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Particle statistics of a capillary specimen in synchrotron
powder diffractometry
T. IdaJ. Appl. Cryst. (2011). 44, 911–920

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J. Appl. Cryst. (2011). 44, 911–920

T. Ida · Particle statistics of a capillary specimenresearch papers
Journal ofApplied
Crystallography

Particle statistics of a capillary specimen insynchrotron powder diffractometry

ISSN 0021-8898

T. IdaReceived 4 April 2011
Accepted 14 July 2011

# 2011 International Union of Crystallography
Printed in Singapore – all rights reserved

Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka 10-6-29, Tajimi, Gifu
507-0071, Japan. Correspondence e-mail: ida.takashi@nitech.ac.jp

Particle statistics in synchrotron powder diffractometry measured in a capillarytransmission mode have been theoretically and experimentally investigated.
General mathematical formulae for statistical analysis of observed intensities
measured by a spinner-scan method are derived. It is shown that an additionalparameter to characterize the size distribution of crystallites can be evaluatedfrom the skewness of the distribution of measured diffraction intensities
obtained by the spinner-scan method applied to capillary specimens.

1. Introduction
Analysis of observed powder diffraction intensities generally
assumes a sufﬁciently large number of crystallites that satisfy
the diffraction condition. However, it becomes difﬁcult tojustify this assumption when the size of crystallites in a powder
sample is not small enough and/or the diffractometer has highangular resolution. It is known that the experimental errors
caused by particle statistics for a crystalline powder with aparticle size of several micrometres are comparable to thosecaused by counting statistics about intensity data measured
under typical experimental conditions (Alexander et al.,1948).
Recently, the author has found that the effect of particle
statistics can be evaluated quantitatively by analysis of thediffraction intensities observed by step-scan measurements
about the rotation angle of the sample spinner attached to alaboratory powder diffractometer (Ida et al., 2009). It has beenconcluded that the results of the spinner-scan measurements
include information about the texture of the specimen, andevaluation of crystallite sizes of about several micrometres canbe achieved by the method, while it is difﬁcult to evaluate
crystallite sizes over 0.1 mm by conventional line broadening
analysis of a powder diffraction peak proﬁle.
It is expected that the effect of particle statistics will become
more pronounced in synchrotron powder diffraction
measurements as a result of the high brilliance of synchrotron
light sources. On the other hand, higher transparency of thespecimen for the tentatively higher energy (shorter wavelength) of synchrotron X-ray photons may reduce the effect ofparticle statistics owing to the extension of the penetration
depth and the effective irradiated volume of specimen.
It is also expected that the axial rotation of a capillaryspecimen in transmission-mode geometry applicable insynchrotron powder diffractometry will provide more information about the particle statistics than in-plane rotation of aﬂat specimen. The number of statistically independent data
available on the axial rotation of a capillary specimen is estiJ. Appl. Cryst. (2011). 44, 911–920

mated at about several tens of thousands, because the angularresolution about the axial rotation is of the order of 0.01 ,while in-plane rotation of a ﬂat specimen gives a number ofabout several hundred, which is restricted by the aspect ratioof the tolerable axial divergence to the equatorial divergence,
determined by the geometry of the goniometer.
The skewness of the statistical distribution of the diffraction
intensities can be evaluated from such a large number of
independent data. This suggests that not only the average sizeof crystallites but also the broadness of the size distribution
can be estimated from a set of diffraction intensity data.
Statistical models for size distribution deﬁned by two characteristic parameters, such as lognormal, Gamma or Weibull
distributions, can be determined fully by the two characteristic
parameters.
Although the fundamental theory of particle statistics about
diffraction intensities measured with a laboratory X-ray
diffractometer has been established by the pioneering work of
Alexander et al. (1948) and De Wolff (1958), no practicalformulae applicable to the data measured in capillary transmission mode or analysis of crystallite size distribution havebeen reported. In this study, the author intends to construct a
generalized theoretical framework about particle statistics andderive practical mathematical formulae that can be applied toanalysis of diffraction intensity data measured with a highresolution synchrotron powder diffractometer.

2. Theory of particle statistics
In this section, general theoretical formulae for particle
statistics are presented. The theoretical framework not only
includes the formulae equivalent to those derived by Alexander et al. (1948) and De Wolff (1958), but is also extended to
discussions about higher-order cumulants of the statistical
distribution of the observed diffraction intensities.
It will be shown that the statistical variance of the observed
diffraction intensities is determined by the effective number ofdiffracting crystallites, and the skewness of the statistical
doi:10.1107/S002188981102824X

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intensity distribution mainly depends on the dispersion of
crystallite size distribution in powder specimens.
2.1. Hypothetical case of a non-absorbing sample of uniform
particle size

