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Evaluation of particle statistics in powder diffractometry by aspinner-scan method
T. Ida, T. Goto and H. Hibino

J. Appl. Cryst. (2009). 42, 597–606

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J. Appl. Cryst. (2009). 42, 597–606

Ida, Goto and Hibino · Particle statistics

research papers
Evaluation of particle statistics in powder
diffractometry by a spinner-scan method

Journal ofApplied
Crystallography
ISSN 0021-8898

T. Ida,* T. Goto and H. Hibino
Received 2 March 2009
Accepted 28 May 2009

Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi 507-0071,
Japan. Correspondence e-mail: ida.takashi@nitech.ac.jp# 2009 International Union of Crystallography
Printed in Singapore – all rights reserved

The uncertainty in measured diffraction intensities caused by particle statistics,
which originates from the limited number of crystallites satisfying the diffraction
condition, has been evaluated by a step-scan measurement about the rotation
angle of a specimen-spinning attachment of a laboratory powder X-ray
diffractometer. The residual statistical variance of the spinner-scan intensity
data, after subtraction of periodic drift and variance caused by counting
statistics, was assigned to the variance caused by particle statistics. Particle
statistics for a standard Si powder (NIST SRM640c) and three size fractions(nominally 3–7, 8–12 and 18–22 mm in Stokes diameter) of quartz powder
separated by a sedimentation method have been analysed by scanning electron
microscopy (SEM) and the spinner-scan method using a powder X-ray
diffractometer. It has been conﬁrmed that the observed ratio of the squared
diffraction-peak intensity to the variance caused by particle statistics is
proportional to the multiplicity of reﬂections predicted by the crystal structure.
The spinner-scan intensity data for the standard Si powder (NIST SRM640c),
the effective particle diameter of which was estimated at 5.6 mm by SEM image
analysis, was used as the standard for crystallite-size evaluation of quartz
powder based on analysis of spinner-scan data. The effective crystallite
diameters of the three quartz powder samples have been estimated at 6.5 (2),11.7 (2) and 22.8 (2) mm by the analysis of the spinner-scan data, while theeffective particle diameters evaluated by SEM image analysis are 7.1, 12 and25 mm, respectively. Other possible applications of the analysis of particle
statistics based on the spinner-scan method are also discussed.

1. Introduction
It is known that the statistical uncertainty in measured X-ray
diffraction intensity data originates mainly from counting
statistics and particle statistics. The errors caused by counting
statistics can be simply approximated by the square root of themeasured number of counts, when the count rate is sufﬁciently
lower than the reciprocal of the response time of the detection
system. One of the authors recently proposed a practicalmethod to evaluate the statistical errors affected by the ﬁnite
response time of X-ray detection systems (Ida, 2008).
In contrast to counting statistics, which are a general issue invarious ﬁelds using nuclear counting, photon counting andneural counting (Teich, 1985), particle statistics are a problem
peculiar to powder diffractometry.
In the pioneering work on this subject by Alexander et al.
(1948), a theoretical framework for particle statistics has
almost been established. The relative deviation of thediffraction intensity caused by the restricted number ofdiffracting crystallites is simply given by
Iparticle =hIi ¼ 1=n1=2
eff ;J. Appl. Cryst. (2009). 42, 597–606

where neff is the effective number of crystallites that satisfy thediffraction condition. The effective number neff is connected tothe total number of irradiated crystallites N and the probability p that each crystallite satisﬁes the diffraction condition,that is, neff = Np when p is much less than unity.
The total irradiated number of crystallites N is given by
N ¼ f V=veff ;ð2Þwhere f is the ﬁlling factor of the powder sample and V is theirradiated volume, which is given by
V ¼ A1

ð3Þ

for the cross section of the X-ray beam A and the linear
absorption coefﬁcient of the specimen , when the diffraction
intensity data are measured using a divergence slit with a ﬁxed
open angle in symmetric reﬂection mode.
The effective particle volume veff in equation (2) is deﬁned
by the ratio of the mean squared volume to the mean volumeof the crystallites,

ð1Þveff ¼ hv2 i=hvi;
doi:10.1107/S0021889809020378

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ð4Þ

597

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when the crystallite size distribution is taken into account(Alexander et al., 1948).
Since the linear absorption coefﬁcient  of a powder withﬁlling factor f is given by  = f0 for the bulk linear absorption
coefﬁcient 0, the total irradiated number N is independent of
the ﬁlling factor, and equation (2) can be reduced to
N ¼ A=0 veff :ð5ÞThe probability p for randomly oriented crystallites instationary specimens can be approximated by
p ¼ meff !=4;

ð6Þ

where meff is the effective multiplicity of reﬂection, and !
and  are the tolerance angles for the normal orientation of
the diffraction plane to deviate along the equatorial and axial
directions, respectively (de Wolff, 1958).
When the effect of a rotating specimen is simpliﬁed as the
expansion of the tolerance area from that of a rectangle,
!, to a circle, ()2/4, the probability p for crystallites
in a rotating specimen will be given byp ¼ meff ðÞ2 =16:

ð7ÞThe effective multiplicity meff is deﬁned for an overlapped
reﬂection with component multiplicity mj and intensity Ij by
the following equation (Alexander et al., 1948):
2 

