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Journal ofApplied
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ISSN 0021-8898

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New measures of sharpness for symmetric powder diffraction
peak profiles
Takashi Ida

J. Appl. Cryst. (2008). 41, 393–401

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J. Appl. Cryst. (2008). 41, 393–401

Takashi Ida · New measures of sharpness for peak profilesresearch papers
New measures of sharpness for symmetric powder
diffraction peak profiles

Journal ofApplied
Crystallography
ISSN 0021-8898

Takashi Ida
Received 21 May 2007
Accepted 19 December 2007

Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi 507-0071,
Japan. Correspondence e-mail: ida.takashi@nitech.ac.jp# 2008 International Union of Crystallography
Printed in Singapore – all rights reserved

New measures of sharpness for symmetric powder diffraction peak proﬁles areproposed. The sharpness parameter is deﬁned through the th-order momentof the Fourier transform of the proﬁle function. Analytical expressions for the
sharpness parameter for empirical model proﬁle functions, namely the Gaussian,logistic distribution, hyperbolic secant, Lorentzian, Voigt, Pearson VII and
pseudo-Voigt functions, and theoretical size-broadening proﬁles with statisticalsize distribution are presented. Theoretical diffraction proﬁles with complicatedformulae can be approximated by empirical model functions assuming
equivalent values of the sharpness parameter. The concept of the sharpness
parameter provides a simple way to deﬁne an approximation for a theoretical
diffraction peak proﬁle with empirical model functions.

1. Introductionor

The main features of powder diffraction peaks are speciﬁed by(i) the integrated intensity, (ii) the peak position, (iii) the line
width, (iv) the asymmetry and (v) the sharpness of the proﬁle.
Only the integrated intensities of the peaks are necessary to
determine the most probable average crystallographic structure, while precise evaluation of peak positions is also
important to determine the dimensions of the unit cell, which
can be the deciding factor in identifying the crystallographic
symmetry from powder diffraction data. Since the intrinsic
line width of a diffraction peak is related to the coherence
length along the diffraction vector, it can be used as a direct
measure of the crystallinity along the corresponding crystallographic direction. Asymmetry of experimental peak proﬁles
is usually attributed to instrumental aberrations or asymmetric
spectroscopic distribution of the source X-rays.
Hereafter, we restrict our attention to a normalized
symmetric peak proﬁle function f ðkÞ located at the origin, forwhich the following equations hold:R1f ðkÞ dk ¼ 1;

ð1Þ

f ðkÞ ¼ f ðkÞ:

ð2Þ

1

Recently, Langford et al. (2000) have reported that the
sharpness of the intrinsic diffraction peak proﬁle is related tothe distribution of the crystallite size, and suggested the ratio(B/W) of the integral breadth B, given by
B ¼ 1=f ð0Þ;

ð3Þfor the normalized peak proﬁle function f(k), to the FWHM
W, deﬁned byf ðW=2Þ ¼ f ð0Þ=2
J. Appl. Cryst. (2008). 41, 393–401

W ¼ 2f 1 ½f ð0Þ=2;ð4Þ

as a measure of the sharpness of a symmetric powder
diffraction peak proﬁle. However, it is generally difﬁcult toobtain an explicit formula of FWHM for a statistically
distributed size-broadening proﬁle.
The author has previously proposed the ratio (A/B) of theFourier initial slope, deﬁned byd
A ¼  lim
x!þ0 dx

Z1
f ðkÞ expð2ikxÞ dk;

ð5Þ

1

to the integral breadth B to be used as a measure of sharpness(Ida et al., 2003); this approach is consistent with the simultaneous application of two major conventional methods forline-broadening analysis, the Warren–Averbach (Warren,1969) and Williamson–Hall (Williamson & Hall, 1953)
methods, and is similar to the method applied by Ungár et al.
(2001). It has also been shown that both A and B can easily beevaluated for the log-normally distributed size-broadening
proﬁle for a given median and logarithmic standard deviation.
Sánchez-Bajo et al. (2006) have pointed out that the use of
the parameter (A/B) may cause systematic deviation on
application of curve ﬁtting analysis with the pseudo-Voigt
function to the size-broadening proﬁle. It has been shown thatthe pseudo-Voigt proﬁle optimized to approximate the sizebroadening proﬁle has less sharpness than that determined by
the same (A/B). Even though the validity of the approximation using the pseudo-Voigt proﬁle has not yet been fully
certiﬁed, it should be worth examining other measures for the
sharpness of peak proﬁle, which can provide a straightforward
method to deﬁne an approximation for the size-broadening
proﬁle with empirical model proﬁle functions.
doi:10.1107/S0021889807067659

electronic reprint393

research papers
In this paper, the author deﬁnes another type of line-widthparameter, C , with a tuneable parameter  through the thorder moment of the Fourier transform, and examines thevalidity of using the ratio of the integral breadth to this widthparameter, ðB=C Þ, as a measure of sharpness for a powder
diffraction peak proﬁle.