First, a hypothetical situation is formalized, where all the
crystallites have the same size and the absorption of thespecimen is negligible, similarly to the theoretical work of
Alexander et al. (1948).
It is assumed that crystallites are randomly oriented in a
powder specimen, but it will not be difﬁcult to incorporate theeffect of preferred orientation, if necessary.
The reﬂection multiplicity m is ﬁrst treated as a uniquevalue for a diffraction peak, and next the effect of overlapping
non-equivalent reﬂections is formalized. The formulae for theeffects of the Soller slits that restrict the axial divergence ofthe diffracted beam and the fundamental resolution function
connected with the equatorial divergence are also presented inthis section.
2.1.1. Effective number of crystallites and skewness ofintensity distribution. The probability  that each crystallite

contributes to the diffraction intensity of a reﬂection withmultiplicity m is given by ¼ mX=4;ð1Þ

where  and X denote the tolerance angles in the unit ofradians for the normal orientation of the diffraction plane todeviate along the equatorial and axial directions, respectively.
The product ðXÞ is considered to be a tolerable solid
angle, while the whole solid angle is given by 4.
It has been assumed that  is restricted by the ratio of theeffective focal width of X-ray source to the goniometer radius,and X is restricted by the open angle of the Soller slits, in thecase of a laboratory Bragg–Brentano diffractometer (Ida et al.,2009). The axial tolerance angle X is related to the restriction of the axial divergence of the beam , by the following
equation:
X ¼


;
2 sin T. Idapj ðIj Þ ¼ ð1  Þ ðIj Þ þ  ðIj  I0 Þ;

ð3Þ

where I0 is the diffraction intensity from a crystallite satisfying
the diffraction condition, and ðxÞ is the Dirac delta function.
The average, variance and third central moment of thediffraction intensity from each crystallite are then given by
hIj i ¼

R1

Ij pj ðIj Þ dIj ¼ I0 ;ð4Þ0hðIj Þ2 i ¼ ð1  ÞI02 ;ð5ÞhðIj Þ3 i ¼ ð1  Þ½2 þ ð1  Þ2 I03 :ð6Þ

The observed diffraction intensity I is given by
I¼

N
P

Ij ;

ð7Þ

j¼1

which means that the statistical distribution of the total
diffraction intensity I is identical to the multiple convolutionof distribution of intensity from each crystallite, that is, theprobability density function of the total diffraction intensityPðIÞ is given by
!
N
R1
R1
P
PðIÞ ¼     I  Ij p1 ðI1 Þ    pN ðIN Þ dI1    dIN : ð8Þ
0

j¼1

0

Since the cumulant of the convolution is equal to the sum of
the cumulants of the component functions, the average,
variance and third central moment of the total intensity areexactly given by
hIi ¼

R1

IPðIÞ dI ¼ NI0 ;ð9ÞhðIÞ2 i ¼ Nð1  ÞI02 ;ð10ÞhðIÞ3 i ¼ Nð1  Þ½2 þ ð1  Þ2 I03 :

ð11Þ

0

ð2Þ

for the diffraction angle 2. In a typical case of the source
width wsource = 0.1 mm, goniometer radius RG = 185 mm, axial
divergence angle  = 2.5 , diffraction angle 2 = 30 andmultiplicity m = 2, the probability is estimated as  ’ 1.5 
105 .
It has been suggested that the solid angle of tolerance(X) or an instrumental constant () can be calibrated by measurements of powder with known crystallite size
distribution (Ida et al., 2009).
It is more difﬁcult to connect each of  and X directlywith the optical geometry of a synchrotron beamline and ahigh-resolution diffractometer with crystal analyser, as
compared with the case of a laboratory Bragg–Brentano
diffractometer. Hereafter, it is assumed that the parameterðÞ can be regarded as an instrumental constant determined by the setup of the synchrotron powder diffraction
measurement system. It should be noted that the validity of

912

this assumption is not obvious and should be experimentally
examined.
The probability density function pj ðIj Þ for the statistical
distribution of diffraction intensity Ij from the jth crystallite
can be expressed by

The effective number of diffracting crystallites Neff, deﬁned
as the ratio of the square of the average to the variance of thetotal intensity I (Alexander et al., 1948), is given by thefollowing equation:
Neff 

hIi2
N
’ N;
¼
hðIÞ2 i 1  

ð12Þ

where the last approximation will be justiﬁed for   1. The
quantity Neff can be evaluated experimentally and regarded as
the effective number of diffracting crystallites, because it isequal to the expectation value of the number of particles thatsatisfy the diffraction condition in the case of the nonabsorbing sample of uniform crystallite size assumed in thissection.

Particle statistics of a capillary specimen

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J. Appl. Cryst. (2011). 44, 911–920

research papers
The skewness of the distribution of diffraction intensity
expected to be observed for the non-absorbing crystalline
powder of uniform particle size is given byhðIÞ3 i2 þ ð1  Þ2
1¼’ 1=2 ;
S¼
2 3=2
1=2
hðIÞ i½Nð1  ÞNeffkfor overlapping reﬂections with component multiplicity mk
and intensity Ik (Ida et al., 2009). The concept of the effective
multiplicity certainly simpliﬁes the analytical procedures
related to the variance of the observed intensities, because thetreatment for overlapping non-equivalent reﬂections only
requires replacement of m by meff for equivalent reﬂections.
However, this concept should be modiﬁed to treat the skewness of the diffraction intensity distribution.
Consider that two groups of non-equivalent reﬂections ofintensities I1 and I2 with multiplicity m1 and m2 are located at
the same peak position. The total probability  to satisfy thediffraction condition should be divided into component
probability 1 and 2 as follows: ¼ 1 þ 2 ;ð15Þ

1 ¼ m1 X=4;ð16Þ

2 ¼ m2 X=4:

ð17Þ

The average, variance and third-order moment of the total
intensity are given byhIi ¼ Nð1 I1 þ 2 I2 Þ;ð18ÞhðIÞ2 i ’ Nð1 I12 þ 2 I22 Þ;ð19ÞhðIÞ3 i ’ Nð1 I13 þ 2 I23 Þ:

