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Journal ofApplied
Crystallography
ISSN 0021-8898

Extended pseudo-Voigt function for approximating the Voigt profile
T. Ida, M. Ando and H. Toraya

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J. Appl. Cryst. (2000). 33, 1311–1316

Ida et al.  Extended pseudo-Voigt functionresearch papers
Journal ofApplied
Crystallography

Extended pseudo-Voigt function for approximating
the Voigt profile

ISSN 0021-8898

T. Ida,* M. Ando and H. Toraya
Received 7 May 2000
Accepted 21 July 2000

# 2000 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

The formula of the pseudo-Voigt function expressed by a weighted sum ofGaussian and Lorentzian functions is extended by adding two other types ofpeak functions in order to improve the accuracy when approximating the Voigt
pro®le. The full width at half-maximum (FWHM) values and mixing parametersof the Gaussian, the Lorentzian and the other two component functions in theextended formula can be approximated by polynomials of a parameter  =ÿL/(ÿG + ÿL), where ÿG and ÿL are the FWHM values of the deconvoluted
Gaussian and Lorentzian functions, respectively. The maximum deviation of theextended pseudo-Voigt function from the Voigt pro®le is within 0.12% relative
to the peak height when sixth-order polynomial expansions are used. The
systematic errors of the integrated intensity ÿG and ÿL, estimated by ®tting theextended formula to Voigt pro®les, are typically less than 1/10 of the errors
arising from the application of the original formula of the pseudo-Voigt
approximation proposed by Thompson et al. [J. Appl. Cryst. (1987), 20, 79±83],
while the time required for computation of the extended formula is only about
2.5 relative to the computation time required for the original formula.

1. Introduction
The Voigt function, which is de®ned by the convolution ofGaussian (normal distribution) and Lorentzian (Cauchy
distribution) functions, is the most important description of
the symmetric feature of a diffraction peak pro®le because of
its theoretical signi®cance as well as the practical goodness of®t to experimental data (Langford, 1992). Not only are the Xray emission spectra of transition metals naturally Lorentzians(HoÈlzer et al., 1997), but the particle-size broadening is alsosuggested to be described by a Lorentzian (de Keijser et al.,1982), while the instrumental contributions can roughly beapproximated by a Gaussian (Rietveld, 1969), according to thecentral limit theorem, and strain broadening may also possessa Gaussian form (David & Matthewman, 1985). Therefore, theVoigt pro®le is a theoretically natural description of diffraction peak shape.
Although experimentally observed diffraction peak pro®lesare often asymmetric, by multiply convoluting asymmetric
instrumental functions with the symmetric pro®le function,
asymmetric model functions can be constructed systematically(Ida & Kimura, 1999). However, the computation of pro®le®tting applying the Voigtian-based function may generally
become quite time consuming, especially when it is calculated
by multiple numerical convolutions with asymmetric instrumental functions.
J. Appl. Cryst. (2000). 33, 1311±1316

The pseudo-Voigt function, which is a linear combination ofGaussian and Lorentzian functions with the same full width at
half-maximum (FWHM) values, is a simple approximation
for the Voigt pro®le function (Wertheim et al., 1974). It has
been claimed that the maximum deviation between the exact
Voigt pro®le and the optimized pseudo-Voigt approximation
is 0.77% relative to the peak height, and it is adequate fordata containing 2  104 counts channel 1, for which one
standard deviation is 0.7% of the counts. Furthermore,
it is advantageous that an ef®cient algorithm (Ida & Kimura,1999) is applicable to the numerical evaluation of the
convolutions of the pseudo-Voigt-based functions with
instrumental functions, because for the Gaussian andLorentzian functions, the primitive function and its inverse areboth available.
Thompson et al. (1987) have proposed a convenient
expression of the pseudo-Voigt function for approximating the
Voigt pro®le, in which the FWHM and Lorentzian weight ofthe pseudo-Voigt function are related to the FWHM values ofthe deconvoluted Gaussian and Lorentzian functions, ÿG andÿL. However, the accuracy of the approximation may not beadequate for analysing high-precision data collected with asynchrotron radiation source.
In this paper, we propose an extended formula of thepseudo-Voigt function for precisely approximating the Voigt
pro®le and demonstrate that it can signi®cantly improve theelectronic reprint