2. Definition of line-width parameter
If the cumulants of a peak proﬁle function are deﬁned up to
the fourth order, the sharpness can be unambiguously determined by the kurtosis, which is given by the ratio of the fourthorder cumulant to the squared second-order cumulant of thefunction. The ith cumulant of a function f ðkÞ is deﬁned bydii ¼ lim i ln
!0 d

FðxÞ 

expðkÞ f ðkÞ dk:
1

R1

R1
1

R1
1

k f ðkÞ dk;

R1
1FðxÞ dx:

ð7Þ

1

A parameter C , which is proportional to the width of the
proﬁle function f ðkÞ, can be deﬁned for arbitrary order of  by
the following equations:
 1
1=
R

jxj FðxÞ dx
ð8ÞC  Bðk  1 Þ2 f ðkÞ dk;

 1
1=
R

jxj FðxÞ dx
C0  lim B
!0
1


R1
¼ exp BFðxÞ ln jxj dx :

ð9Þ

1

ðk  1 Þ3 f ðkÞ dk:ðk  1 Þ4 f ðkÞ dk  322 ;and the kurtosis, which can be a measure of the sharpness ofthe peak shape, is deﬁned as 4 =22.
However, it is well known that any moments or cumulantsof order higher than zero cannot be deﬁned for the Lorentzian(Cauchy distribution) function, while Lorentzian-like peak
proﬁles are often observed, perhaps for the following reasons:(i) the spectroscopic proﬁle of the characteristic X-ray naturally has a Lorentzian character, (ii) the size-broadening
proﬁle of small crystallites is likely to have a Lorentzian-likeshape (Popa & Balzar, 2002; Ida et al., 2003), and (iii) the
broadening caused by stacking faults with low mutual correlation has intrinsically Lorentzian character as the result of
Fourier transform of the exponential distribution of coherent
domain size. Therefore, we should deﬁne alternative measuresof sharpness instead of the moments or cumulants for analysisof experimental diffraction peak proﬁles.

394

R1

B1 ¼

for  6¼ 0, and1

The skewness, which can be used as a measure of asymmetryof peak proﬁle, is deﬁned as the ratio of the third-order
cumulant to the cubic standard deviation, 3 =3=2
2 . The fourthorder cumulant of the normalized function f ðkÞ is given by
4 ¼

ð6Þ

1

and the third-order cumulant is identical to the third-order
central moment given by3 ¼

f ðkÞ expð2ikxÞ dk;

is also real and symmetric. When the proﬁle function f ðkÞ isnormalized, the value of FðxÞ at the origin x ¼ 0 becomes
unity, that is, Fð0Þ ¼ 1. Since the area of the Fourier transformFðxÞ is identical to the value f ð0Þ, the integral breadth B for thenormalized function f ðkÞ is related to the Fourier transformFðxÞ by the following equation,

the second-order cumulant is identical to the variance given by
2 ¼

R1
1

Z1

When the function f ðkÞ is assumed to be normalized forsimplicity, the zeroth cumulant is zero (0 ¼ 0), the ﬁrst-order
cumulant is identical to the mean position given by1 ¼

Fortunately, the cumulants of the Fourier transform can bedeﬁned for most of the peak proﬁle functions, including theLorentzian function. The Fourier transform FðxÞ of a real
symmetric function f ðkÞ located at the origin, deﬁned by

Note that the integral breadth B can be considered as theconstant for normalization about the Fourier transform FðxÞ,as shown in equation (7). The parameter C2 is also related tothe curvature of f ðkÞ at the origin, by
C22 ¼ ðB=42 Þ f 00 ð0Þ:

ð10Þ

It is expected that the ratio of the integral breadth B to thewidth parameters C deﬁned in this section, ðB=C Þ, can be ameasure of sharpness, similarly to (A/B) or (B/W) described in
the preceding section.

3. Sharpness parameters of elementary peak profile
functions
In this section, the validity of the use of ðB=C Þ deﬁned in thepreceding section as the sharpness parameter is examinedby comparing several peak proﬁles described by elementaryfunctions. Analytical solutions of ðB=C Þ for the Gaussian,logistic distribution, hyperbolic secant, modiﬁed Lorentzian,intermediate Lorentzian and Lorentzian functions are given in
Appendix A.
The proﬁles of the normalized elementary functions with an
integral breadth of unity are shown in Fig. 1. The sharpness
parameters ðB=WÞ, ðA=BÞ and ðB=C Þ of the six peak functions
are listed in Table 1.
The ratios of the Fourier initial slope to the integral
breadth, (A/B), are zero for all the listed functions except theLorentzian function, while the apparent sharpness of the peak

Takashi Ida  New measures of sharpness for peak profiles

electronic reprint

J. Appl. Cryst. (2008). 41, 393–401

research papers
Table 1
Sharpness parameters for the Gaussian, logistic distribution, hyperbolic
secant, modiﬁed and intermediate Lorentzian, and Lorentzian functions.
Function

B=W

Gaussian
Logistic
Hyperbolic
secantModiﬁedLorentzianIntermediate
Lorentzian
Lorentzian

1.06447 0
1.13459 0
1.19275 00.13484
0.137700.140190.21138 0.26968 0.31831 0.39894
0.22037 0.28694 0.34553 0.45016
0.26968 0.30290 0.37123 0.5