ð20ÞThen the formula of skewness for an overlapping reﬂection
should be modiﬁed according to the following formula:
J. Appl. Cryst. (2011). 44, 911–920

hðIÞ3 i
m1=2
eff
’
;
hðIÞ2 i3=2 ðm0eff Neff Þ1=2

ð21Þ

where m0eff is deﬁned by

ð13Þ

where the last approximation is justiﬁed for   1 again. The
above equation implies that the statistical distribution of theobserved diffraction intensity is intrinsically asymmetric, but
the asymmetry will be reduced by increasing the number ofdiffracting crystallites.
2.1.2. Effect of overlapping non-equivalent reflections. The
reﬂection peaks of a Friedel pair always appear at the same
position in powder diffractometry, but the component
diffraction intensities may not have a common value when
there is no inversion centre in the crystallographic symmetry.
Other types of systematic overlap of non-equivalent reﬂections can also be caused by the space-group symmetries. The
relative intensities of the 101 and 011 reﬂections of quartz(P31 21 or P32 21) are approximately 2:1, for example (Alexander et al., 1948).
The author has introduced the effective multiplicity meff,deﬁned by

2 
1
P
P
meff 
mk I kmk Ik2
;
ð14Þ
k

S¼

m0eff  ðm1 I12 þ m2 I22 Þ3 ðm1 I13 þ m2 I23 Þ2 :ð22Þ

It is easy to derive a more generalized deﬁnition of m0eff ,given by

3 
2
P
P
0
2
3
meff 
m k Ikm k Ik:ð23Þ
kk

Since the effect of reﬂection multiplicity on the skewness ofthe observed intensity distribution is generally expressed by
the factor ðmeff =m0eff Þ1=2 , it will be omitted in the following
discussions about other contributions to observed diffraction
intensities.
2.1.3. Effect of Soller slits. It is assumed that the axial(horizontal) divergence of the incident synchrotron X-ray
beam is negligible, but the orientation of a diffraction plane ina crystallite is allowed to deviate along the axial direction, thisdeviation being restricted only by the opening angle of theSoller slits attached on the diffracted-beam side of the highresolution synchrotron powder diffractometer.
The dependence of the diffracted beam intensity on theaxial deviation angle is given byðI0 =Þð1  j j=Þ ½j j < ;ð24ÞIð Þ ¼
0½ j j;where  is the angle determined by the ratio of the distance tothe length of metallic foils in the Soller slits.
The effective axial divergence angle  is then given by 
2  
1
RR
Ið Þ d½Ið Þ2 d¼ 3=2:
ð25Þ
 ¼




The contribution of the Soller slits to the skewness of theintensity distribution is formalized by introducing anotherparameter, 0 , given by)2 
3 (  0
R
R
2
3
0
 
½Ið Þ d½Ið Þ d¼ 32=27: ð26Þ0The effect of the Soller slit geometry on the skewness of theobserved intensity distribution is expressed by a factor ofð=0 Þ1=2 ¼ 9=8 ¼ 1:125.
2.1.4. Effect of equatorial resolution of the diffractometer.
The equatorial deviation of the orientation of a diffraction
plane is restricted by the fundamental resolution function,
which is affected by the spectroscopic intensity distribution,
the equatorial (vertical) divergence of the source X-ray andthe diffraction condition at the analyser crystal on thediffracted-beam side of the high-resolution goniometer.
It is assumed that the proﬁle of the resolution function ismodelled by the Pearson VII function given by


ðÞ
!2
1þ 2
;
ð27Þ
fP7 ð!; P7 ; Þ ¼ 1=2
 ð  1=2ÞP7
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Particle statistics of a capillary specimen913

research papers
where P7 and  are the width and shape parameters, respectively. ðxÞ is the  function deﬁned byðxÞ 

R1 x1
t expðtÞ dt:

ð28Þ

0

The full width at half-maximum (FWHM) of the Pearson VII
proﬁle is given by wP7 ¼ 2ð21=  1Þ1=2 P7.
The effective equatorial divergence parameter  for thePearson VII proﬁle should be given by 1
2  1
1
R 
R
2
fP7 ð!Þ d!
fP7 ð!Þ d!
 ¼
1

¼

2

ð29Þ

In the Gaussian ( ! 1) and Lorentzian ( ! 1) limits, thevalues of  are reduced to ð ! 1Þ ¼ 1=2 P7 andð ! 1Þ ¼ 2P7 .
The modiﬁed effective equatorial angle of the Pearson VII
proﬁle is then given by 1
3  1
2
R 
R 
2
3
fP7 ð!Þ d!
fP7 ð!Þ d!
0 
1

¼

1

1=2 ½ð3Þ2 ½ð2  1=2Þ3
P7 ;½ð2Þ3 ½ð3  1=2Þ2

When the resolution function is expressed by the Gaussian( ! 1), the skewness of the observed intensity distribution
is not changed from that of the uniform distribution, that is,ð=0 Þ1=2 ¼ 1. In the Lorentzian ( ¼ 1) case, the effect isexpressed by ð=0 Þ1=2 ¼ 3=2.
2.2. Effect of absorption in capillary transmission mode

In this section, the effects of absorption in capillary transmission mode upon the observed intensity distribution areformalized.
The diffraction intensity from a crystallite in a cylindrical
specimen of linear absorption coefﬁcient  and radius R isgiven byn 
1=2
Ij ðrj ; ’j Þ ¼ I0 exp  R2  r2j sin2 ð’j þ Þo
1=2  R2  r2j sin2 ð’j  Þ
 exp 2rj sin  sin ’j ;