Ida et al.  Extended pseudo-Voigt function1311

research papers
accuracy of the integrated intensity, ÿG and ÿL, estimated by®tting to a Voigt pro®le.

fG x; G   1=1=2 G  exp

2. The Voigt profile
The Voigt pro®le de®ned by the convolution of Gaussianand Lorentzian functions, with FWHM values of ÿG and ÿL, isexpressed by the following equations:fV x; ÿG ; ÿL   2=ÿG  ln 2=1=2
 K2 ln 21=2 x=ÿG ; ln 21=2 ÿL =ÿG ;1
R1
1

x2 = G2 ;7

fL x; L   1= L  1  x2 = L2  1 ;

8

and

2.1. The Voigt and complex error functions

K x; y  y=where fG(x; G) and fL(x; L) are the normalized Gaussianand Lorentzian functions, with FWHM ÿ = 2(ln 2)1/2 G = 2 L,and  is a parameter which mixes the two functions.
Thompson et al. (1987) have proposed the following
expression for the pseudo-Voigt approximation for the
convolution of the Gaussian and Lorentzian functions, with
FWHM values of ÿG and ÿL, respectively:  1:36603 ÿL =ÿ

0:47719 ÿL =ÿ2  0:11116 ÿL =ÿ3 ;
9

exp

2

2

t =y  x2t  dt

 <w x  iy;w z  expÿ  ÿ5G  2:69269ÿ4G ÿL  2:42843ÿ3G ÿ2L

z2  erfc

iz;

3

where K(x, y) is usually referred to as the Voigt function, w(z)is a scaled complex error function called the Faddeeva function, and erfc(z) is the complementary complex error function.
<[z] means the real part of z.
An ef®cient algorithm to evaluate the Faddeeva function
has been proposed by Poppe & Wijers (1990), the accuracy ofwhich is 14 signi®cant digits throughout almost the whole ofthe complex plane.
2.2. Specification of the Voigt profile

Although the variable shape of the Voigt function K(x, y) is
speci®ed by a single parameter y taking x as abscissa, it seems
numerically inconvenient because we need x ! 1 and y !
1 to reproduce a pure Lorentzian pro®le using the formula ofK(x, y). Thus here we de®ne a parameter   ÿL/(ÿG + ÿL)instead of y and take the Voigt pro®le for ÿG + ÿL = 1 as areference pro®le, without loss of generality. From the reference Voigt pro®le de®ned by
p x;   fV x; 1

 4:47163ÿ2G ÿ3L  0:07842ÿG ÿ4L  ÿ5L 1=5 ;2; ;

10

which we abbreviate as the TCH formula.
Fig. 2 is a plot of the difference relative to the peak height
between the pro®le calculated by the TCH formula and thereference Voigt pro®le; we ®nd the maximum deviation to beabout 1.2% at  ' 0.5. It is expected that the goodness of ®t to
the Voigt pro®le would be improved by the addition of otherfunctions, with intermediate shape between Gaussian andLorentzian, into the formula of the pseudo-Voigt function.
3.2. Application of intermediate functions

We can ®nd only a few functions that satisfy both of thefollowing requirements: (i) intermediate shape between
Gaussian and Lorentzian; (ii) availability of both the primitive
function and its inverse. The latter is required for applicationof the ef®cient numerical method for evaluating the convolutions with instrumental functions (Ida & Kimura, 1999). One
candidate is an irrational function with the following
normalized formula:fI x; I   1=2 I 1  x= I 2  3=2 :

11

4

the general Voigt pro®le for arbitrary ÿG and ÿL is derived
straightforwardly as
fV x; ÿG ; ÿL   1= ÿG  ÿL px= ÿG  ÿL ; :5

The Voigt pro®les calculated by the algorithm of Poppe &Wijers (1990) on variation of the parameter  are shown in
Fig. 1.