1.22033 00.141470.23142 0.30680 0.375

1.30477 00.145450.24290 0.32726 0.40529 0.55133

1.57080 2

0.159160.28073 0.39270 0.5

Figure 1

4. Sharpness parameters for the peak profile functions
with variable shape
4.1. Voigt function

The Voigt function is deﬁned as the convolution of theGaussian and Lorentzian functions. When the integral
breadths of the component functions are given by bG and bL ,respectively, the convolution is expressed bygV ðk; bG ; bL Þ ¼

R1
1

fG ðk  t; bG Þ fL ðt; bL Þ dt

1=2
1=2
bL =bG Þ;
gV ðk; bG ; bL Þ ¼ b1G Vð k=bG ; Vðx; yÞ  ðy=Þ

R1

ð12Þ

½y2 þ ðx  tÞ2 1 expðt2 Þ dt

1

¼ Re½wofzðx þ iyÞ;ð13ÞwofzðzÞ  expðz2 Þ erfcðizÞ;

ð14Þ

where wofzðzÞ is a scaled complex error function called theFaddeeva function, and erfcðzÞ is the complementary complex
error function. Re½z means the real part of the complex
number z. The formulae for the normalized Gaussian andLorentzian functions, fG ðk; bG Þ and fL ðk; bL Þ, are given inequations (33) and (34) in Appendix A.
The Fourier initial slope AV, the integral breadth BV , andthe width parameters ðC1 ÞV and ðC2 ÞV of the Voigt functiongV ðk; bG ; bL Þ are given by
J. Appl. Cryst. (2008). 41, 393–401

B=C1

B=C2

0.5

0.70711BV ¼ bG expðb2L =b2G Þ ½erfcðbL =1=2 bG Þ1
¼ bG ½wofzðibL =1=2 bG Þ1 ;1 2
ðC1 Þ1
V ¼  bG ðBV  bL Þ;
1 2
2
2 1=2
ðC2 Þ1
V ¼  bG ðbG =2  bL BV þ bL Þ ;while no simple expression for the FWHM is available.
The normalized Voigt proﬁle for given value of integral
breadth BV can be calculated for any value of bL =bG as
follows. A parameter deﬁned by the following equation is
introduced,V  bL =ðbG þ bL Þ;

ð15Þ

so that the parameter V in the range 0  V  1 could cover
all the variable shape of the Voigt proﬁle. The parameters bG
and bL can uniquely be determined from the parameters BVand V bybG ¼ BV exp½1 2V =ð1  V Þ2 erfc½1=2 V =ð1  V Þ; ð16ÞbL ¼ bG V ð1  V Þ1 :ð11Þ

in integral formula, or by the following equations:

B=C1=2

AV ¼ 2bL ;Proﬁles of normalized elementary peak proﬁle functions: the Gaussian,logistic distribution, hyperbolic secant, modiﬁed Lorentzian, intermediate
Lorentzian and Lorentzian functions.

proﬁle shown in Fig. 1 corresponds well to the parameters
ðB=WÞ or ðB=C Þ except that the modiﬁed Lorentzian function
may appear to be slightly sharper than the hyperbolic secant
function, while the parameters ðB=C2 Þ for the two functions
have the same value. It is also conﬁrmed that the curvaturesof the hyperbolic secant and modiﬁed Lorentzian functions
coincide at the origin, in accordance with the common value of
ðB=C2 Þ for the two functions and the relation shown inequation (10).

A=B B=C1=2 B=C0

ð17Þ

Then the normalized Voigt proﬁle explicitly including the
integral breadth as an argument deﬁned by the following
equation,fV ðk; BV ; V Þ  gV ðk; bG ; bL Þ;

ð18Þ

can be calculated by applying equations (12), (13), (16) and(17).
The normalized Voigt proﬁles with an integral breadth ofunity, BV ¼ 1, are plotted for V ¼ 0; 0:2; 0:4; . . . ; 1:0 in Fig.
2. The component widths, bG and bL , for the normalized Voigt
proﬁles and the sharpness parameters, including the parameter ðB=WÞV numerically evaluated by a bisection method,
are also listed for V ¼ 0; 0:1; 0:2; . . . ; 1:0 in Table 2.
4.2. Pearson VII function

The Pearson VII function can be expressed as
1
gP7 ðk; P7 ; Þ ¼ 1=2 ð  1=2Þ1 P7ðÞ½1 þ ðk=P7 Þ2  ;ð19Þ

Takashi Ida  New measures of sharpness for peak profiles

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research papers
Table 2
Sharpness parameters for the Voigt proﬁle.
The integral breadths of the component function, bG and bL , of the normalized
Voigt proﬁle with an integral breadth of unity are also listed.

