REVIEW OF SCIENTIFIC INSTRUMENTS

VOLUME 69, NUMBER 6

JUNE 1998

Formula for the asymmetric diffraction peak profiles based on double
Soller slit geometry
Takashi Ida
Department of Material Science, Faculty of Science, Himeji Institute of Technology, Kanaji, Kamigori-cho,
Ako-gun, Hyogo 678-1297, Japan~Received 20 January 1998; accepted for publication 6 March 1998!
The asymmetric diffraction peak profiles and peak shift of conventional powder diffractometry
systems caused by the angular divergence along the vertical axis are reproduced by asymmetrized
peak profile functions. The asymmetrization is achieved by convoluting the angular dispersions ofboth incident and scattered beams along the vertical axis. A general method for mapping vertical
window functions to the horizontal direction is proposed, and the formulas of two types ofhorizontal window functions mapped from symmetric double vertical Bartlett ~triangular! andGaussian window functions are presented. Both formulas incorporate a single asymmetry parameter
connected with the open width of the Soller slits along the vertical direction. When experimental
diffraction peak profiles are fitted by asymmetrized pseudo-Voigt functions, the asymmetry
parameter gives good coincidence with the Soller slit angle, which is clearly specified in a given
diffractometer. © 1998 American Institute of Physics. @S0034-6748~98!00806-5#

I. INTRODUCTION

II. MODEL FUNCTIONS
A. General procedure for mapping vertical windowfunctions to horizontal direction

It is generally accepted that the experimental diffraction
peak profiles are the convolution of the wavelength spectrumwith various functions arising from instrumental factors andspecimen defects.1 In powder diffractometry with large vertical divergence, the diffraction peaks are shifted from their
original positions and peak profiles become asymmetric. Rietveld has introduced a ‘‘semiempirical’’ asymmetry factor
intended to correct peak shapes for this effect.2 Howard has
proposed an approximation of asymmetric diffraction peaks
by a sum of Gaussians which incorporates a single asymmetry parameter.3 The Howard method is based on a convolution of a Gaussian profile with a rectangular window profile
along the vertical direction. However, the assumed geometry
does not match the double Soller slit geometry shown in Fig.
1, which is commonly adopted in commercial powder x-ray
diffractometry systems with divergent beam sources.
Although considerable improvements in analysis of neutron or synchrotron diffractometry have recently beenachieved,4 mathematically clear formalism applicable to the
double Soller slit geometry of conventional powder x-ray
diffractometers has not yet been reported to the author’s
knowledge. Considering the rapid development of computersand computing methods, it is worth establishing a precise
mathematical model for commonly used diffractometry systems, even if the exact form might need more computation
time than approximated ones.
In this article, the author presents mathematical models
for diffraction peak profile asymmetrized by vertical divergence of both incident and scattered beams limited by double
Soller slit geometry, and an example of fitting to experimental diffraction peak profiles.
0034-6748/98/69(6)/2268/5/$15.00

Figure 2 illustrates the general geometry of powder diffractometry including vertical divergence. Here a and b are,respectively, the angles of the deviations of the incident andscattered beams from the horizontal plane. The diffraction
angle 2 u 0 for given a and b is exactly related to the horizontal angle 2 u by the following equation:cos 2u 0 5cos 2u cos a cos b 1sin a sin b~1!

or
2 u 22 u 0 5arccos~ cos 2u 0 sec a sec b 2tan a tan b !
22 u 0 [z ~ a b ! .

~2!

The function z( a , b ) can be approximated byz ~ a , b ! >2

a 21 b 2
cot 2u 0 1 a b cosec 2u 0 ,2~3!

on the assumption that a and b are sufficiently small. Bychanging the variables ( a , b ) to (x,y) by a [(x1y)/ A2 andb [(x2y)/ A2, the above relation can be simplified asz>2 21 ~ x 2 t2y 2 /t ! ,~4!

where t[tan u 0 .
We assume that the vertical window profiles of the incident and scattered beams to be f V1( a ) and f V2( b ), and thehorizontal profile to be f H(2Q22 u ), where 2Q is the horizontal angle which defines the position of the receiving slitor the detector. When we define D2 u [2Q22 u 0 , the convoluted profile function P(D2 u ) has a general form of2268© 1998 American Institute of Physics

Rev. Sci. Instrum., Vol. 69, No. 6, June 1998

T. Ida2269

FIG. 1. Schematic view of a conventional powder diffractometry systemwith double Soller slit geometry.