P
P
m j Ijmj Ij2 :ð8Þmeff ¼jj

The tolerance angles ! and  are given by
! ¼ w=R;ð9Þ ¼ h=2R sin ;

ð10Þ

where R is the goniometer radius,  is the Bragg angle, and w
and h are the effective width and length, respectively, of theline focus of the X-ray source (de Wolff, 1958).
Although the above formulae were originally intended todescribe the statistical properties of the integrated intensity ofa diffraction peak (Alexander et al., 1948), they can also be
applied to peak intensity, by modifying only the interpretationof the effective width w (de Wolff, 1958). The effective width wof the peak intensity is predominantly determined by thegeometry of the X-ray source, but it may also be affected by
the spectroscopic width of the X-ray beam and the width of
the receiving slit under the restrictions of the diffraction
condition. The effective length h is considered to be determined by the open angle A of the Soller slits, which arecommonly adopted in modern Bragg–Brentano diffractometers, as h = RA (Smith, 2001).
In a typical case of R = 185 mm, w = 0.1 mm and A = 5.0 ,the probabilities of crystallites in stationary and rotating
specimens satisfying the diffraction condition at a ﬁxed
rocking angle are estimated at p = 6.0  105 and 1.5  102,respectively, for the 111 reﬂection of Si with multiplicity m = 8
at the diffraction angle 2 = 28.4 , and the relative errors
caused by particle statistics for 5 mm crystallites in an irra-

598

diated volume of V = 3 mm3 will be about Iparticle/hIi = 1.9
and 0.12%, respectively.
It should be emphasized that the improved accuracy
obtained by rotating the specimen is mainly caused by thegeometry of the diffractometer, where the tolerance angle forthe normal orientation of the diffraction plane along the axial
direction,  = A/2sin, is much more generous than thatalong the equatorial direction, ! = w/R.
Since the aspect ratio /! is greater than 100 for lowerangle diffraction peaks, it is expected that a slight rotation of
the specimen by about 1 has a similar effect to reﬁlling the
crystalline powder in the sample holder. It is suggested thatquantitative analysis of particle statistics can be achieved bysimply recording the variation in the diffraction intensities on
rotation of the specimen.
In this study, we have conducted step-scan diffraction
intensity measurements of Si and quartz crystalline powder
samples with rotation of the specimen at ﬁxed diffraction
angles, and examined the validity of the application of thetheory proposed by Alexander et al. (1948) to intensity datacollected at a ﬁxed rocking angle. It will be shown thatquantitative evaluation of crystallite sizes larger than 1 mm isenabled by applying this method. Other possible applicationsof the method are also discussed.

2. Experimental
2.1. Samples

Standard Si powder (NIST SRM640c) was used without
further grinding or sieving. The median particle size of the Si
powder, determined by a laser scattering method, wasreported to be 4.9 mm in the certiﬁcate (Freiman & Trahey,2000).
Three fractions of quartz powder samples were prepared byseparating crushed and ground Brazilian quartz crystals by asedimentation method. The nominal Stokes diameters of thethree quartz samples were 3–7, 8–12 and 18–22 mm.
2.2. SEM image analysis

Scanning electron microscopy (SEM) images of powder
samples were taken with a ﬁeld-emission-type scanning electron microscope (Jeol JSM-7000F). Particle images wereextracted from SEM images with the aid of computer software
for image analysis (Scion Image; Scion Corporation, Maryland, USA). The numbers of extracted particle images were1049 for Si, and 1134, 1049 and 1391 for the 3–7, 8–12 and 18–
22 mm fractions of quartz powder, respectively.
The size of each crystallite was speciﬁed as the diameter of acircle with the same area as the particle image.
2.3. Spinner-scan measurements

A sample holder with a cylindrical hollow of ’ = 30 mm indiameter and 0.6 mm in depth was ﬁlled with the powder
samples. A home-made specimen spinner attached to aconventional powder diffractometer (Rigaku RAD-2C) with agoniometer radius of R = 185 mm was used for step-scan

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FW ðDÞ ¼  þ ð1 þ Þ P  þ 3=; ðD=D0 Þ ;Table 1
Reﬂections measured by the spinner-scan method.
hkl is the index of reﬂection, meff is the effective multiplicity of reﬂection, FT0
is the measurement time per step for Si, and FT1, FT2 and FT3 are those for the
3–7, 8–12 and 18–22 mm quartz samples, respectively.
Si (NIST SRM640c)

-quartz

hkl

meff

FT0 (s)hkl

meff

FT1 (s)FT2 (s)FT3 (s)