ð32Þwhen the crystallite is located at the position
ðxj ; yj Þ ¼ ðrj cos ’j ; rj sin ’j Þ in a coordinate system with theorigin on the centre of the cylinder.
T. Ida2 ( )

0

30

60

90

120

150

AðR; Þ
fcap
gcap

0.435
0.963
1.049

0.438
0.946
1.075

0.446
0.906
1.120

0.459
0.865
1.146

0.473
0.833
1.152

0.484
0.814
1.152

Since the overall transmittance AðR; Þ of a cylinder,
related to the absorption correction factor A ðR; Þ asAðR; Þ ¼ 1=A ðR; Þ, is given by (Dwiggins, 1975)
AðR; Þ ¼

n 
R RR
2 =2
1=2
r
exp
 R2  r2 sin2 ð’ þ Þ
R2 =2 0o
1=2  R2  r2 sin2 ð’  Þ
 expð2r sin  sin ’Þ dr d’;

ð33Þ

the average of the kth power of the diffraction intensity from acrystallite is generally given by
hIjk i ¼ I0k AðkR; Þ:

ð34Þ

Therefore, the effective number of diffracting crystallites
Neff should be modiﬁed by the factor of

ð30Þ

and the values for the Gaussian and Lorentzian limits are0 ð ! 1Þ ¼ 1=2 P7 and 0 ð ! 1Þ ¼ 8P7 =9.
The factor ð=0 Þ1=2 for the equatorial distribution
expressed by the Pearson VII proﬁle is given by

1=2

½ð2Þ2 ð3  1=2Þð  1=2Þ
¼
:
ð31Þ
0
½ð2  1=2Þ2 ð3ÞðÞ

914

Overall transmittance AðR; Þ and modiﬁcation factors fcap, gcap for acylinder of R ¼ 0:5.

1

 ð2Þ½ð  1=2ÞP7 :½ðÞ2 ð2  1=2Þ
1=2Table 1

fcap ðR; Þ ¼ ½ AðR; Þ2 =Að2R; Þ;ð35Þ

the value of which lies in the range 0 < fcap ðR; Þ < 1. Thisimplies that the absorption by the specimen not only reduces
the irradiated volume but also causes additional variation inmeasured diffraction intensities.
A modiﬁed factor given by0ðR; Þ ¼ ½ Að2R; Þ3 =½ Að3R; Þ2
fcap

ð36Þ

is introduced in a similar way to the formulations given in x2.1.
The skewness should then be modiﬁed by the factor of"#1=2
fcap ðR; ÞAð3R; Þ AðR; Þ¼: ð37Þ
gcap ðR; Þ ¼ 0
fcap ðR; Þ½ Að2R; Þ2
Table 1 lists the values of overall transmittance AðR; Þ,and factors fcap and gcap calculated for a cylinder of R ¼ 0:5
and diffraction angles 2 ¼ 0; 30; . . . ; 150. The functionAðR; Þ can be evaluated efﬁciently by a method recently
developed by the author (Ida, 2010).
It should be noted that the distribution of the observed
diffraction intensities should certainly be affected by theabsorption, but the difference from the non-absorbing case is
not very large in the case of R ¼ 0:5.
2.3. Effect of particle size distribution

It is assumed that the statistical distribution of the volume vj
of a crystallite is expressed by a probability density function
psize ðvj Þ. Since the diffraction intensity from a crystallite is
proportional to the particle volume, the probability density
function for the diffraction intensity from a crystallite is given
byParticle statistics of a capillary specimen

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J. Appl. Cryst. (2011). 44, 911–920

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pj ðIj Þ ¼ ð1  Þ ðIj Þ þ 

R1ðIj  Iv vj Þ psize ðvj Þ dvj

0¼ ð1  Þ ðIj Þ þ  psize ðIj =Iv Þ=Iv ;ð38Þ

where Iv is the diffraction intensity per unit volume.
The statistical average, variance and third central momentof the intensity from a crystallite are then given by
hIj i ¼

R1

Ij pj ðIj Þ dIj ¼ Iv hvj i;ð39ÞhðIj Þ2 i ¼ Iv2 hv2j i  ðvj i2 ;

ð40Þ

0


hðIj Þ3 i ¼ Iv3 hv3j i  3hv2j ihvj i þ 22 ð2  Þhvj i3 ;R1 k
vj psize ðvj Þ dvj :khvkj i ¼ D3m =6 exp 9k2 !2 =2 :

ð42Þ

ð48Þ

The effective particle volume veff and diameter Deff forparticle statistics and the modiﬁcation factor eff for the
skewness of intensity distributions are given by

ð41Þ

where hvj i, hv2j i and hv3j i are the average, squared and cubic
averages of the particle volume, respectively, which aredeﬁned by the following common formula:hvkj i 

where Dm is the median diameter and ! is the logarithmic
standard deviation. When the particles are assumed to havespherical shape, the average of the kth power of the particle
volume is given byveff  hv2j i=hvj i ¼ ðD3m =6Þ exp 27!2 =2 ;ð49ÞDeff  ð6veff =Þ1=3 ¼ Dm exp 9!2 =2 ;

ð50Þ

3
2 2
2
eff  hvj ihvj i=hvj i ¼ expð9! Þ:ð51Þ

It will not be difﬁcult to derive formulae for nonsphericalshapes and/or another model for size distribution, if anappropriate model is available.