Figure 2

V bG

bL

B=W

A=B

B=C1=2 B=C1=2

B=C1

B=C2

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

0
0.10367
0.21470
0.33278
0.45678
0.584200.710310.82695
0.921960.981641

1.06447
1.09623
1.13354
1.17765
1.22996
1.291641.362631.439071.509481.55633
1.5708000.20734
0.429410.66556
0.91355
1.168411.42062
1.65389
1.84393
1.9632820.134840.140030.144800.14898
0.152420.155070.156930.158110.15877
0.15908
0.159160.318310.32775
0.33891
0.35225
0.36833
0.38779
0.411220.43856
0.46757
0.491280.5

0.39894
0.412800.429400.449540.474280.50501
0.54325
0.58995
0.64260
0.68898
0.707111
0.933020.858810.776480.685170.584200.473540.354410.23049
0.1090700.269680.290410.311340.33155
0.349910.365360.36536
0.385210.38992
0.392130.39270

Proﬁles of the normalized Voigt function.

for  > 1=2, where P7 is a parameter proportional to thewidth of the peak. The parameter  speciﬁes the sharpness ofthe peak proﬁle of the Pearson VII function. The function
coincides with the Lorentzian function for  ¼ 1, intermediate Lorentzian for  ¼ 3=2 and modiﬁed Lorentzian for ¼ 2, and approaches the Gaussian function for  ! 1. Itshould be noted that the Pearson VII function can model
proﬁles sharper than the Lorentzian function for  in therange 1=2 <  < 1.
The integral breadth of the Pearson VII function BP7 isrelated to P7 as follows:
BP7 ¼ 1=2 ðÞ1 ð  1=2ÞP7 :P7  1is introduced, so that all possible shapes of the Pearson VII
function can be represented by varying the parameter P7
within the range 0 < P7 < 2.
The sharpness parameters of the Pearson VII function for
the values P7 ¼ 0; 0:2; 0:4; . . . ; 2 are listed in Table 3. The
proﬁles of fP7 ðk; BP7 ; P7 Þ for several values of P7 are shownin Fig. 3. As shown in Fig. 3, the Pearson VII function for thevalue 1=2 <  < 1 corresponds to a shape sharper than theLorentzian proﬁle (P7 ¼  ¼ 1). The sharpness of thePearson VII function becomes inﬁnite for  ! 1=2.
4.3. Pseudo-Voigt function

Then the normalized Pearson VII function explicitly including
the integral breadth as an argument is given byn2 o
:fP7 ðk; BP7 ; P7 Þ ¼ B1
P7 1 þ  ð  1=2Þk=ðÞBP7
ð20Þ
The sharpness parameter ðB=WÞP7 from the FWHM of thePearson VII function WP7 is related to the parameter  by thefollowing equation:
ðB=WÞP7 ¼ ð1=2 =2Þð21=  1Þ1=2 ð  1=2Þ=ðÞ:The parameter ðA=BÞP7 from the Fourier initial slope AP7
takes only three values depending on the parameter  asfollows:8
ð1 < Þ< 02
ð ¼ 1 Þ ;ðA=BÞP7 ¼:
1 ð1=2 <  < 1Þ

The pseudo-Voigt function is given by
gpV ðk; WpV ; Þ ¼
ð1  ÞfG ½k; ð= ln 2Þ1=2 WpV =2 þ fL ðk; WpV =2Þ;ð21Þ

where WpV is the FWHM of the function and  is the fractionof the Lorentzian component that is called the mixing parameter. The formulae for the normalized Gaussian andLorentzian functions, fG ðk; bG Þ and fL ðk; bL Þ, are given inequations (33) and (34) in Appendix A.
The integral breadth BpV of the pseudo-Voigt function is
given by
BpV ¼ ð=2ÞWpV ½ð ln 2Þ1=2 ð1  Þ þ 1 :ð22Þ

Then the normalized pseudo-Voigt function can be expressedasfpV ðk; BpV ; Þ  gpV ðk; WpV ; Þ;ð23Þ

which means that the Pearson VII function can model only
three types of decay in the tail of the peak proﬁle.
The sharpness parameter for the Pearson VII function
ðB=C ÞP7 is generally given by
 


1=

B
ð  1=2Þ
1  þ 1


þ
¼ 1=2
;

C P7
 ðÞ 1=2 ðÞ22

the values of which can be calculated for any values of theintegral breadth BpV and the mixing parameter  by using
equation (22).
The sharpness parameters of the pseudo-Voigt proﬁle are
given by
1
ðB=WÞpV ¼ ð=2Þ ð ln 2Þ1=2 ð1  Þ þ  ;for any values of  and  in the ranges  > 1=2 and  >  1.
A shape parameter for the Pearson VII function deﬁned by

ðA=BÞpV ¼ 2½ð ln 2Þ1=2 ð1  Þ þ ;

396

Takashi Ida  New measures of sharpness for peak profiles

electronic reprint

J. Appl. Cryst. (2008). 41, 393–401

research papers
Table 3
Sharpness parameters for the Pearson VII proﬁle.
P7 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2

1
5
2.5
1.666671.25
10.83333
0.714290.625
0.55556
0.5

B=W

A=B B=C1=2 B=C0

B=C1=2

B=C1

B=C2

1.06447
1.11385
1.179421.268211.39177
1.57080
1.846542.31525
3.26558
6.14089
1

0
0
0
0
0
2
1
1
1
1
1

0.26968
0.28134
0.296960.31838
0.348540.39270
0.461280.57852
0.81719
1.540891

0.318310.336460.36025
0.392190.436350.5
0.59763
0.76301
1.09747
2.10760
10.39894
0.43234
0.474510.529280.60301
0.707110.86431
1.12769
1.65640
3.2462310.13484
0.13683
0.13963
0.14370
0.14977
0.159160.174480.201810.25935
0.4380110.211380.217540.22600
0.237890.25505
0.280730.32136
0.391850.536930.98000
1

Figure 3
Table 4Proﬁles of the normalized Pearson VII function.