3. Extension of the pseudo-Voigt approximation
3.1. Pseudo-Voigt function

The normalized pseudo-Voigt function is given by
fpV x  1

1312

fG x; G   fL x; L ;

Ida et al.  Extended pseudo-Voigt function

Figure 1

6

The Voigt pro®les on variation of the Gaussian and Lorentzian FWHM
values, ÿG and ÿL.

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J. Appl. Cryst. (2000). 33, 1311±1316

research papers

Figure 3Pro®les of the Gaussian, Lorentzian, irrational and hyperbolic functions
with FWHM = 1.

Figure 2

Three-dimensional plot of the deviation of the pseudo-Voigt function
calculated by the formula proposed by Thompson et al. (1987) from the
Voigt pro®le relative to the peak height.

The FWHM values of the components are given by WG =2(ln 2)1/2 G, WL = 2 L, WI = 2(22/3 1)1/2 I and WP = 2[ln 21/2
+ 1)] P.

The FWHM of this function is given by (22/3
primitive function and its inverse are

3.3. Least-squares optimization of the FWHM and mixing
parameters

1)1/2 I, and theFI x  x= I =21  x= I 2 1=2

12

and
FI 1   2 I = 1

42 1=2 ;13

respectively. Another candidate is the squared hyperbolic
secant function de®ned by
fP x; P   1=2 P  sech2 x= P ;

14sech x  2=exp x  exp

15

x;
1/2

of which the FWHM is given by 2[ln (2 + 1)] P, and theprimitive function and its inverse are given by
FP x  1=2 tanh x= P 

16

and
FP 1  

P =2 ln 1  2= 1

2;

17

3.4. Approximation of dependence of wi and gi on q

respectively, withtanh x  exp x

expx=exp x  exp

x:

18

The pro®les of the above two functions are shown in Fig. 3.
Since the irrational and hyperbolic functions are roughly
equidistantly spaced between Gaussian and Lorentzian functions, it is suggested that addition of both terms would
considerably improve the goodness of ®t to the Voigt pro®le.
Thus we propose a formula of the extended pseudo-Voigt
function:
fepV x  1

L

I

As the ®rst step, we determine the least-squares optimized
values of the FWHM parameters WG, WL, WI and WP and themixing parameters L, I and P of the extended pseudo-Voigt
function for approximating the reference Voigt pro®les,varying the parameter  = 0.01, 0.02, . . . , 0.99. We denote the
FWHM optimized in this process as wi (i = {G, L, I, P}) insteadof Wi to clarify that it is to be optimized for the Voigt pro®lewith ÿG + ÿL = 1. Once wi is determined, the FWHM Wi for
arbitrary ÿG and ÿL is simply given by Wi = (ÿG + ÿL)wi.
The reference Voigt pro®le data {yi} are calculated at xi = 0,0.05, 0.010, . . . , 9.95 (200 data points) by using the Poppe±
Wijers algorithm, and numerical least-squares optimization of
the extended pseudo-Voigt function is achieved by theLevenberg±Marquardt method (Press et al., 1986), applying
the weight yi 1=2 . All FWHM parameters, wG, wL, wI and wP,
and mixing parameters, L, I and P, in the extended formulaof the pseudo-Voigt function are treated as independent
variable parameters.

The dependence of the optimized wi and i on  is, as shownin Figs. 4 and 5, suggested to be quite well approximated bypolynomials of .
Taking the behaviour of wi and i towards  ! 0, 1 into
account, we apply the following polynomial expansion
formulae:P fG x; G   L fL x; L  I fI x; I   P fP x; P :
J. Appl. Cryst. (2000). 33, 1311±1316

19electronic reprintwG  1wL  1



1

N
P
i0



ai i ;NPi0bi  i ;Ida et al.  Extended pseudo-Voigt function

20

21

1313

research papers

Figure 4

The optimized values of extended pseudo-Voigt parameters wG, wL, wI
and wP.

Figure 6

Three-dimensional plot of the deviation of the extended pseudo-Voigt
function from the Voigt pro®le relative to the peak height.