0

The effective number of diffracting crystallites for a nonabsorbing sample is given by
Neff ¼

Nhvj i2Nhvj i2;’hv2j ihv2j i  hvj i2

ð43Þ

where the approximation holds for small . The above equation can be rewritten asf Vhvj i;
Neff ’
hv2j ið44Þ

for the irradiated volume V and ﬁlling factor of powder f ,because the number of particles is given by N ¼ f V=hvj i. Thismeans that the effective number of diffracting crystallites ofany size distribution is equivalent to the case of uniform
particle volume of hv2j i=hvj i.
The ﬁlling factor f relates the linear absorption coefﬁcientsof the powder specimen  and bulk material 0 by  ¼ f 0.
The skewness of the statistical distribution of the diffraction
intensity is given by
S¼

hv3j i  3hv2j ihvj i þ 22 hvj i3
ðNÞ

1=2

3=2
hv2j i  hvj i2

’

hvi
:
ðNÞ1=2 hvi3=2

ð45Þ

It is related to the effective number of diffracting crystallites
by the following equation:S’

hv3j ihvj i
1=2 2 2
Neff
hvj i

;

ð46Þ

which means that the skewness should be varied proportionally to the value hv3j ihvj i=hv2j i2, even if the effective number ofdiffracting crystallites Neff is unchanged.
The density function of lognormal distribution for theparticle diameter Dj is given by



Dj 2
1
1
pLN ðDj Þ 
exp  2 ln
;
ð47Þ
2!
Dm
ð2Þ1=2 !Dj
J. Appl. Cryst. (2011). 44, 911–920

2.4. Combined formulae for particle statistics

Finally, the combined formulae for the effective number ofdiffracting crystallites ðNeff Þcap and the skewness of theintensity distribution Scap measured in a capillary transmission
mode are presented in this section.
The effective number of diffracting crystallites is given byeff R2 Lhvj i½ AðR; Þ2;ðNeff Þcap ¼
0 hv2j iAð2R; Þð52Þ

where  and 0 are the linear absorption coefﬁcients of thepowder specimen and bulk material, respectively, R is theradius of the capillary, L is the width of the cross section of the
X-ray beam along the axial direction, and the functionAðR; Þ is the transmission coefﬁcient of the capillary at the
Bragg angle . The parameter eff is the effective probability
that each crystallite satisﬁes the diffraction condition, given byeff ¼

meff 
;8 sin 

ð53Þ

for the effective multiplicity meff and the allowed deviation
angles along the axial and equatorial directions,  and ,
respectively. The formulae for meff ,  and  have been
given in equations (14), (25) and (29). The deﬁnition of hvkj i isgiven by equation (42).
The skewness of the intensity distribution is given by
Scap ¼ðmeff Þ1=2 hv3j ihvj iAð3R; Þ AðR; Þ½ðNeff Þcap m0eff 0 0 1=2 hv2j i2 ½Að2R; Þ2

;

ð54Þ

where m0eff , 0 and 0 are the parameters given by equations (23), (26) and (30), respectively.

3. Experimental
3.1. Powder samples

Three fractions of quartz powder separated by a sedimentation method were used without further grinding or
T. Idaelectronic reprint

Particle statistics of a capillary specimen915research papers
Table 2

Table 3

Parameters of size distribution evaluated by SEM image analysis of threefractions of quartz powder separated by a sedimentation method.

Mixing ratio and parameters of size distribution for three samples ofquartz powder.

Stokes diameter (mm)

3–7

8–12

18–22

w(3–7), w(8–12) and w(18–22) are the weight fractions of component powders
with the nominal Stokes diameter 3–7, 8–12 and 18–22 mm, respectively.

Dm (mm)!
Deff (mm)4.831 (6)0.2836 (11)
7.0
2.06

9.239 (18)0.2400 (12)
12.0
1.68

21.03 (2)0.1937 (11)
28
1.40

eff

sieving. The nominal Stokes diameters of the fractions were 3–7, 8–12 and 18–22 mm. The crystallite size distributions of thefractions were determined by analysis of scanning electron
microscopy (SEM) images (Ida et al., 2009). The characteristic
parameters for the optimized lognormal distributions arelisted in Table 2. The effective diameter Deff is calculated by
Deff ¼ ð6veff =Þ1=3 from the effective particle volume veff .
Three powder samples (1, 2 and 3) with similar averages anddifferent degrees of dispersion about the size distribution wereprepared as controlled mixtures of the separated fractions.
The effective particle diameter Deff and modiﬁcation factor3
2 2
eff ¼ hvj ihvj i=hvj i for the mixture are calculated by

1=3
P
3
2
Deff ¼
wn ðDm Þn expð27!n =2Þ;ð55Þ

Sample123

w(3–7) (%)
w(8–12) (%)
w(18–22) (%)
Deff (mm)

0
100
0
12.0
1.68

54.2
42.2
3.6
11.3
6.67

92.6
0
7.4
11.3
11.8

eff

Table 4Characteristic parameters of capillary specimens.
2R is the diameter of the capillary,  is the linear absorption coefﬁcient, andfcap (101/011) and gcap (101/011) are the modiﬁcation factors for the 101/011
reﬂection of quartz at the diffraction angle 2 = 20.625 .
Sample12

3

2R (mm) (cm 1)
R
fcap (101/011)
gcap (101/011)