Sharpness parameters for the pseudo-Voigt proﬁle.


B=W

A=B

B=C1=2

B=C0

B=C1=2

B=C1

B=C2

0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2

1.06447
1.13782
1.222031.31970
1.43434
1.57080
1.73594
1.93989
2.198142.53572
2.995790.00000
0.552211.02832
1.42832
1.752212.00000
2.17168
2.26726
2.286722.230092.09734

0.13484
0.13932
0.144020.14891
0.15397
0.159160.16437
0.169450.174150.178020.180350.21138
0.22252
0.234840.24850
0.26373
0.28073
0.299770.321070.344780.37075
0.398030.26968
0.287710.30832
0.332110.35988
0.392700.432100.480270.540500.617930.721110.318310.343420.37276
0.407460.449110.50000
0.563480.644750.752190.90030
1.116120.39894
0.439900.48852
0.54701
0.618410.707110.81962
0.96600
1.16250
1.43698
1.84099

4.4. Sharpness parameters for size-broadening profile
Figure 4
Proﬁles of the normalized pseudo-Voigt function.

ðB=C ÞpV ¼ ½1=2 ðln 2Þ1=2 ð1  Þ þ 11= ðln 2Þðþ1Þ=2 ð1  Þ½ð þ 1Þ=2 þ ð=2 Þð þ 1Þ1=:

The sharpness parameters of the pseudo-Voigt function for
the values  ¼ 0; 0:2; 0:4; . . . ; 2 are listed in Table 4. The
proﬁles of the normalized pseudo-Voigt function with BpV ¼ 1
and  ¼ 0; 1=3; 2=3; . . . ; 5=3 are shown in Fig. 4.
It should be noted that the parameter ðA=BÞpV has the
maximum value
12 ðln 2Þ½ð ln 2Þ

1=2

 1

1

¼ 2:28899

at ¼ 21 ð ln 2Þ1=2 ½ð ln 2Þ1=2  11 ¼ 1:55116:
Practical use of the pseudo-Voigt function will be limited to
the range  < 1:55, which corresponds to ðB=WÞpV < 2:13,
ðB=C1 ÞpV < 0:72 and ðB=C2 ÞpV < 1:11.
The parameter  can inversely be evaluated from the
sharpness parameters ðB=WÞ, ðA=BÞ, ðB=C1=2 Þ and ðB=C1 Þ bysolving linear or quadratic equations, while numerical solutionof higher-order equations is required in the case of a generalvalue of .
J. Appl. Cryst. (2008). 41, 393–401

4.4.1. Spherical crystallite. When the shape of a crystalliteis spherical, spheroidal or ellipsoidal, the size-broadening
diffraction proﬁle from the crystallite becomes identical tothat of a sphere, according to the theory of Stokes & Wilson(1942). The normalized size-broadening proﬁle is given by thefollowing formula:(3Ds2 ½1  2s1 sin s þ 4s2 sin2 ðs=2Þ ðk 6¼ 0Þ;
fS ðk; DÞ ¼
3D=4
ðk ¼ 0Þ;

ð24Þ
s  2kD;

ð25Þ

where D is the effective dimension along the diffraction
vector, which is identical to the diameter in the case of aspherical crystallite (Langford & Wilson, 1978). When theangles between the diffraction vector and the three principal
axes of an ellipsoidal crystallite are 1, 2 and 3 , the effective
dimension is given by2
D ¼ ðD2
1 cos

2
2
1 þ D2 cos

22
2 þ D3 cos

3Þ

1=2

;

ð26Þ

where the diameters along the principal directions are
assumed to be D1, D2 and D3 (Popa, 2005). When thediffraction vector deviates by an angle of from the uniqueaxis of a spheroid, the effective dimension is given by2
D ¼ ðD2
k cos

2
þ D2
Þ1=2 ;? sin

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where Dk and D? are the diameters along the axial andequatorial directions, respectively.
The Fourier transform of the size-broadening proﬁle of aspherical crystallite is given by(1  3jxj=2D þ jxj3 =2D3 ðjxj < DÞ;ð28Þ
FS ðx; DÞ ¼
0
ðD  jxjÞ:The integral breadth BS of the function fS ðk; DÞ is given by
BS ¼ ð4=3ÞD1 ;and the sharpness parameters are given byðA=BÞS ¼ 9=8 ¼ 1:125;
ðB=C ÞS ¼ ð4=3Þ 8=½ð þ 1Þð þ 2Þð þ 4ÞTable 5Logarithmic standard deviation ! of the SLN proﬁles with integral
breadth B and sharpness parameter ðB=C Þ equivalent to the pseudoVoigt function, with the mixing parameter  ¼ 0:8; 1; 1:2, varied on theorder .
! ( ¼ 0:8)