111
220
311
400
331
422
333/511
440
531
620
533

8
12
24
6
24
24
32
12
48
24
24

0.5
1
2.5
3
6.5
5.5
12.5
8.5
18
10
19

100
101/011110
102/012111/111200
201/021112/112202/022103/013211/121

6
9.85
6
8.23
12
6
10.39
12
9.24
6.01
23.72

1
0.5
2
3
5
3
5
1.5
5
12
3

1
0.5
2
2
4
3
5
1.5
5
9
3

1
0.5
2
2
5
3
5
1.5
5
10
3

measurements about the rotation angle of the specimen. A Cu
target sealed tube operating at 40 kV and 30 mA was used as
the X-ray source. The take-off angle of the source X-ray beamfrom the Cu target was about 4 and the effective width of the
X-ray source was estimated at w = 0.12 mm from the 2-scanintensity proﬁle of the direct beam. The divergence/scatteringslit open angles were ﬁxed at DS/SS = 1 and a receiving slit ofwidth 0.15 mm was used. The width of the X-ray beam,measured by locating a ﬂuorescent plate at the specimen
position, was Wbeam = 10 mm. The cross section of the beam at
the specimen position was estimated at A = WbeamRDS/SS =32 mm2. A curved graphite monochromator attached on thediffracted-beam side of the goniometer was adjusted for Cu K
wavelength.
400 diffraction intensity data were recorded by rotating thespecimen stepwise with an interval of 0.9 over 360 , for eachof 11 diffraction peaks at 2/ angles ﬁxed at the peak top
positions. The measurement time per step was varied for
different reﬂections, so that at least several hundred counts
were collected for each measurement step. The hkl indices ofthe measured reﬂections, the effective multiplicity meff and themeasurement time are listed in Table 1. The effective multiplicity of reﬂection was estimated based on the results ofRietveld analysis using the program RIETAN developed by
Izumi & Ikeda (2000).

ð11Þ

where  is the missing volume fraction of particles smaller
than the measurable size, D0 is a scale parameter, and  and 
are shape parameters for the model function (see Appendix
A). The normalized incomplete  function P(; x) is deﬁnedby1Pð; xÞ ðÞZ1ðÞ 

tZx

t1 expðtÞ dt;

0

1

ð12Þ
expðtÞ dt:

0

It should be noted that the formula of equation (11) reduces tothe usual  distribution for  = 1 and to the Weibull distribution for  = 1.
The optimized values of the ﬁtting parameters are  =0.0077 (4), D0 = 5.63 (7) mm,  = 0.210 (17) and  = 3.57 (9),where the error values in parentheses are estimated by

3. Results and discussion
3.1. SEM images

Typical SEM images of the Si (NIST SRM640c) and threefractions of -quartz powder samples are shown in Figs. 1 and
2. Extracted particle images of the Si sample are illustrated in
the lower panel of Fig. 1.
The cumulative volume distribution of Si powder obtained
by SEM image analysis is plotted in Fig. 3. Since the observed
size distribution could not be reasonably ﬁtted by any of the
log-normal,  or Weibull distributions, it was ﬁtted by a
modiﬁed formula of a -distribution function given by
J. Appl. Cryst. (2009). 42, 597–606

Figure 1
SEM image of standard Si powder (NIST SRM640c) (upper panel) andthe extracted particle proﬁles (lower panel).
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assuming a ﬁxed value of error D = 0.1 mm in the measured
diameter. As a result, it can be assumed that the missing

fraction  is negligible, similar to the assumption applied in thework of Alexander et al. (1948). The effective diameter of the
Si powder, deﬁned by Deff  ð6veff =Þ1=3, is estimated directlyfrom the raw data {Dj} by the equation
P 6 .P 3 1=3
Dj
Dj
;
ð13Þ
ðDeff Þraw ¼
jj

and from the ﬁtting parameters D0,  and  by the equation
1=3ð þ 6=Þ:
ð14ÞðDeff Þfit ¼ D0
ð þ 3=ÞThe effective diameter of the Si powder was estimated at(Deff)raw = 5.6 mm from the raw data and also at (Deff)ﬁt =5.6 mm from the ﬁtting parameters.
The median diameter for the volume distribution of
SRM640c numerically evaluated from the raw data ofmeasured diameter, and from the curve calculated using the
optimized ﬁtting parameters, were 5.3 and 5.2 mm, respectively, while the value reported in the certiﬁcate of SRM640c is
4.9 mm.
The diameter evaluated by the current method may beslightly overestimated, because the unknown diameter along
the viewing direction on capturing a SEM image is assumed tobe equal to the diameter of a circle with the same area as theparticle image, while it is expected that non-spherical particles
tend to adhere, making the shortest diameter normal to theface of the substrate. However, the effective diameter of Si isassumed to be (Deff)Si = 5.6 mm in this study.
The cumulative volume distributions of quartz powder
obtained by SEM image analysis are plotted in Fig. 4. The
volume distributions of the quartz samples are satisfactorily
ﬁtted by a model based on the log-normal distribution, which
is given by1þln D  ln Dm  3!2
1 þ erf;
FLN ðDÞ ¼  þ
2
ð2!Þ1=2
ð15Þ
where  is the missing volume fraction of particles smaller
than the measurable size, Dm is the median diameter and ! isFigure 3
Figure 2

SEM images of (a) 3–7 mm, (b) 8–12 mm and (c) 18–22 mm fractions of
-quartz powder.

600

Cumulative particle-size distribution of Si (NIST SRM640c), plotted as
black dots, and the optimized modiﬁed -distribution curve (grey line)
calculated with the parameters D0 = 5.63 mm,  = 0.210 and  = 3.57.