0.575 (1)
17.11 (9)
0.492
0.970
1.060

0.547 (1)17.34 (8)
0.474
0.972
1.054

0.508 (1)15.50 (8)
0.394
0.982
1.034



n

P
eff ¼ 

2 ;

n

P

I2 ðzÞ ¼

wn ðDm Þ6n expð36!2n Þ
ð56Þ

wn ðDm Þ3n expð27!2n =2Þnwhere wn, ðDm Þn and !n are the weight fraction, median
diameter and logarithmic standard deviation of the nthcomponent, respectively. The weight fractions and calculated
values of Deff and eff are listed in Table 3.
The three powder samples 1, 2 and 3 were ﬁlled into Lindemann glass capillaries with a nominal diameter of 0.5 mm.
3.2. Measurements

A high-resolution synchrotron powder diffractometerequipped with a ﬂat Ge(111) crystal analyser on the beamline
BL4B2 at the Photon Factory in Tsukuba was used for
diffraction measurements. The peak wavelength of the source
X-ray beam calibrated with the peak positions of a standard Si
powder (NIST SRM640c) was 1.20670 (3) Å.
The transmittance intensity proﬁle was recorded by scanning the vertical position of the capillary samples at the step
interval of 0.01 mm. The cross section of the incident beam
was restricted to be 0.05 mm along the vertical direction withthe entrance slit of the diffractometer.
The observed intensity proﬁle IðzÞ was ﬁtted by thefollowing model:IðzÞ ¼ I1 ðzÞ I2 ðzÞ;
I1 ðzÞ ¼

916

I0 exp 2½R2  ðz  z0 Þ2 1=2
I0

T. Ida

1=H
0

for
for

jzj < H=2;
H=2 jzj;

ð59Þ

where I0 is the intensity of the incident beam, R the radius of
the capillary, z0 the centre position and H the height of thecross section of the incident beam, restricted by the entrance
slit.
The results of ﬁtting for the intensity proﬁle of sample 1 areshown in Fig. 1, and the values of optimized parameters for thethree capillary specimens are listed in Table 4. The values of
modiﬁcation factors fcap and gcap for cylindrical samples,
calculated for the 101/011 reﬂection of quartz at the diffraction
angle 2 = 20.625 , are also listed in Table 4.
The 2 scan peak proﬁles of the quartz 101/011 reﬂection
were recorded for capillary specimens 1, 2 and 3. The step
interval was 0.01 in 2, and the measurement time was 0.5 s

ð57Þ
for
for

jz  z0 j < R;
R jz  z0 j;
ð58Þ

Figure 1
Intensity proﬁle on a vertical scan of capillary specimen 1. The observed
intensities are marked by crosses, and the ﬁtted curve is drawn as a solidline.

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per step. The capillary specimens were continuously rotated ata speed of 2 r s1 during the measurements.
Five thousand peak-top intensities of the quartz 101/011
reﬂection were recorded for each sample on stepwise rotationof the capillary specimens over 360 at an interval of 0.072 .

4. Results and discussions
4.1. Peak profile

Fig. 2 shows the 2 scan peak proﬁles of the 101/011
reﬂection of the quartz powder specimens 1, 2 and 3. The
observed proﬁles were ﬁtted by the convolution of thePearson VII proﬁle fP7 ðx; P7 ; Þ shown in equation (27) andthe axial-divergence aberration function wA ðx; Þ (Ida &Hibino, 2006). The formula of the ﬁtting function pP7 A ð2Þ isgiven by
pP7 A ð2Þ ¼ b þ IfP7 ð2  2; P7 ; Þ wA ð2; Þ;wA ðx; Þ ¼

ð60Þj A j1 ½ðx= A Þ1=2  1 for  1 < x= A < 0;0elsewhere;
ð61Þ

Table 5
Optimized parameters of proﬁle ﬁtting.
b is the constant background, I is the integrated intensity, 2 is the peaklocation, and wP7 and  are the FWHM and shape parameters of the symmetric
Pearson VII function. The errors estimated for 2 and wP7 are omitted,
because they are smaller than the mechanical precision of about 0.0001 .
Sample

1

2

3

b
I
2 ( )
wP7 ( )

143 (3)828 (2)20.6255
0.01901.480 (9)143 (4)784 (2)20.62530.02011.453 (9)143 (4)703 (2)20.62480.02051.432 (10)2A ¼ ð =2Þðcot 2 þ tan A Þ;

ð62Þ

where b is the constant background, I the integrated intensityand A the Bragg angle of the analyser crystal.
The optimized values of ﬁtting parameters are listed inTable 5. The smaller values of the shape parameter  for the
samples with broader size distribution may be related to thetheories (Langford et al., 2000; Popa & Balzar, 2002; Ida et al.,2003) that predict that the intrinsic diffraction peak proﬁlefrom small crystallites becomes more ‘Lorentzian-like’ for amore dispersed crystallite size distribution.
The effects on the equatorial intensity distribution as
calculated by equation (31) are ð=0 Þ1=2 ¼ 1:321, 1:326
and 1:330 for samples 1, 2 and 3, respectively.
4.2. Spinner-scan data

Figure 2
The 2 diffraction proﬁles of the 101/011 reﬂection of quartz powder
samples (a) 1, (b) 2 and (c) 3. The observed intensities are marked bycrosses, and the ﬁtted curves are drawn as solid lines in the lower panels.
The upper panels show the difference plots (crosses) and errors predictedby counting statistics (grey area).
J. Appl. Cryst. (2011). 44, 911–920