! ( ¼ 1)! ( ¼ 1:2)

2
3/2
1
1/2
0
1/2
1
3/2
2

0.665660.60068
0.55837
0.529360.508860.494100.48332
0.47532
0.469270.75853
0.70675
0.670390.64238
0.619610.60043
0.58389
0.56935
0.55639

0.81100
0.776510.754250.73587
0.71772
0.69857
0.67857
0.658390.638761=

for  6¼ 0, andðB=C0 ÞS ¼ ð4=3Þ expð7=4Þ ¼ 0:231699:
4.4.2. Lognormal size distribution of spherical crystallites.
The density function of a lognormal distribution with a medianm and logarithmic standard deviation ! is given byfLN ðD; m; !Þ ¼ ½ð2Þ1=2 D!1 expf½lnðD=mÞ2 =2!2 g: ð29ÞThe normalized formula of diffraction intensity proﬁles from
log-normally size-distributed crystallites is given by
Z1
fSLN ðk; m; !Þ ¼
fS ðk; DÞ fLN ½D; m expð3!2 Þ; ! dD; ð30Þrelated to the logarithmic standard deviation parameter ! of
the log-normal distribution.
As has been previously reported (Ida et al., 2003), the SLN
proﬁle will become close to the Lorentzian proﬁle when thevalue of the logarithmic standard deviation ! in determining
the sharpness of the proﬁle has certain values corresponding
to similar sharpness as the Lorentzian function.
Sánchez-Bajo et al. (2006) have suggested that the SLN
proﬁle with the least-squares deviation from the Lorentzianfunction corresponds to ! ¼ 0:70, while the values of !
evaluated through sharpness parameters equivalent to theLorentzian function are given by! ¼ ½2 lnð4=3Þ1=2 ¼ 0:758528

0which is abbreviated as the ‘SLN’ proﬁle.
The Fourier transform of the SLN proﬁle (Ungár et al.,2001) is given by

for the same ðA=BÞ,FSLN ðx;m; !Þ ¼ ð1=2Þerfcf½lnðjxj=mÞ  3!2 =21=2 !g ð3jxj=4mÞ expð2:5!2 Þerfc lnðjxj=mÞ  2!2 =21=2 !

for the same ðB=C0 Þ, andþ ðjxj3 =4m3 Þ expð4:5!2 Þerfc ½lnðjxj=mÞ=21=2 ! : ð31ÞThe integral breadth BSLN of the function fSLN ðk; m; !Þ isgiven by
BSLN ¼ ð4=3Þm1expð3:5! Þ;2and the sharpness parameters are given byðA=BÞSLN ¼ ðA=BÞS expð!2 Þ;
ðB=C ÞSLN ¼ ðB=C ÞS exp ð þ 1Þ!2 =2 :4.4.3. Approximation with the pseudo-Voigt function to
the SLN profile. When the size distribution of crystallites ismodelled by the log-normal distribution, the line-width andthe sharpness parameters of the size-broadening proﬁleuniquely determine all of the characteristics of the statistical
distribution.
Sánchez-Bajo et al. (2006) have proposed the use of thepseudo-Voigt function instead of the theoretical size-broadening proﬁle in least-squares curve ﬁtting analysis. The mixing
parameter  of the pseudo-Voigt function has been empirically398! ¼ ½ð7=2Þ þ 2 lnð3=8Þ  2 1=2 ¼ 0:619605

 1=2
2
3 1 ð þ 4Þð þ 3Þ
ln þ ln
!¼
þ1
8 
8
for the same ðB=C Þ with  6¼ 0. The above relations clearlyshow that the parameters for the ‘Lorentzian-like’ SLN proﬁledepend on the deﬁnition of the sharpness parameter.
The variation of ! for ‘Lorentzian-like’ proﬁles evaluatedthrough ðB=C Þ is listed in the third column of Table 5, wherethe values for the equal sharpness to the pseudo-Voigt function with  ¼ 0:8 and  ¼ 1:2 are also listed in the second andfourth columns. The values of ! corresponding to  ¼ 2; 1
are numerically evaluated as the limit values for  ! 2; 1.
Fig. 5 compares the pseudo-Voigt functions for the mixing
parameters (a)  ¼ 0:8, (b)  ¼ 1 (Lorentzian) and (c)  ¼ 1:2
with the SLN proﬁles that have the equivalent integral
breadth and sharpness parameters for  ¼ 3=2; 1=2; 1=2;
3=2. The difference curves show the tendency that the deviation in the tail and peak regions is greater for smaller andlarger values of , respectively. It is suggested that a value
around  ¼ 1=2 gives the minimum deviation on theapproximation of the SLN proﬁle by the pseudo-Voigt function through the sharpness parameter ðB=C Þ, and the uncer-

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Table 6Proﬁle parameters of the SLN proﬁle function optimized by a leastsquares method to ﬁt the pseudo-Voigt functions deﬁned in different data
ranges.
Range (I), jxj < 2B

Range (II), jxj < 4B

Range (III), jxj < 6B

 ¼ 0:8  ¼ 1  ¼ 1:2  ¼ 0:8  ¼ 1  ¼ 1:2  ¼ 0:8  ¼ 1  ¼ 1:2
b0 0.0078
S 0.943m 0.695! 0.449

