REVIEW OF SCIENTIFIC INSTRUMENTS

VOLUME 69, NUMBER 11

NOVEMBER 1998

An efficient method for calculating asymmetric diffraction peak profilesTakashi Idaa)
Department of Material Science, Faculty of Science, Himeji Institute of Technology, Kanaji, Kamigori-cho,
Ako-gun, Hyogo 678-1297, Japan~Received 6 May 1998; accepted for publication 15 July 1998!
An efficient method for evaluating asymmetric diffraction peak profile functions based on theconvolution of the Lorentzian or Gaussian function with any asymmetric window function isproposed. When this method is applied to approximate the convolution with the Howard’s windowfunction @J. Appl. Crystallogr. 15, 615 ~1982!#, only a few terms of numerical integration givesatisfactory results, even if the asymmetry is very strong. © 1998 American Institute of Physics.
@S0034-6748~98!00611-X#~1!.6,8 However, it seems difficult to achieve good approximation with those formulas, when the asymmetry is strong
and f (x) has a narrow profile. This note is intended to propose useful formulas to approximate asymmetrized Lorentzian and Gaussian profile functions.
First, let us examine the case where f (x) is expressed by
the following Lorentzian function:

The peak profiles in angular dispersive powder diffractometry generally become asymmetric, which is mainly
caused by the effect of vertical ~axial! divergence.1 The
asymmetric profile function can be expressed by the following standard formula of convolution:
P~ y !5Ef ~ y2z ! w ~ z ! dz,~1!

w H~ z ! 5u z u 21/2

for

2 Au z minu

z min,z,0,S

Au z minuEAu z minu0f ~ y1u 2 ! du,DD~6!

in order to reduce the peak-like behavior of the integrandnear z;y, as well as to remove the singularity at z→0. The
parameter m should be chosen in correspondence with thedegree of the singularity of w(z) at z→0, for example, m52 will be appropriate for w H (z) given by Eq. ~2!. The
convolution of f L (x) with any window function w(x) isgiven by~2!

P L~ y ! 55EEbaf L ~ y2z ! w ~ z ! dzb L m j m21

aLp

FSw y1 g L tan 2 j m 2arctanygLDG

dj,~7!

~4!

whereand utilizing Simpson’s rule to evaluate the integral, whilevan Laar and Yelon and Finger et al. have suggested direct
application of Gauss–Legendre quadrature procedure to Eq.
a!

Electronic mail: ida@sci.himeji-tech.ac.jp0034-6748/98/69(11)/3837/3/$15.00

ygL

y2z
y 1/m
⇔ j [ arctan2arctan,gL
gL~3!

1Sz[y1 g L tan 2 j m 2arctan

which gives the following formula:
P H~ y ! 5~5!

where g L specifies the width of the peak. Instead of Eq. ~3!,
the author proposes utilizing the following substitution of thevariable:and w H (z)50, elsewhere. Here, z min is the parameter which
specifies the degree of asymmetry. The main feature of thewindow function proposed by van Laar and Yelon6 is similar
to this model.
Howard has proposed a removal of the singularity fromthe integrand in Eq. ~1! by the substitution
z[2u 2 ,F S DG

1x 2 21
11,
pgL
gL

f L~ x ! 5

where y is the deviation of the horizontal angle of the receiving slit from the diffraction angle, f (x) is a symmetric profile
function, and w(z) is an asymmetric window function. The
Voigt,2 pseudo-Voigt,3,4 or Pearson VII ~Ref. 5! function has
been used as f (x), and some formulas for w(z) have also
been proposed.1,6,7 Although the convolution can be evaluated by any numerical integration, a rapid method with afixed formula is strongly desired for application to refinement programs.
In Howard’s model1 the asymmetric window function isexpressed by3837

SSDDa L [ arctan

y2by 1/m2arctan
,
gL
gL

~8!

b L [ arctany2ay 1/m2arctan.
gLgL~9!

© 1998 American Institute of Physics

3838

Rev. Sci. Instrum., Vol. 69, No. 11, November 1998

T. IdaFIG. 1. Approximation of the convolution of the Lorentzian function f L (x)with the Howard’s window function w H (x) for the case z min525 and g L
51. Solid circles are the exact values calculated by Eq. ~11!, and the lines
are the approximations by three-term numerical integrations. Solid line iscalculated by Eqs. ~7!–~10!, while dashed and dotted lines are calculated bythe methods of Howard and Finger et al., respectively.