The spinner-scan intensity proﬁles measured for samples 1,
2 and 3 are shown in Fig. 3. The statistical independence of thedata is supported by the lack of systematic variation in theobserved intensity proﬁles. Evolution of asymmetry in thestatistical distribution of the observed diffraction intensities
for samples with more dispersed particle size distribution canbe viewed directly in the proﬁle in Fig. 3. The histograms of
the observed intensities are shown in Fig. 4.
The average, variance, third-order central moment andskewness of the observed intensity distributions are listed inTable 6. The errors in parentheses are estimated from thehigher-order central moments of the observed data (seeAppendix A).
The estimated relative error of the average is equal to1=ðNNeff Þ1=2 , which is expected for a Poisson distribution withthe expectation value of Neff by deﬁnition. The relative errorsof the variance estimated at 0.032, 0.040 and 0.062 for samples
1, 2 and 3 are of the same order as the value 2=N 1=2 ¼ 0:028
expected for a Poisson distribution. The relative errors for thethird-order moment estimated at 0.16, 0.15 and 0.15 forsamples 1, 2 and 3 are also of the same order as the values
ð6Neff =NÞ1=2 = 0.28, 0.30 and 0.26, respectively. The resemblance of the observed statistics to the Poisson distribution
with the expected value of Neff suggests that the variation of
the measured diffraction intensities is certainly dominated by
the ﬁnite number of diffracting crystallites.
The effective multiplicities of the quartz 101/011 reﬂection
are estimated at meff ¼ 10:27 and m0eff ¼ 8:45 from theT. Idaelectronic reprint

Particle statistics of a capillary specimen917

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Table 6
Characteristics of spinner-scan data.
The value of the effective diameter Deff of sample 1 estimated by SEM image
analysis is treated as the reference to determine the instrumental constant.
Sample

1

2

3

hI i
hðIÞ2 ihðIÞ3 iS
Neff
Deff (mm)28370 (50)
1.29 (4)  107
2.6 (4)  1010
0.56 (9)62 (3)
12.0 (ﬁxed)
2.8 (5)25430 (40)
8.7 (4)  106
3.1 (5)  1010
1.22 (19)74 (5)
11.0 (7)6.7 (1.0)22380 (40)
8.8 (5)  106
7.5 (11)  1010
2.9 (5)56 (6)
11.1 (1.2)
14 (2)

eff

component multiplicities m101 ¼ m011 ¼ 6 and the calculated
relative intensities of I011 =I101 ¼ 2:393 obtained by the preliminary Rietveld analysis using RIETAN-FP developed by
Izumi & Momma (2007).
The effective crystallite diameter of sample 1 estimated at
Deff ¼ 12:0 mm by the SEM image analysis is treated as thereference value to calibrate the instrumental constant in thisstudy. The effective equatorial tolerance angle is estimated at
X = 0.061 (2) from the effective number of diffracting
crystallites Neff ¼ 62 (3) evaluated from the spinner-scan data,assuming R ¼ 0:492, R ¼ 0:288 mm, ½AðR; Þ2 =Að2R;Þ ¼ 0:970, =0 ¼ 17:11=48:49 ¼ 0:3529, L ¼ 10 mm and = 1.5
The effective diameters of samples 2 and 3 listed in Table 6
are evaluated from the observed average and variance of thespinner-scan data and the instrumental constants. The values
coincide with those estimated from the SEM image analysis
listed in Table 3 within the estimated experimental errors.
The crystallite size dispersion parameters eff evaluated forsamples 1, 2 and 3 by statistical analysis of the observed
spinner-scan data are listed in the last row of Table 6. Goodagreement with the values estimated by the SEM image
analysis have been found, except that the value of eff determined for sample 1 by the spinner-scan method is slightly
larger than that predicted by the SEM image analysis.
The relative errors of the skewness estimated for samples 1,
2 and 3 are 0.16, 0.16 and 0.17, while those predicted for the
Poisson case (see Appendix A4) for N ¼ 5000 are about 0.3.
The area-weighted average diameter deﬁned by
hDiA  hD3 i=hD2 i can be connected with the experimentallyobtained values of Deff and eff by the equation
hDiA ¼ Deff 2=9
eff , when the lognormal distribution is assumed.
The values of hDiA for samples 1, 2 and 3 are then estimated at
9.5 (8), 7.2 (7) and 6.2 (8) mm, while the values calculated fromthe mixing ratios wj and the parameters Dm and ! for each

Figure 3
The spinner-scan proﬁles of the 101/011 reﬂection of quartz powder
samples (a) 1, (b) 2 and (c) 3. The observed intensities are drawn as solidlines.

918

T. IdaFigure 4
The histograms of the 101/011 reﬂection intensity distribution of quartz
powder samples (a) 1, (b) 2 and (c) 3.

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fraction determined from the SEM image analysis are 10.7, 7.5
and 6.3 mm.

and the expected value of X coincides with the true meanvalue, that is, hXi ¼ x.
The unbiased estimation of the variance is given by
V ¼ NðN  1Þ1 M2 ;5. Conclusion
Observed powder diffraction intensities are considered to be
the multiple convolution of the diffraction from each crystallite. Quantitative descriptions of the statistical properties ofthe observed intensities have been derived on the basis of theadditivity of cumulants with respect to convolution.
Practical formulae for the particle statistics of capillaryspecimens in synchrotron powder diffractometry arepresented. The skewness of the observed intensity distribution
is predominantly affected by a size dispersion parameter,
which can be connected with the broadness of the crystallite
size distribution.
The errors in the experimentally estimated skewness were
evaluated by utilizing the sample central moments up to sixthorder. The experimental values of skewness evaluated fromcollections of diffraction intensity data measured by a spinnerscan method agree well with the values predicted from theresults of SEM image analysis.