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the logarithmic standard deviation. The error function erf(x) is
deﬁned by2erfðxÞ  1=2


Zxexpðt2 Þ dt:

ð16Þ

0

Table 2 lists the values of the optimized ﬁtting parameters andeffective diameters, (Deff)raw and (Deff)ﬁt. The effective
diameter (Deff)ﬁt is calculated from the optimized ﬁtting
parameters Dm and ! for the log-normal distribution by theequationðDeff Þfit ¼ Dm exp 9!2 =2

ð17Þ

Table 2
Results of SEM image analysis of quartz powder.
D is the assumed error in the measured diameter,  is the missing volume
fraction below the measurable size, Dm is the median diameter, ! is thelogarithmic standard deviation, and (Deff)raw and (Deff)ﬁt are the values of theeffective diameter evaluated from raw data and optimized ﬁtting parameters,
respectively.
Stokes diameter (mm)

3–7

8–12

18–22

D (mm)
Dm (mm)!
(Deff)raw (mm)(Deff)ﬁt (mm)0.2
0.0009 (0)4.831 (6)0.2836 (11)
7.1
7.0

0.4
0.0000 (0)9.239 (18)0.2400 (12)
11.8
12.0

0.8
0.0003 (0)21.03 (2)0.1937 (11)
25.3
27.7

3.2. Spinner-scan data of Si powder(see Appendix B).

The observed spinner-scan intensity proﬁle of the Si 111
reﬂection is shown in Fig. 5. Periodic drift in the observed
intensity proﬁle, which is likely to be caused by slight misalignment of the sample face, is modelled by Fourier expansion up to the second order. The Fourier coefﬁcients {ck} are
calculated byck ¼ n1

nP
1

Ij expð2ikj=nÞ

ð18Þ

j¼0

from the observed intensity data {Ij} (j = 0, . . . , n  1), and the
proﬁle of the periodic drift {(Idrift)j} is approximated byðIdrift Þj ¼

2
P

ck expð2ikj=nÞ:

ð19Þ

k¼2

The calculated drift proﬁle is also shown in Fig. 5.
The average intensity hIi is straightforwardly given by the
zeroth-order Fourier coefﬁcient c0. The statistical variance ofthe residuals ( I)j = Ij  (Idrift)j is calculated byðIobs Þ2 ¼ ðn  5Þ1

n1
P

ð IÞ2j ;

ð20Þ

j¼0

Figure 5
Figure 4

Cumulative particle-size distribution of (a) 3–7 mm, (b) 8–12 mm and (c)18–22 mm fractions of -quartz powder. Measured diameters are plottedas black dots, and the optimized log-normal distribution curves,
calculated with optimized parameters, are drawn as grey lines.
J. Appl. Cryst. (2009). 42, 597–606

Spinner-scan intensity proﬁle of the Si 111 reﬂection. The raw intensity
data I and assumed drift Idrift are drawn as grey and black lines,
respectively, in the upper panel. The lower panel shows the deviation I =I  Idrift (grey line) and the standard deviation predicted by counting
statistics (broken lines).
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Table 3Results of spinner-scan measurements for Si powder (NIST SRM640c).
hIi is the average intensity, (Iobs)2 is the observed statistical variance,(Iparticle)2 is the statistical variance assigned to particle statistics, neff is theeffective number of diffracting crystallites, D0eff is the effective diameter
calculated from assumed values of geometric parameters of the diffractometerand Deff is the calibrated value of the effective diameter.
hkl

2
( )

hIi

(Iobs)2(Iparticle)2

neff

D0eff
(mm)

Deff
(mm)

111
28.59 2304 (3) 3440 (260)1140 (260)4670 (1100) 4.1 (3) 5.0 (4)22047.44 2579 (3) 4890 (340)2310 (340)2900 (440) 4.6 (2) 5.2 (3)311
56.26 3266 (4) 7140 (470)3880 (470)2750 (340) 5.6 (2) 6.1 (3)400
69.24 1211 (3) 2720 (180)1510 (180)970 (120)
4.7 (2) 5.0 (2)33176.48 4157 (5) 9680 (690)5520 (690)3130 (400) 4.9 (2) 5.2 (2)42288.12 4532 (5) 10770 (760) 6240 (760)3290 (410) 4.7 (2) 5.0 (2)333/511 95.05 5028 (6) 12310 (840) 7290 (840)
3470 (410) 4.9 (2) 5.4 (2)440
106.79 2344 (4) 5930 (400)3590 (400)1530 (180) 4.5 (2) 5.2 (2)531
114.16 9705 (8) 25800 (2000) 16100 (2000) 5860 (740) 4.6 (2) 5.4 (2)620
127.60 5385 (6) 13510 (960) 8100 (960)3570 (430) 4.2 (2) 5.4 (2)533
136.92 6821 (6) 16200 (1100) 9400 (1100) 4950 (600) 3.7 (2) 5.2 (2)The values of the observed average intensity hIi, thestatistical variance (Iobs)2, the statistical variance assigned tothe particle statistics (Iparticle)2 and the effective number ofdiffracting crystallites neff = hIi2/(Iparticle)2, evaluated for the
spinner-scan diffraction intensity data of Si powder (NIST
SRM640c), are listed in Table 3.
Fig. 6 shows the values of neff sin  calculated from thespinner-scan data, and also the known multiplicity meff for all
measured reﬂections from Si powder. It is clearly shown thatthe variation of the experimental values of neff sin  is very
similar to that of meff. The observed similarity conﬁrms thatthe particle statistics have certainly been evaluated by the
spinner-scan measurement, because no origin of statisticalvariation except the particle statistics is likely to cause such
behaviour as is proportional to the multiplicity of reﬂections.
The effective crystallite diameter D0eff is ﬁrst calculated by
D0eff ¼

