Lorentzian components extracted by the pro®le ®tting
are plotted as dotted curves in the upper parts of Figs. 4 and 5.
Although the experimental pro®les observed in the two
measurement runs are quite different, the estimated Lorentzian widths wj show very good coincidence within the standard
uncertainties as listed in Table 1; that is, the shapes of theLorentzian components extracted by the pro®le ®ttings for thetwo different measurement runs are virtually identical.
If the value of the tilt angle is negligible, the analytical formof the pro®le function, equation (39), will be applicable. The
results of ®tting by the analytical pro®le model to theexperimental data collected with a well aligned analyser are
listed in the fourth column of Table 1. Though the R factors
have been slightly increased, the estimated intensities Ij havecoincided with the results obtained by the general pro®lemodel within the standard uncertainties. Although systematicdifferences beyond the standard uncertainties are found in thepeak location 2j and the Lorentzian width wj, the angulardifference, about 0.0001 , will usually be negligible.
Since the computation of the analytical formula is fasterthan the numerical form, it will be more convenient for
application to Rietveld analysis or the whole-pattern ®tting
method.

5.3. Accuracy of the numerical calculation

Finally, the accuracy of the numerical evaluation of the
convolution is tested by applying a decreased number of termsof the numerical integral. The differences between theexperimental data measured with a well aligned analyser andthe curves calculated with the same pro®le parameters aslisted in the third column of Table 1, are evaluated in terms of
the R factors. The dependence of Rwp and Rp values on thenumber of terms N is represented in Table 2.

electronic reprint

J. Appl. Cryst. (2001). 34, 144±151

research papers
Table 2The dependence of Rwp and Rp values on the number of terms N ofnumerical integration for each divided region.
The pro®le parameters listed in the third column of Table 1 were used for
calculation.
N

Rwp (%)

Rp (%)

1
2
4
8
16

8.41
3.95
2.27
2.07
2.04

6.90
2.84
1.50
1.41
1.39

The R values rapidly decrease and approach to convergenceon increasing N. It is suggested that N = 4 is useful for roughestimation and N = 8 will give practically suf®cient precision.
As the convolution is calculated by two or three divided
regions, high-precision evaluation of the total pro®le function
can by achieved only by use of a 16- to 24-term numericalintegration.

6. Conclusions
High-resolution synchrotron X-ray powder diffraction peakpro®les are satisfactorily reproduced by the convolution of aLorentzian function with a correct instrumental function
incorporating the tilt angle of the analyser crystal.

J. Appl. Cryst. (2001). 34, 144±151

When the analyser is well aligned and the tilt angle isnegligible, a rapid analytical formula of the model pro®lefunction is applicable. Even if the pro®le is signi®cantly
distorted as a result of misalignment of the analyser crystal,
the general model pro®le function, precisely calculated bynumerical integration employing only 16 to 24 terms, will be
useful for Rietveld analysis or the whole-pattern ®tting
method.

References
Finger, L. W., Cox, D. E. & Jephcoat, A. P. (1994). J. Appl. Cryst. 27,
892±900.
Hastings, J. B., Thomlinson, W. & Cox, D. E. (1984). J. Appl. Cryst. 17,85±95.
Howard, C. J. (1982) J. Appl. Cryst. 15, 615±620.
Ida, T. (1998). Rev. Sci. Instrum. 69, 2268±2272.
Ida, T. & Kimura, K. (1999). J. Appl. Cryst. 32, 982±991.
Keijser, Th. H. de, Langford, J. I., Mittemeijer, E. J. & Vogels, A. B. P.
(1982). J. Appl. Cryst. 15, 308±314.
Laar, B. van & Yelon, W. B. (1984). J. Appl. Cryst. 17, 47±54.
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T.
(1986). Numerical Recipes. Cambridge University Press.
Rietveld, H. M. (1969). J. Appl. Cryst. 2, 167±173.
Toraya, T., Hibino, H. & Ohsumi, K. (1996). J. Synchrotron Rad. 3,
75±83.

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Ida, Hibino and Toraya  Peak profile function151