Figure 5

The optimized values of extended pseudo-Voigt parameters L, I and P.

wI 
wP 

N
P
i0
N
P
i0

ci i ;

22di i ;

23hL   1  1I   1P   1



N
P
i0

N
P
i0
N
P
i0

i
fi  i ;The optimized values of the coef®cients {ai}, {bi}, {ci}, {di}, {fi},{gi} and {hi} for N = 6 are listed in Table 1, and the deviation of
the extended pseudo-Voigt function, calculated with thesecoef®cients, from the Voigt pro®le is plotted in Fig. 6. The
maximum deviation relative to the peak height is found to beabout 0.12% at  ' 0.46, which is just 1/10 of the maximum
deviation according to the TCH formula.
The computing speed of the algorithms was compared byusing a compiled macro procedure (WaveMetrics, Igor Pro) ona personal computer (Apple, Power Macintosh 7600/200). The
computing time required for evaluating the extended formulaof the pseudo-Voigt function is roughly 2.5 relative to thecomputing time required by the TCH formula, while therelative time required for the evaluation of the Voigt pro®lesby the Poppe±Weijers algorithm is about 20.

24

gi i ;

25hi i :

26

4. Errors in estimation of profile parameters by a fitting
methodThe accuracy of pro®le parameters estimated by least-squares®ttings by applying the TCH form and the extended form of
the pseudo-Voigt function to a Voigt pro®le is discussed in thissection.

Table 1The optimized polynomial coef®cients {ai}, {bi}, {ci}, {di}, {fi}, {gi} and {hi}.
i
0
1
2
3
4
5
6

1314

{ai}
0.66000
0.150211.249844.74052
9.48291
8.48252
2.95553

{bi}

{ci}

0.42179
1.2569310.30003
23.45651
29.14158
16.50453
3.19974

1.199131.4302115.36331
47.06071
73.61822
57.92559
17.80614

Ida et al.  Extended pseudo-Voigt function{di}1.101860.477450.686882.766224.55466
4.05475
1.26571

electronic reprint{fi}
0.30165
1.389279.31550
24.10743
34.96491
21.18862
3.70290{gi}
0.25437
0.141073.23653
11.09215
22.10544
24.12407
9.76947{hi}
1.01579
1.50429
9.21815
23.59717
39.71134
32.83023
10.02142

J. Appl. Cryst. (2000). 33, 1311±1316

research papers
Sample data {yi} were prepared by using the reference Voigt
pro®le as follows:yi  B  p xi ; ;xi Figure 7

Errors in estimation of the integrated intensity I from Voigt pro®le data.
The data labelled `TCH' were calculated by the formula proposed byThompson et al. (1987), and those labelled `epV' were calculated by theextended pseudo-Voigt function. B is the height of the background added
to the Voigt pro®le.

5:0; 4:9; 4:8; . . . ; 5:0;

the data points of which roughly satisfy the Rietveld re®nement guidelines proposed by McCusker et al. (1999). In order
to elucidate the effect of background, two sets of data, B = 0(no background) and B = 1 (large background), were examined. Note that the peak height of the reference Voigt pro®levaries from 0.64 to 0.94 depending on ; that is, B = 1
means that the background is similar to or higher than the topof the peak function.
The constant background b, integrated intensity I, andGaussian and Lorentzian FWHM values ÿG and ÿL, aretreated as variable ®tting parameters. The function to beminimized is
S

P
i

yi

b

I f xi ; ÿG ; ÿL 2 =yi ;27

where f(xi; ÿG, ÿL) is the pseudo-Voigt function based on the
TCH formula or the extended formula.
Table 2

Relative errors in pro®le parameters estimated by a pro®le ®tting basedon the TCH formula of the pseudo-Voigt function to Voigt pro®les
without background (B = 0).
I , ÿG and ÿL are the relative errors in the estimated integrated intensity,and in the Gaussian and Lorentzian FWHM values, respectively. RP is the R
factor for pro®le ®tting.

Figure 8

Errors in estimation of the Gaussian FWHM ÿG from Voigt pro®le data.