where the degree of freedom is assumed to be decreased byﬁve, because the second-order Fourier expansion of real data
includes ﬁve independent coefﬁcients determined by thesource data. The errors in the evaluated variance, [(Iobs)2],
were calculated byf½ðIobs Þ2 g2 ¼

n1
Xð IÞ4jj¼0n2

ðIobs Þ4
:
n

ð21Þ

3meff AwA
42 neff 0 R sin 

1=3
;

ð24Þ

according to the formula for stationary specimens given inequation (6). The values of D0eff for each reﬂection peak of Si,
calculated by assuming 0 = 142.6 cm1 and with instrumental
parameters A = 32 mm2, w = 0.12 mm, A = 5 and R =185 mm, are also listed in Table 3.
Fig. 7(a) plots the values of D0eff evaluated for all themeasured reﬂection peaks of Si versus the diffraction angle 2.

The variance caused by the particle statistics, (Iparticle)2, iscalculated from the observed variance (Iobs)2 by the equation
ðIparticle Þ2 ¼ ðIobs Þ2  ðIcount Þ2 ;

ð22Þ

where (Icount)2 is the variance caused by the counting
statistics, which is approximated by (Icount)2 ’ hIi.
Then, the effective number of diffracting crystallites neff iscalculated byneff ¼ hIi2 =ðIparticle Þ2

ð23Þ

for each reﬂection.

Figure 7

Figure 6

Product of the effective number of crystallites neff with sin  (open circles)
and the known effective multiplicity of reﬂection meff for Si (dotsconnected by broken lines).

602(a) Effective diameter D0eff of Si powder calculated with assumed
instrumental parameters (circles) and the ﬁtted curve of D0fit =(5.6 mm)(0.62/tan   0.41 + 1.33 tan )1/3 (broken line). (b) Calibrated
effective diameter Deff of Si powder (circles) and the average value of
5.6 mm (broken line).

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It should be noted that the relative error for the particle
diameter is suppressed by a factor of 1/3 from that of theparticle volume evaluated directly by the spinner-scanmeasurement. The weighted average of D0eff is nominallyestimated at D0 eff = 4.52 (6) mm, while the corresponding value
estimated by the SEM image analysis is (Deff)Si = 5.6 mm.
At the moment, it is difﬁcult to derive an a priori formulafor the systematic behaviour of the evaluated effective
diameter D0eff depending on the diffraction angle. However,such dependence is likely to be caused by a combination of theneglected spectroscopic distribution of the source X-ray, theﬁnite receiving-slit width and instrumental aberrations, as has
been suggested by de Wolff (1958). When the systematicdeviation is assumed to be caused predominantly by instrumental effects, it is expected that a common calibration curve
can be applied to the intensity data from different samples
when they are measured under the same conditions.
Taking into account the uncertainties of the instrumental
parameters of the diffractometer on the evaluation of D0eff , andthe above systematic deviation, the formula for evaluating theeffective diameter Deff should be modiﬁed using the equation
Deff ¼ D0eff ðDeff ÞSi =ðD0eff Þfit ;effective number of diffracting crystallites neff, the known
effective multiplicity meff of -quartz (see Table 1) is used forthe calculation of D0eff , and the calibration curveðDeff ÞSi =ðD0eff Þfit determined from the Si data is used for theestimation of Deff.
The values of the observed average intensity hIi, thestatistical variance (Iobs)2, the statistical variance assigned toparticle statistics (Iparticle)2, the effective number ofdiffracting crystallites neff = hIi2/(Iparticle)2, and the crude andcalibrated effective diameters D0eff and Deff, evaluated for thethree fractions of quartz powder, are listed in Tables 4–6.
The calibrated values of the effective diameter Deff for 11
reﬂections of the three quartz powder samples are plotted inFig. 8. No signiﬁcant systematic deviation of Deff is detected,
except that the value estimated for the 100 reﬂection seems toð25Þ

where (Deff)Si = 5.6 mm is the value determined by the SEM
image analysis, and ðD0eff Þfit is the correction that should beapplied to the observed values of D0eff .
In this study, the systematic behaviour of D0eff for the Si data
is modelled by the following formula depending on thediffraction angle 2,ðD0eff Þfit ¼ ðDeff ÞSi ðt0 = tan  þ t1 þ t2 tan Þ1=3 ;ð26Þ