When we define the integrand of the second equation of Eq.
~7! as g( j ), the approximated formula is given byEbaFIG. 2. Approximation of the convolution of the Gaussian function f G (x)with the Howard’s window function w H (x) for the case z min525 and g G
51. Solid circles are the exact values obtained by increasing the terms ofthe numerical integration, and the lines are the approximations by three-term
numerical integrations. Solid line is calculated by Eqs. ~15!–~17! and ~10!,
while dashed and dotted lines are calculated by the methods of Howard andFinger et al., respectively.

Fz[y1 g G erf21 2 h m 2erf

Ng~ j !dj;~ b2a !

( c i g„a 1 ~ b 2 a ! x i …,~10!

i51

where c i and x i are the Gauss–Legendre weights and abscissa associated with the ith point.9
Now, the convolution of the Lorentzian function f L (x)given by Eq. ~5! with the Howard’s window function w H (x)given by Eq. ~2! is examined. Although Howard has suggested applying numerical integration for evaluating theconvolution,1 the analytical solution of the convolution iscertainly available in this case, which is given by
P LH ~ y ! 5

Av 1u
4& p g L z v
21v 1uSFlnz 2 1& Av 2u z 1 v
z 22 v

parctan
1& Av 1u z 2

DG,

f G~ x ! 5

1Ap g GF S DGexp 2x 2.
gG

In this case, we can apply the following substitution:

~12!

y2z
y2erf
gG
gG

1/m

~13!

,

where erf(x) is the error function defined byerf~ x ! [EAp2x0

exp@ 2t 2 # dt,~14!

and erf21(x) is the inverse function of erf(x). The asymmetrized profile function is given by5~11!

where z [ Au z minu/gL, u[y/ g L , and v [ A11u 2 . Figure 1
compares the exact solution with the approximate functions
based on Eqs. ~7!–~10! and the methods of Howard1 andFinger et al.,8 for the case z min525 and g L 51. Here, thenumber of terms of numerical integration is set to be N53 toclarify the difference. The approximation by Eqs. ~7!–~10! isalmost indistinguishable from the exact solution with only
three terms, while the methods by Howard and Finger et al.
give rather poor results with vibrational structures, which isparticularly unfavorable for application to fitting experimental data.
Next, let us examine the case where f (x) has the following form of a Gaussian function:ygG

F S D S DG

⇔ h [ erf

P G~ y ! 5z 2 2& Av 2u z 1 v

S DG

EEbaf G ~ y2z ! w ~ z ! dzb G m h m21aG2HF

3w y1 g G erf21 2 h m 2erf
whereF S D S DG
F S D S DG
y2by2erf
gGgG

1/ma G [ erfy2ay2erf
gG
gG

1/mb G [ erf

S D GJ
ygL

dh,

~15!

,~16!

.

~17!

The approximated formula is given by Eq. ~10!, again.
The results of various methods for evaluating the convolution of f G (x) with w H (x) for the case z min525 and g G
51 are shown in Fig. 2. Since the analytical formula of theconvolution is not available, the exact solution is evaluatedby increasing the terms of the numerical integral. As isshown in Fig. 2, the approximation based on the formulas
given by Eqs. ~15!–~17! is clearly better than the methods by
Howard and Finger et al., again.

Rev. Sci. Instrum., Vol. 69, No. 11, November 1998

Since the pseudo-Voigt function is a sum of Lorentzianand Gaussian functions,3,4 the asymmetrized pseudo-Voigtfunction is simply given by the sum of the asymmetrized
Lorentzian and Gaussian functions. Furthermore, it will
be easy to construct the formula of the substitution likeEq. ~6! or Eq. ~13! for any f (x), when the primitive function
F(x)[ * f (x)dx and its inverse function F 21 (x) areboth available. Even if F(x) or F 21 (x) is not available, it isworth trying the substitution given by Eq. ~6! or Eq. ~13!,
when f (x) is approximated by a Lorentzian or Gaussianfunction.

T. Ida3839

C. J. Howard, J. Appl. Crystallogr. 15, 615 ~1982!.
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3
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4
P. Thompson, D. E. Cox, and J. B. Hastings, J. Appl. Crystallogr. 20, 79~1987!.
5M. M. Hall, Jr., J. Appl. Crystallogr. 10, 66 ~1977!.
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9W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes ~Cambridge University Press, Cambridge, 1986!,Chap. 12.
12