P ~ D2 u ! 5

EEE

f H~ 2Q22 u ! f V1~ a ! f V2~ b ! d ~ 2 u

22 u 0 2z ! d a d b d ~ 2 u ! ,~5!

FIG. 3. Profiles of normalized Bartlett and Gaussian window functions with
FWHM (F B , F G) of 1.

where d (x) is the Dirac delta function. The integration of Eq.
~5! on 2 u gives
P ~ D2 u ! 5

EE

w~ z !5f H~ D2 u 2z ! f V1~ a ! f V2~ b ! d a d b .

E

f H~ D2 u 2z ! w ~ z ! dz,~7!

where w(z) is the window function to be convoluted to theoriginal profile function f H(z), w(z) is found to be given by
the following integration:
w~ z !5E

f V1~ a ! f V2~ b !

f V1

x1y
2

f V2

S D
x2y

A2

dy
dx.
dz

~9!

~6!

When we compare the above formula with the following
standard form of convolution:
P ~ D2 u ! 5

E SA Ddbda,dz

B. Mapping of vertical Bartlett windows to thehorizontal direction

If the Soller slits are ideally designed and the beamsource or the detector have sufficient length in the verticaldirection, the vertical window functions should have the profile of a triangle as illustrated in Fig. 3, which is known as a
Bartlett window in the field of time series analysis.5 The
normalized Bartlett window function f B( w ) with the full
width at half maximum ~FWHM! of F B is given by~8!
f B~ w ! 5

or alternatively,

S

uwu
1
12
FB
FB

D

for u w u ,F B ,~10!

and f B( w )50, elsewhere. Usually the Soller slits for the incident and scattered beams have symmetric geometry, that is,f V1~ w ! 5 f V2~ w ! 5 f B~ w ! .

~11!

The horizontal window function w BB(z) mapped from Eqs.
~4!, ~10!, and ~11! is derived by solving the integration in Eq.
~9!. In case t<1(2 u 0 ,90°), the analytical solution of w BB
has the following forms:
F 2B
4

w BB52S~ 11t 2 u !@ 3t1 A12 ~ 12t 2 ! u #1 12
FIG. 2. Geometry of powder diffractometry including vertical divergence.
The overall diffraction angle and its horizontal component are denoted by
2 u 0 and 2 u . a and b are the angles of the deviations of the incident andscattered beams from the horizontal plane.

2 ~ 11t A12 ~ 12t 2 ! u !

D

11t 2
11 A12 ~ 12t 2 ! uu ln
2~ 11t ! A2u1
12t 2
for 2 2 <u<2
,
t
4t 2

~12!

2270

Rev. Sci. Instrum., Vol. 69, No. 6, June 1998

T. IdaFIG. 4. Horizontal window function w BB mapped from symmetric double
vertical Bartlett window functions with the FWHM F B51 for various t(5tan u 0 ) values.

F 2B
4

w BB52~ 11t 2 u !@ 3t1 A12 ~ 12t 2 ! u #2 ~ 11t A12 ~ 12t 2 ! u !

S

12 A2 ~ 12t 2 ! u1 11
11 A12 ~ 12t 2 ! u

3ln

for2
F 2B
4

~ 11t ! A2u

1

11t 2
u
2

D

12t
11t 2
u ln
2
11t

<u,0,
2

w BB52

vertical Gaussian windows, the analytical solution of whichis easier owing to the infinite integral range. Furthermore, amodel based on Gaussian windows will be appropriate, if thereal Soller slits have random error in geometry. The normalized Gaussian window function f G( w ) with the FWHM of
F G is given by

1t u

12t 2
4t

f G~ w ! 5~13!

S

2 ~ 11t A12 ~ 12t 2 ! u !

D

C G[
~14!

where
u[

z
F 2B t~15!

.