0.0070 0.0008 0.0022
0.944
0.996 0.971
0.405
0.181 0.578
0.599
0.760 0.501

0.0020 0.0007
0.973 0.988
0.326 0.192
0.646 0.750

0.0010
0.980
0.534
0.523

0.0009 0.0004
0.982 0.990
0.298 0.180
0.665 0.755

tainty in the evaluated ! will be within about 0:1, independent of the order  of the sharpness parameter ðB=C Þ.
Finally, ambiguity about the least-squares-based optimization of approximation with empirical functions for theoretical
proﬁles, which has been commonly adopted in the literature
treating this subject (Thompson et al., 1987; Ida et al., 2000;
Popa & Balzar, 2002; Sánchez-Bajo et al., 2006), is brieﬂy
discussed. Table 6 lists the results of least-squares ﬁtting by the
SLN proﬁle to the pseudo-Voigt functions with the integral
breadth of unity (B ¼ 1) and the mixing parameters ¼ 0:8; 1; 1:2. Three different data sets in the ranges jxj < 2B
(I), jxj < 4B (II) and jxj < 6B (III) are separately analysed. A
model function of the following form is applied:f ðxÞ ¼ b0 þ SfSLN ðx; m; !Þ;ð32Þ

where the constant background b0 , integrated intensity S,
median diameter m and logarithmic standard deviation ! inthe SLN proﬁle function are treated as the adjustable parameters. As is listed in Table 6, the optimized values of theparameters vary considerably in the data range for the
analysis. Furthermore, it is expected that the results of theleast-squares ﬁtting will also be affected by the backgroundlevel and assumed statistics, when it is applied to experimentally obtained diffraction peak intensity proﬁles.
The concept of approximation through the sharpness
parameter ðB=C Þ seems favourable because of the unambiguous deﬁnition of formulae and the portability to various
model function systems, including theoretical size-broadening
proﬁles with statistical distribution.

5. Conclusion

Figure 5Comparison of the pseudo Voigt functions for the mixing parameters (a) ¼ 0:8, (b)  ¼ 1 (Lorentzian) and (c)  ¼ 1:2 with the SLN proﬁlefunctions with equivalent integral breadth and sharpness parameters
ðB=C Þ for  ¼ 3=2; 1=2; 1=2; 3=2. Difference curves are plotted in thelower part of each panel.
J. Appl. Cryst. (2008). 41, 393–401

A new deﬁnition of the sharpness parameter, which is deﬁned
through the th-order moment of the Fourier transform, forsymmetric peak proﬁle functions has been examined. Analytical expressions for the sharpness parameter for empiricalmodel proﬁle functions, namely the Gaussian, logistic distribution, hyperbolic secant, Lorentzian, Voigt, Pearson VII
and pseudo-Voigt functions, and theoretical size-broadening
proﬁles with statistical size distribution have been derived.
Theoretical diffraction proﬁles of spherical crystallites with
log-normal size distribution can be approximated by thepseudo-Voigt function with equivalent values of the sharpness
parameter for  ¼ 1=2. The concept of the sharpness parameter will provide a simple way to deﬁne an approximation
Takashi Ida  New measures of sharpness for peak profiles

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ðB=WÞSH2 ¼ lnð21=2 þ 1Þ ¼ 1:13459;

for theoretical diffraction peak proﬁle by empirical model
functions.

ðA=BÞSH2 ¼ 0:

APPENDIX A
Analytical expressions of elementary peak profile
functions
A1. Gaussian function

The normalized Gaussian function with integral breadth BG
is given by22
fG ðk; BG Þ ¼ B1G expðk =BG Þ:ð33Þ

Although a simple expression of ðB=C Þ for the general valueof  is not always available, some analytical solutions for
integer or half-integer values of  are as follows:
 
2
3 1
;
¼ 0:137697;
ðB=C1=2 ÞSH2 ¼ 
2 2
ðB=C0 ÞSH2 ¼

The sharpness parameters from the FWHM W and Fourier
initial slope A of the Gaussian function are given by9
ðB=C1 ÞSH2 ¼ 54

ðB=WÞG ¼ 21 1=2 ðln 2Þ1=2 ¼ 1:06447;

The sharpness parameters ðB=C ÞG of the Gaussian function
are given by
ðB=C ÞG ¼ 

½ð þ 1Þ=2

1=

ðB=C2 ÞSH2 ¼ 21=2 1 ¼ 0:450158;
where ðsÞ is the Riemann Zeta function deﬁned by

;

ðsÞ 

for  6¼ 0, where ðxÞ is the gamma-function deﬁned byðxÞ 

R1 x1
t expðtÞ dt:

ð34Þ

ks ;

and ðs; vÞ is the Hurwitz Zeta function (generalized Riemann
Zeta function) deﬁned by

The values for  ¼ 1=2; 0; 1=2; 1; 2 are given by

ðs; vÞ 

1Pðv þ kÞs :k¼0

ðB=C1=2 ÞG ¼ 1=2 ½ð1=4Þ2 ¼ 0:134838;
ðB=C0 ÞG ¼ 21 1=2 expð=2Þ ¼ 0:211375;