I (%)

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

0.50
0.90
1.13
1.19
1.08
0.85
0.54
0.25
0.06

ÿG (%)
0.15
0.15
0.03
0.54
1.35
2.15
2.62
2.95
5.32

ÿL (%)

RP (%)

1.23
1.56
1.45
1.29
1.12
0.91
0.63
0.31
0.08

0.43
0.72
0.94
1.05
1.02
0.84
0.55
0.25
0.06

Table 3

Relative errors in pro®le parameters estimated by a pro®le ®tting basedon the TCH formula of the pseudo-Voigt function to Voigt pro®les withlarge background (B = 1).
See Table 2 for de®nitions.

Figure 9

Errors in estimation of the Lorentzian FWHM ÿL from Voigt pro®le data.
J. Appl. Cryst. (2000). 33, 1311±1316I (%)

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

0.70
1.20
1.47
1.52
1.35
1.02
0.63
0.28
0.06

electronic reprintÿG (%)
0.53
0.87
1.27
1.82
2.40
2.70
2.48
2.05
3.68

ÿL (%)

RP (%)

6.96
5.52
4.24
3.12
2.17
1.39
0.76
0.32
0.07

0.03
0.06
0.09
0.09
0.09
0.07
0.05
0.02
0.00

Ida et al.  Extended pseudo-Voigt function1315

research papers
Table 4

Relative errors in pro®le parameters estimated by a pro®le ®tting basedon the extended pseudo-Voigt function to Voigt pro®les withoutbackground (B = 0).
See Table 2 for de®nitions.
I (%)

ÿG (%)

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

0.006
0.008
0.004
0.008
0.010
0.005
0.002
0.003
0.000

0.014
0.058
0.046
0.037
0.091
0.200
0.208
0.042
0.231

ÿL (%)

RP (%)

0.220
0.158
0.072
0.115
0.094
0.043
0.031
0.019
0.001

0.038
0.057
0.070
0.073
0.072
0.064
0.045
0.024
0.006

Table 5

Relative errors in pro®le parameters estimated by a pro®le ®tting basedon the extended pseudo-Voigt function to Voigt pro®les with large
background (B = 1).
See Table 2 for de®nitions.
I (%)

ÿG (%)

ÿL (%)

RP (%)

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

0.021
0.027
0.020
0.023
0.022
0.013
0.006
0.007
0.001

0.026
0.003
0.010
0.028
0.020
0.131
0.134
0.082
0.106

0.216
0.095
0.069
0.019
0.034
0.012
0.014
0.008
0.001

0.004
0.005
0.006
0.007
0.006
0.006
0.004
0.002
0.001

The relative errors of the estimated pro®le parameters (I,ÿG, ÿL) are plotted in Figs. 7±9 and summarized in Tables 2±5.
The R factors for the pro®le ®tting (RP), de®ned by
P
P
yi ;
28
RP  jyi b I f xi ; ÿG ; ÿL jii

are also listed in Tables 2±5.
The extended pseudo-Voigt function clearly gives improved
accuracy in the estimation of the pro®le parameters for any
shape of the Voigt pro®le. The errors in the pro®le parameters
estimated by ®tting based on the extended pseudo-Voigt
function are typically less than 1/10 of the errors arising from1316

Ida et al.  Extended pseudo-Voigt function

application of the TCH formula. When a large background isincluded in the experimentally observed pro®le, application of
the TCH formula may result in a considerable amount ofsystematic error, especially in the estimation of ÿL, while nosigni®cant reduction of accuracy is found in the estimation
based on the extended formula.

5. Conclusions
We propose a practical formula of an extended pseudo-Voigt
function optimized for approximating the Voigt pro®le. The
systematic errors of the pro®le parameters estimated by ®tting
the extended formula to Voigt pro®les are typically less than
1/10 of the errors resulting from the application of the original
formula of the pseudo-Voigt approximation proposed byThompson et al. (1987), while the time required for computation of the extended formula is only about 2.5 relative to thecomputation time required for the original formula.

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J. Appl. Cryst. (2000). 33, 1311±1316