In case t.1, w BB can be derived from the above form andthe following relation:

S D

w BB~ z,t ! 5w BB 2z,

1.
t

exp 24 ~ ln 2!

Ap F G

F G

1

exp 2

Ap C G

w2

C 2G

w2
F 2G

G

~17!

~18!

,

where

11t 2
11 A12 ~ 12t 2 ! uu ln
2~ 11t ! Au

for 0,u<1,F

2 Aln 2orf G~ w ! 5~ 11t 2 u !@ 3t1 A12 ~ 12t 2 ! u #1 11

FIG. 5. Horizontal window function w GG mapped from symmetric double
vertical Gaussian window functions with the FWHM F G51 for various t(5tanu 0 ) values.

~16!

FG
2 Aln 2

~19!

.

The profile of f G( w ) is illustrated in Fig. 3. When the Soller
slits for the incident and scattered beams have symmetricgeometry again,f V1~ w ! 5 f V2~ w ! 5 f G~ w ! .

~20!

The horizontal window function w GG mapped by Eqs. ~4!
and ~9! has a much more simple analytical form than w BB ,that is,
w GG5

2

F

exp
2

pCG

GS

D

t 2 21 z
t 2 11 u z uK,0t C 2Gt C 2G~21!

Figure 4 plots the profiles of the w BB(z) function for various
t values. It should be noted that w BB(z) has finite width at
t51, which means that the vertical divergence in conventional diffractometry system contributes to the width of the
convoluted diffraction peak, even if t51.

where K0 (x) is the modified Bessel function of the second
kind. Figure 5 plots the profiles of the w GG(z) function forvarious t values. The profile of w GG(z) is found to be quite
similar to that of w BB(z) shown in Fig. 4, while the analytical forms are considerably different.

C. Mapping of vertical Gaussian windows to thehorizontal direction

III. APPLICATION TO FIT EXPERIMENTAL PEAK
PROFILES

Although the geometry of Soller slits is naturally modeled by the double vertical Bartlett windows discussed in thepreceding section, we here examine the formula for mapping

In this section, the above models for powder diffraction
profiles are tested by fitting to experimental powder x-ray
diffraction data. Experimental diffraction profiles of Si wereRev. Sci. Instrum., Vol. 69, No. 6, June 1998

T. IdaTABLE I. Fitting parameters for Si diffraction data on the model based onsymmetric double vertical Bartlett ~triangular! windows.

2271

TABLE II. Fitting parameters for Si diffraction data on the model based onsymmetric double vertical Gaussian windows.

hkl

2 u 0 ~deg!

G G ~deg!

G L ~deg!

F B ~deg!

ff0

R P(%)hkl

2 u 0 ~deg!

G G ~deg!

G L ~deg!

F G ~deg!

ff0

R P(%)

111
220
311
400
331
422
333
440
531
620
533

28.43547.29656.11369.11576.36688.01194.930106.684
114.067
127.509136.857

0.043
0.044
0.048
0.052
0.047
0.051
0.046
0.042
0.044
0.039
0.048

0.019
0.023
0.025
0.026
0.026
0.045
0.046
0.059
0.072
0.086
0.100

2.38
2.61
2.58
2.93
2.97
2.68
2.78
2.81
2.52
2.89
2.88

1276
613
446
112
216
191
163
82
191
190
106

60
33
26
12
19
19
15
14
14
17
18

5.2
3.3
3.1
3.3
2.9
2.6
2.3
3.0
2.6
2.6
3.2

111
220
311
400
331
422
333
440
531
620
533

28.43547.29656.11369.11576.36688.01194.930106.684
114.067
127.509136.857

0.043
0.045
0.049
0.054
0.050
0.053
0.049
0.045
0.046
0.043
0.050

0.019
0.023
0.025
0.024
0.023
0.041
0.045
0.057
0.071
0.085
0.100

2.36
2.56
2.54
2.94
3.03
2.82
2.74
2.81
2.44
2.84
2.80

1287
616
447
112
215
189
163
82
191
191
106

46
29
24
12
19
20
14
14
14
16
17

4.2
3.0
3.0
3.4
3.0
2.6
2.3
2.9
2.5
2.6
3.3

collected with a commercial powder x-ray diffractometer,
Rigaku RINT-2000, with a Cu K a radiation tube as the x-ray
source. The radius of the goniometer circle is 185 mm, and0.15 mm width receiving slit, 1°-open divergence and scattering slits were used. The specified Soller slit angle 5° is
identified with twice the FWHM of the vertical windowfunctions. No corrections of the instrumental error were applied. The overall profile function is calculated by numericalintegration of Eq. ~7!, assuming that the horizontal profile isexpressed by a pseudo-Voigt function,6,7
f H~ x ! 5 ~ 12 h !