A3. Hyperbolic secant function

ðB=C1=2 ÞG ¼ 3=2 ½ð3=4Þ2 ¼ 0:269676;

The normalized hyperbolic secant function with the integral
breadth BSH is given by

ðB=C1 ÞG ¼ 1 ¼ 0:318310;

fSH ðk; BSH Þ ¼ B1SH sech ðk=BSH Þ:

ð36Þ

The sharpness parameters for the function are given by

ðB=C2 ÞG ¼ ð2Þ1=2 ¼ 0:398942;

ðB=WÞSH ¼ =½2 lnð2 þ 31=2 Þ ¼ 1:19275;

where  is the Euler constant, deﬁned by!
nPj1  ln n ¼ 0:57721566490153286060 . . . ;  limj¼1

ðA=BÞSH ¼ 0;
1
ðB=C1=2 ÞSH ¼
4

 


2
1 1
1 3
;
;

¼ 0:140194;
2 4
2 4

"

A2. Logistic distribution function

1

The logistic distribution function is equivalent to the squareof the hyperbolic secant function deﬁned bysechðxÞ  2½expðxÞ þ expðxÞ1 :ðB=C0 ÞSH ¼ 

2
fSH2 ðk; BSH2 Þ ¼ B1SH2 sech ð2k=BSH2 Þ:#
1
4Xð1Þk lnð1 þ 2kÞ
¼ 0:228473;
exp   k¼0
1 þ 2kðB=C1=2 ÞSH ¼

The normalized formula for the logistic distribution function
with the integral breadth BSH2 is given by

1
162

 


2
3 1
3 3
;
;

¼ 0:302896;
2 4
2 4

ðB=C1 ÞSH ¼ 42 G ¼ 0:371227;

ð35Þ

The sharpness parameters ðB=WÞ and ðA=BÞ for the function
are given by

400

1
P
k¼1

0

n!1

 
2
5 1
;
¼ 0:28694;
2 2

ðB=C1 ÞSH2 ¼ 284 ð3Þ ¼ 0:345528;

ðA=BÞG ¼ 0:

ð1þ1=Þ=2

 Z1

2
4
t ln tdt¼ 0:220372;
exp
2
2 0 sinh t

ðB=C2 ÞSH ¼ 21 ¼ 0:5;
where G is the Catalan constant given by

Takashi Ida  New measures of sharpness for peak profiles

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G

1
P

2 2
2 1
fL ðk; BL Þ ¼ B1
L ð1 þ  k =BL Þ :ð1Þk ð2k þ 1Þ1 ¼ 0:915965594177 . . . ;

k¼0

ð39Þ

The sharpness parameters from the FWHM and Fourier initialslope are given byðB=WÞL ¼ =2 ¼ 1:5708;

A4. Modified and intermediate Lorentzian functions

The modiﬁed and intermediate Lorentzian functions are
special cases of the Pearson VII function (Young & Wiles,
1982).
The normalized formula of the modiﬁed Lorentzian function for integral breadth BML is given by22 2
2fML ðk; BML Þ ¼ B1
ML ð1 þ  k =4BML Þ :ð37Þ

ðA=BÞL ¼ 2:
The sharpness parameters ðB=C ÞL of the Lorentzian function
are given for general values of  byðB=C ÞL ¼ 21 ½ð þ 1Þ1= ;for  6¼ 0; the values for  ¼ 1=2; 0; 1=2; 1; 2 are

The sharpness parameters for the modiﬁed Lorentzian function are

ðB=C1=2 ÞL ¼ 21 1 ¼ 0:159155;

ðB=WÞML ¼ ð=4Þð21=2  1Þ1=2 ¼ 1:22033;

ðB=C0 ÞL ¼ 21 expðÞ ¼ 0:28073;

ðA=BÞML ¼ 0;

ðB=C1=2 ÞL ¼ 81  ¼ 0:392699;

ðB=C ÞML ¼ ð1=2Þ 1=2 ð=2 þ 2Þ½ð þ 1Þ=2

1=;

ðB=C1 ÞL ¼ 21 ¼ 0:5;

for  6¼ 0, andðB=C2 ÞL ¼ 21=2 ¼ 0:707107:

ðB=C0 ÞML ¼ ð1=4Þ expð1=2   Þ ¼ 0:231423:
The normalized formula of the intermediate Lorentzianfunction for integral breadth BIL is given by2
2 3=2:fIL ðk; BIL Þ ¼ B1
IL ð1 þ 4k =BIL Þ

ð38Þ

The sharpness parameters for the intermediate Lorentzianfunction are given byðB=WÞIL ¼ ð22=3  1Þ1=2 ¼ 1:30477;
ðA=BÞIL ¼ 0;
ðB=C ÞIL ¼ 21 21 ½ð þ 3Þ=2 ½ð þ 1Þ=2

1=for  6¼ 0, andðB=C0 ÞIL ¼ 21 1 expð1  Þ ¼ 0:242903;

A5. Lorentzian function

The normalized Lorentzian function with integral breadth
BL is given by

J. Appl. Cryst. (2008). 41, 393–401

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