A1. Cumulants@

where Mk is the kth-order sample central moment deﬁned by
Mk  N 1

R1lnexpðxÞ pðxÞ dx;
k  lim!0 @ k1k ðx  xÞ pðxÞ dx;

ð63Þ

T ¼ N 2 ðN  1Þ1 ðN  2Þ1 M3 ;so that hTi ¼

k

ð64Þ

x pðxÞ dx:

ð65Þ

where x is the average given by
R1

ð69Þ

3.

A3. Errors in estimation of statistical parameters

The statistical variance of the sample average X is given byðXÞ2  hðX  xÞ2 i ¼

2=N ’ M2 =N:ðVÞ2  hðV 

Þ i ’ ðM4  M22 Þ=N:

2 2

ð70Þ

ð71Þ

The statistical variance of the unbiased third-order cumulant T calculated by equation (69) is similarly approximatedbyðTÞ2  hðT  3 Þ2 i
’ ðM6  6M4 M2  M32 þ 9M23 Þ=N:

1

x

ð68Þ

Note that the above formula for the unbiased sample varianceis determined to satisfy that the expected value of V should beequal to 2, that is, hVi ¼ 2 .
The unbiased estimation of the third-order cumulant issimilarly given by

while the kth-order central moment  is deﬁned by
R1ðXj  XÞk :j¼0

k for a probability density func-

k

N1
P

The statistical variance about the unbiased sample variance
V calculated by equation (67) can be estimated as the
expected value of the squared difference of V from the truevariance 2, which is approximated by

APPENDIX A
Estimation of cumulants up to the third order
The kth-order cumulanttion pðxÞ is deﬁned by

ð67Þ

ð72Þ

The variance in the estimation of the skewness S ¼ T=V 3=2
is formally given by

2 
 2
T 2 T
3 V
2ðSÞ ’ 3
;
ð73Þ
þ
T
2 V
V

1

The ﬁrst, second and third-order cumulants are equivalent
to the mean x, the variance 2 and the third-order central
moment 3 , that is, 1 ¼ x, 2 ¼ 2 ¼ 2 , 3 ¼ 3 . The
cumulants from the fourth to sixth order are related to thecentral moments k by 4 ¼ 4  322, 5 ¼ 5  103 2 and23
6 ¼ 6  154 2  103 þ 302 .

when the correlation between the variance and third-order
cumulant is neglected. Even though there should be somecorrelation between V and T, the value of S calculated by
the above equation will still be a measure of the error inestimation of the skewness S, because the relative error T=T
will be much larger than V=V.
A4. Poisson case

A2. Unbiased estimation

Consider N independent sample data fXj g ( j = 0, . . . , N  1)
about a stochastic variable with the mean x, variance 2 andthird-order cumulant 3 . The sample average is deﬁned by
X  N 1

N1
P

Xj ;

ð66Þ

Since all of the cumulants of the Poisson distribution are
equal to the expected value of the stochastic variable, theerrors in estimations of the average, variance and third-order
cumulant for the Poisson distribution for the expected value of
Neff will be approximated by
X ’ ðNeff =NÞ1=2 ;j¼0

J. Appl. Cryst. (2011). 44, 911–920

T. Idaelectronic reprint

Particle statistics of a capillary specimen

ð74Þ

919

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V ’
T ’





2
Neff þ 4Neff
=N

1=2

;

2
3
þ 6Neff
Neff þ 18Neff
=N

ð75Þ
1=2

:

ð76Þ

When Neff is sufﬁciently large, the relative errors for X, V, T
and the skewness S can be further approximated by
X=X ’ 1=ðNNeff Þ1=2 , V=V ’ 2=N 1=2 and T=T ’
S=S ’ ð6Neff =NÞ1=2 .
This work has been performed under the approval of the
Photon Factory Program Advisory Committee (proposal No.
2009G131).

920

T. IdaReferences
Alexander, L., Klug, H. P. & Kummer, E. (1948). J. Appl. Phys. 19,
742–753.
De Wolff, P. M. (1958). Appl. Sci. Res. 7, 102–112.
Dwiggins, C. W. (1975). Acta Cryst. A31, 146–148.
Ida, T. (2010). J. Appl. Cryst. 43, 1124–1125.
Ida, T., Goto, T. & Hibino, H. (2009). J. Appl. Cryst. 42, 597–606.
Ida, T. & Hibino, H. (2006). J. Appl. Cryst. 39, 90–100.
Ida, T., Shimazaki, S., Hibino, H. & Toraya, H. (2003). J. Appl. Cryst.
36, 1107–1115.
Izumi, F. & Momma, K. (2007). Solid State Phenom. 130, 15–20.
Langford, J. I., Louër, D. & Scardi, P. (2000). J. Appl. Cryst. 33, 964–
974.
Popa, N. C. & Balzar, D. (2002). J. Appl. Cryst. 35, 338–346.

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J. Appl. Cryst. (2011). 44, 911–920