because it is expected that the effect of spectroscopic broadening is proportional to tan , and the dependence proportional to 1/tan  is dominant in the effect of instrumental
aberrations (Ida & Toraya, 2002). The optimized values of theﬁtting parameters are estimated at t0 = 0.62 (19), t1 =
0.41 (45) and t2 = 1.33 (23) by a nonlinear least-squares
ﬁtting method. The ﬁtting curve drawn in Fig. 7(a) satisfactorily reproduces the observed systematic behaviour.
The formula to evaluate the calibrated value of the effective
diameter Deff is then given by1=3
meff ð138 m2 Þð0:62= tan   0:41 þ 1:33 tan ÞDeff ¼:
0 neff sin 
ð27Þ
The values of the effective diameter Deff calculated by equation (27) are listed in the last column of Table 3 and are also
shown in Fig. 7(b). Coincidence of the calibrated values Deff
evaluated from different reﬂections of Si is signiﬁcantly
improved from that of the crude estimation D0eff .
3.3. Spinner-scan data of quartz powder

The spinner-scan intensity data of the three fractions ofquartz powder have been analysed in the same manner as the
Si powder sample, except that the bulk linear absorption
coefﬁcient of 0 = 89.81 cm1 is used for the calculation of the
J. Appl. Cryst. (2009). 42, 597–606

Figure 8

Effective diameters Deff evaluated for (a) 3–7 mm, (b) 8–12 mm and (c)18–22 mm fractions of quartz powder. The values evaluated for each
reﬂection are plotted as circles, and the weighted average values, 6.5 (2),11.7 (2) and 22.8 (2) mm, respectively, are shown as broken lines.
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Table 4

Results of spinner-scan measurements for the nominally 3–7 mm fractionof quartz powder.
See Table 3 for deﬁnitions.
hkl

2
( )

hIi

(Iobs)2(Iparticle)2 neff

D0eff
(mm)

Deff
(mm)

100
101/011110
102/012111/111200
201/021112/112202/022103/013211/121

20.77
26.56
36.46
39.38
40.23
42.37
45.71
50.06
54.79
55.24
59.87

904 (2)2032 (3)636 (1)824 (2)752 (2)661 (2)737 (1)717 (2)751 (2)802 (2)993 (2)1180 (80)3000 (200)710 (50)1020 (70)940 (70)
1000 (70)840 (60)
1090 (70)990 (70)1120 (90)1130 (80)280 (80)990 (200)
80 (50)190 (70)
190 (70)
340 (70)
100 (60)
370 (70)240 (70)
320 (90)130 (80)

5.6 (5)5.4 (4)3.8 (8)4.7 (6)5.6 (7)5.8 (4)4.2 (8)6.8 (4)5.1 (5)4.6 (4)4.6 (9)8.2 (8)7.3 (5)4.7 (1.0)
5.8 (7)
6.8 (8)6.9 (5)5.0 (1.0)
7.8 (5)5.7 (5)5.2 (5)5.2 (1.0)2960 (870)4170 (870)5400 (3600)
3500 (1300)
3000 (1000)
1300 (280)5400 (3100)
1390 (270)2390 (660)2020 (560)
7300 (4300)

Table 5Results of spinner-scan measurements for the nominally 8–12 mm fractionof quartz powder.
See Table 3 for deﬁnitions.
hkl

2
( )

hIi

(Iobs)2(Iparticle)2 neff

D0eff
(mm)

Deff
(mm)

100
101/011110
102/012111/111200
201/021112/112202/022103/013211/121

20.79
26.56
36.46
39.40
40.21
42.37
45.71
50.06
54.79
55.25
59.88

825 (5)1641 (8)550 (4)795 (7)739 (4)560 (4)673 (4)636 (5)663 (6)629 (6)
978 (5)

2200 (150)6840 (490)
1730 (150)1440 (110)1380 (110)2190 (150)2290 (170)2180 (150)2550 (180)2360 (190)2390 (190)1250 (150)4760 (490)
1070 (150)
850 (110)740 (110)
1520 (150)1540 (170)1450 (150)1780 (180)1740 (190)1390 (190)

8.8 (3)9.0 (3)9.0 (4)9.7 (4)9.9 (5)9.5 (3)
10.3 (4)10.6 (4)9.7 (3)9.7 (3)10.0 (5)12.9 (5)12.2 (4)11.1 (5)11.8 (5)11.9 (6)11.4 (4)12.1 (5)12.2 (4)11.1 (4)11.0 (4)11.3 (5)750 (90)920 (100)410 (60)410 (50)560 (80)
290 (30)370 (40)360 (40)330 (30)220 (20)720 (100)

Table 6

Results of spinner-scan measurements for the nominally 18–22 mm
fraction of quartz powder.
See Table 3 for deﬁnitions.
hkl

2
( )

hIi

(Iobs)2(Iparticle)2

neff

D0eff
(mm)

Deff
(mm)

100
101/011110
102/012111/111200
201/021112/112202/022103/013211/121

20.80
26.57
36.47
39.40
40.22
42.38
45.72
50.08
54.80
55.26
59.88

968 (2)2086 (4)660 (2)591 (2)643 (2)668 (2)756 (2)727 (2)770 (3)616 (2)1001 (2)8260 (600)27700 (2100)
5330 (440)16900 (1500)
6340 (570)
7570 (530)7840 (690)
10000 (840)
12400 (890)
15100 (1500)
9900 (1700)