F S DGAF S DGS D
S D

2 Aln 2p G

exp 24 ~ ln 2!

x 2
G

2
x 2 21
1h
114
,
p G
G

h 51.36603

GL 2
GL 3
GL
20.47719
10.11116
,
G
G
G

R P5

~22!

~23!

G5 ~ G 5G12.69269 G 4G G L12.42843 G 3G G 2L
14.47163 G 2G G 3L10.07842 G G G 4L1G 5L! 1/5,

where G G and G L are Gaussian and Lorentzian FWHMs.
Two types of window functions, w BB(z) defined by Eqs.
~12!–~15! and w GG(z) defined by Eq. ~21! are examined. The
position and the integrated intensity of the K a 1 peak
(2 u 0 , f ), Gaussian and Lorentzian FWHMs (G G ,G L), constant background f 0 , and the FWHM of the Soller slits F B or
F G are treated as independent adjustable parameters for eachpeak. The results of the least-squares fit to 111-533 reflections are listed in Tables I and II. The R factor for the profile
fitting R P defined by~24!

FIG. 6. Experimental ~open circles! and calculated ~solid line! peak profiles
and error plot ~dotted line! for Si 111-reflection. Calculation is based on thegeometry of double vertical Bartlett ~triangular! windows.

(i u Y ~ 2 u i ! obs2Y ~ 2 u i ! calc u / (i Y ~ 2 u i ! obs , ~25!

where Y (2 u i ) obs and Y (2 u i ) calc are, respectively, observedand calculated data, are listed in the last columns of thetables. Figures 6 and 7 show the experimental Si 111reflection data, the best fit curves and the deviations. The
observed asymmetry of the peak is well reproduced by themodel functions. Practically no significant difference has
been found in the best fit curve or estimated parameters between models based on Bartlett and Gaussian window functions. The estimated FWHM values of the Soller slits F B andF G are close to 2.5° for all reflection peaks, which can be

FIG. 7. Experimental ~open circles! and calculated ~solid line! peak profiles
and error plot ~dotted line! for Si 111-reflection. Calculation is based on thegeometry of double vertical Gaussian windows.

2272

Rev. Sci. Instrum., Vol. 69, No. 6, June 1998

identified with half of the specified Soller slit open angle.
The estimated FWHM of the horizontal Gaussian component
G G is close to the value 0.046° which is the arctangent of theratio of the receiveng slit width to goniometer radius. It
should be noted that the iteration steps in the curve fitting
procedure normally converged in all cases, and the final values of F B and F G are plausible, even if 2 u 0 is near 90° andthe profile is almost symmetric, which supports that not only
the asymmetry but also the symmetric feature of the convoluted profile is considerably affected by the vertical divergence. Simultaneous estimation of G G , G L and F B or F G foreach peak seems to have uncertainty due to the mutual correlations. However, these uncertainties are expected to be
avoided in a Rietveld analysis, when the angular dependences of G G and G L are properly modeled and a constant

T. Idavalue for F B or F G is assumed. Since F B or F G is a purely
instrumental factor, the value can be fixed, if the geometry of
the Soller slits of a given diffractometer is once precisely
specified.
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H. M. Rietveld, J. Appl. Crystallogr. 2, 65 ~1969!.
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L. W. Finger, D. E. Cox, and A. P. Jephcoat, J. Appl. Crystallogr. 27, 892~1994!.
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W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,
Numerical Recipes ~Cambridge Univesity Press, Cambridge, 1986!, Chap.
12.
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G. K. Wertheim, M. A. Butler, K. W. West, and D. N. E. Buchanan, Rev.
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