7430 (600)26000 (2100)
4780 (440)16100 (1500)
5600 (570)7000 (530)7170 (690)
9400 (840)11740 (890)
14500 (1500)
8920 (700)91 (7)103 (9)63 (6)
39 (4)
98 (10)45 (4)
63 (6)
43 (4)
37 (3)
27 (3)107 (8)17.8 (5)18.6 (5)16.7 (6)21.3 (7)17.7 (6)17.9 (4)18.7 (6)21.7 (6)20.2 (5)19.4 (7)19.0 (5)26.2 (7)25.3 (7)20.7 (6)25.8 (8)21.4 (7)21.4 (5)22.0 (7)25.0 (7)23.0 (6)22.1 (8)21.4 (6)be overestimated, which may be caused by an error in theextrapolation of the calibration curve.
It seems that the errors estimated based on the propagationfrom the value calculated by equation (21) are slightly
underestimated, but acceptable for the results of the 3–7 and8–12 mm fractions of the quartz samples. The uncertainties in
Deff evaluated for the 18–22 mm quartz sample may be partly
caused by the small number of diffracting crystallites neff,604

ranging only from 27 to 107 for the applied measurement
conditions. It is suggested that accuracy in size estimation forlarge crystallites will be improved by expanding the irradiatedvolume V or enhancing the probability p, which may beachieved by changing the measurement conditions.
Even though the calibration formula given by equation (26)does not have a fully concrete theoretical basis, it can beconcluded that the dependence on the diffraction angle has
been successfully removed.
The weighted average values of Deff are estimated at 6.5 (2),11.7 (2) and 22.8 (2) mm for the 3–7, 8–12 and 18–22 mm
fractions of quartz powder, while the values estimated by the
SEM image analysis were 7.1, 11.8 and 25.3 mm. Any discrepancy might be caused by the assumptions of the SEM image
analysis rather than the analysis of the spinner-scan data.
NIST SRM640c Si powder might not be an ideal material as
the standard for evaluation of particle statistics, because of theirregular shape and broad size distribution, as can be seen in
the SEM image (Fig. 1). It is expected that a more reliable
analysis could be achieved if a standard crystalline powder
with a more regular shape and narrower size distribution wereavailable.
The method will also be applicable to a parallel-beamgeometry using synchrotron or multilayer mirror optics, if theeffective focal size is evaluated using a standard powder. Useof synchrotron X-rays may expand the sensitivity of themethod to smaller crystallite sizes, because a smaller focal size
is expected for a synchrotron X-ray source.
3.4. Possible applications of the spinner-scan method

The results of the current study show that the effective
number of crystallites neff that satisfy the diffraction condition
can certainly be measured, simply by rotating the specimen
stepwise and recording the diffraction peak intensities.
As has been suggested in x3.2, information on the multiplicity of the measured reﬂection can be obtained experimentally by this method without any preliminary knowledgeof crystal symmetry or atomic arrangements.
The effective multiplicity of reﬂection is formally calculatedbymeff ¼ neff 0 D3eff f ðÞ;f ðÞ ¼
3
42 R sin  ðD0eff Þfit;3AwA ðDeff ÞSi

ð28Þ
ð29Þ

where the calibration curve f() is determined by themeasurement of the standard sample. The formula for f()applied in this study is simpliﬁed asf ðÞ ¼ sin = ð138 m2 Þð0:62= tan   0:41 þ 1:33 tan Þ :
ð30ÞNote that the effective number of diffracting crystallites isstraightforwardly evaluated by the spinner-scan measurement
as neff = hIi2/(Iparticle)2. The absorption coefﬁcient 0 is
obtained from the chemical composition and the density.
Then, the value of the multiplicity of reﬂection meff can be

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J. Appl. Cryst. (2009). 42, 597–606

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calculated using equation (28) if only the crystallite size Deff isdetermined.
The experimental values of meff, calculated for Si and the 8–
12 mm fraction of the quartz powder sample using equations(28) and (30), and the values predicted by the known crystal
structures, are plotted in Fig. 9. The values of the effective
diameter Deff = 5.6 and 12 mm, determined by SEM image
analysis for the Si and quartz samples, respectively, are used
for the calculation.
Even though the calibration curve has been adjusted for the
Si data, the good coincidence between the predicted andobserved values of meff shown in Fig. 9(a) indicates that therelative values of the effective multiplicity meff can beexperimentally evaluated by this method without any standard
samples or knowledge of particle size.
The values of meff for the 8–12 mm quartz powder evaluatedby the spinner-scan measurement coincide absolutely with thevalues predicted by the crystal structure, almost within theexperimental errors, as can be seen in Fig. 9(b). This meansthat the absolute value of the multiplicity of reﬂection can beevaluated by this method without any knowledge of the crystal
structure, when the effective crystallite diameter is known.
Evaluation of the relative values of meff will be much easier,because the values of 0 and Deff are not needed. It is thus
expected that the method will provide valuable informationon the determination of an unknown structure.
The results presented in x3.3 show that crystallite sizes overa range of several micrometres, which cannot be evaluated byline-broadening analysis, can certainly be evaluated by the
spinner-scan method. The values for different diffraction
peaks coincide well for all the samples examined in this study.
This means that the evaluation can be completed bymeasurement of the strongest diffraction peak, which will only
take a few minutes, when a random orientation of the crystallites can be assumed.
It is further suggested that the preferred orientation of
crystallites may also be evaluated by this method, simply bymeasuring different diffraction peaks, because the number ofdiffracting crystallites neff for each reﬂection should in principle be exactly proportional to the probability that thenormal direction of the diffraction plane coincides with thedirection normal to the face of the specimen.
Since the analysis of spinner-scan data provides additional
information almost independent of that included in onedimensional powder diffraction data, application of themethod can reﬁne any results of an analysis based on powder
diffraction data. For example, the method can be used to test
the validity of structure models from indexing and structure
analysis, and to distinguish a strong impurity diffraction peakfrom weak main-phase diffraction peaks in multiphasemixtures by the different numbers of diffracting crystallites
neff.
Finally, the authors would like to note that the spinner-scanmethod is applicable not only to powder samples but also topolycrystalline materials for practical use, such as sinteredceramics or alloys.

4. Conclusions
The statistical properties of powder X-ray diffraction intensities measured by a step scan about the rotation angle of a
specimen-spinning attachment have been investigated.
The results show that the statistical variance assigned toparticle statistics can be quantitatively evaluated by thismethod, and estimation of crystallite size over a range ofseveral micrometres, which is practically impossible byconventional methods based on line-broadening analysis, hasbeen achieved at considerable accuracy.
It is also suggested that the method is useful for various
applications based on powder diffractometry, including
determination of unknown structures, structure reﬁnement,
evaluation of preferred orientation, and qualitative andquantitative analysis of multiphase mixtures.

APPENDIX A
Modified C distribution

Figure 9

Effective multiplicity of reﬂection meff for (a) Si and (b) 8–12 mm quartz
samples, as predicted by the known crystal structure (dots connected withbroken lines) and as evaluated by the spinner-scan measurement (opencircles).
J. Appl. Cryst. (2009). 42, 597–606

The probability density function of the modiﬁed  distribution
for diameter D used in this study is given by"   # 1

D
D
:
ð31Þ
exp 
fW ðDÞ ¼ðÞD0 D0
D0
The average value of Dj for the distribution is obtained bysolving the equation
Ida, Goto and Hibino  Particle statistics

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hDj iW ¼

R1

Dj fW ðDÞ dD;

ð32Þ

0

fLN ðD; Dm ; !Þ ¼


1
ðln D  ln Dm Þ2
exp

:
2!2
ð2Þ1=2 !

ð39Þ

The average of Dj is given by solving the equation

which gives
hDj iW ¼ ð þ j=ÞDj0 =ðÞ:

ð33Þ

The effective diameter for particle statistics in the modiﬁed
-distribution model is given by 6 1=3

1=3
hD i
ð þ 6=Þ
¼ D0
:
ð34Þ
Deff ¼hD3 i
ð þ 3=Þ
The formula for fW(D) is reduced to the usual  distribution 1


1
D
D
exp 
ð35Þ
f ðDÞ ¼ðÞD0 D0
D0
for  = 1, and to the Weibull distribution"   # 1
 D
D
fW ðDÞ ¼
exp 
D0 D0
D0

RD 3
t fW ðtÞ dt;

R1

Dj fLN ðD; Dm ; !Þ dD;

ð40Þ

0

which gives
hDj i ¼ Djm exp j2 !2 =2 :

ð41ÞThe effective diameter for particle statistics in the log-normal
distribution model is then given by
Deff ¼ hD6 i=hD3 i

1=3¼ Dm exp 9!2 =2 :

ð42ÞThe relative volume fraction for the log-normal distribution isderived by solvingFðDÞ ¼ hD3 iLN

ð36Þ

RD 3
t fLN ðtÞ dt;

ð43Þ

0

for  = 1.
The relative volume fraction based on the distribution isderived by solvingFðDÞ ¼ hD3 i1W

hDj i ¼

which givesFðDÞ ¼
1ln D  ln Dm  3!3
1 þ erf:
21=2 !
2

ð44Þ

ð37Þ

0

which gives

ReferencesFðDÞ ¼ P  þ 3=; ðD=D0 Þ :

ð38ÞAPPENDIX B
Properties of log-normal distribution
The probability density function of the log-normal distribution
for diameter D is given by

606

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742–753.
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Reference Material 640c. National Institute of Standards andTechnology, Gaithersburg, USA.
Ida, T. (2008). J. Appl. Cryst. 41, 1019–1023.
Ida, T. & Toraya, H. (2002). J. Appl. Cryst. 35, 58–68.
Izumi, F. & Ikeda, T. (2000). Mater. Sci. Forum, 321–324, 198–203.
Smith, D. K. (2001). Powder Diffract. 16, 186–191.
Teich, M. C. (1985). Biol. Cybern. 53, 121–124.

Ida, Goto and Hibino  Particle statistics

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J. Appl. Cryst. (2009). 42, 597–606