論文

先進セラミックス研究センター年報（2013）. Vol. 2, 7-11

軌道放射Ｘ線粉末回折における選択配向の影響
井田隆

名古屋工業大学先進セラミックス研究センター〒 507-0071 岐阜県多治見市旭ヶ丘 10-6-29

Effect of Preferred Orientation in Synchrotron X-ray Powder Diffraction
Takashi Ida
Advanced Ceramics Research Center, Nagoya Institute of Technology,
10-6-29 Asahigaoka, Tajimi, Gifu 507-0071, JAPAN
Abstract

The capillary transmission or asymmetric reflection method is usually applied in synchrotron powder diffraction
measurements, while the symmetric reflection method has conventionally been applied in laboratory powder diffraction
measurements with the Bragg-Brentano geometry. Although Dollase [J. Appl. Cryst. 19, 267–272 (1986)] has already pointed
out that the formula of the preferred orientation factor for the capillary transmission mode should be different from that for thesymmetric reflection mode, no appropriate implementation in Rietveld programs is currently available. In this study, ananalytical expression and an efficient numerical algorithm to evaluate the preferred orientation factor based on the pole density
profile of the March-Dollase model for capillary transmission and asymmetric reflection methods have been derived.
Key-words

Preferred orientation, Synchrotron powder diffraction, Rietveld method

1. Introduction

expression of the preferred orientation factor for thecapillary transmission mode in powder diffraction, andalso an effective numerical algorithm to evaluate thepreferred orientation factor for the asymmetric reflectionmode, both of which are based on the March-Dollase
model for the pole density profile function.

It is well known that observed powder diffraction
intensities are often affected by preferred orientation of
crystallites, particularly in the case of a collection ofplaty or rod-like crystallites [1, 2]. Most of theapplication programs for the Rietveld refinement
implement some models for the preferred orientation
factor, which are intended to simulate the change causedby the preferred orientation in the diffraction peakintensities measured in the symmetric reflection mode.
However, (i) a capillary transmission or (ii) an
asymmetric reflection method is usually applied insynchrotron powder diffraction measurements, because
the efficiency of detecting the diffracted x-ray photons isgreatly enhanced by those methods accompanied with aposition-sensitive detector or multiple point-detectors.
Dollase has suggested an integral formula for thepreferred orientation factor based on the March modelfor the capillary transmission mode [2], but noimplementation about capillary transmission or
asymmetric reflection method in Rietveld programs iscurrently available, to the authorʼs knowledge.
In this study, the author has derived an analytical

2. Pole density profile and preferredorientation factor
2.1 Pole density profile of March-Dollase model

Dollase has proposed a formula for the pole density
profile based on the March function, given by(1)where p* represents the preferred orientation direction,
which may be defined by p* = Ha* + Kb* + Lc* for thereciprocal lattice vectors a *, b * and c *, r is theparameter for the preferred orientation, r > 1 for platy
crystallites , r < 1 for rod-like crystallites, and r = 1 forrandom orientation of crystallites in a powder sample[2]. Note that the numbers of index (H K L) may be―7―軌道放射Ｘ線粉末回折における選択配向の影響

2.3 Preferred orientation factor for capillary

non-integer values for rod-like (r < 1) crystallites, when
the crystal structure belongs to the monoclinic or
triclinic crystal system, while it is likely that they areinteger values for platy (1 < r) crystallites. The angle ρis the polar angle between the preferred orientation
direction p * and the direction s * that specifies thedirection of the specimen. Rotational symmetry of thepreferred orientation about s* is assumed in the MarchDollase model, which can be ensured by spinning thesample about the direction s* during the acquisition ofdiffraction intensity data. Since it is essential to keep
spinning the sample during the synchrotron powder
diffraction measurements in most cases, the direction s*
should naturally be identified to the rotation axis of thespinning attachment of the measurement system.
The general formula of preferred orientation factor,
which should be proportional to the observed diffraction
intensity, is given bytransmission mode

Dollase has also suggested that the preferredorientation factor for the capillary transmission mode
can numerically be evaluated by the formula given byeqs. (2)–(4) with Δ = π / 2, that is,(6)
In fact, an analytical solution of eq. (6) is available,
which is given by the following formula:

(2)

(7)
where(3)(8)(4)where α is the angle between the diffraction vector d* =ha* + kb* + lc* and the preferred orientation direction
p*, and Δ is the angle between the vectors d* and s*.
The angle Δ is equal to zero in the symmetric reflectionmode, π / 2 in the capillary transmission mode, and isequal to the difference between the Bragg angle θ andthe incident glancing angle Ω for the flat-plate reflection
measurement, Δ = |θ – Ω|.

is the Legendre complete elliptic integral of the second
kind. As an efficient algorithm to evaluate the elliptic
integral is well known (Appendix A), there will be noreason for applying numerical integral suggested by
Dollase, instead of the analytical solution given by eq.
(7).
Figure 1 compares the preferred orientation factors forsymmetric reflection and capillary transmission modes
on variation of the values of r and α . Note that thevalues for r = 0.8 in the symmetric reflection mode arevery close to the values for r = 2 in the capillarytransmission mode within the range 45º < α < 90º, but
clearly different in the range 0º < α < 45º. Since theMarch-Dollase pole density profile function Pp*(r, ρ)satisfies the normalization condition:

2.2 Preferred orientation factor for symmetricreflection mode

Equations (2)-(4) are reduced to the following
formula:(5)
for the symmetric reflection mode (Δ = 0). It should beemphasized that the formula of eq. (5) is identical to thatof eq. (1), but the meanings are definitely different.
Note that the argument ρ in eq. (1) means the angle
between p* and s*, while α in eq. (5) means the anglebetween p* and d*.

(9)
it is expected that the application of the symmetricreflection formula fd*(r, α, 0) to the data measured in asymmetric reflection mode will be consistent with therequirement of normalization about the probability
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井田隆

density function. On the contrary, application of fd*(r, α,0) to the data measured in a capillary transmission mode
will violate the normalization condition, which will leadsystematic errors in quantitative phase analysis. It isconcluded that the correct formula of fd*(r, α, π/2) should
be used for the data measured in a capillary transmissionmode, even if the difference between fd*(r, α, 0) for r <1 and f d * ( r , α , π /2) for 1 < r could not clearly be
distinguished on application to the observed powder
diffraction pattern.

and accurate evaluation of the preferred orientation
factor by numerical integral will become difficult withan ordinary computing system.
In order to avoid the above quasi-singularity, theauthor proposes an alternative formula of numericalintegral given by the following equations,
(11)

(12)
(13)

(14)

(15)

Fig. 1. Comparison of the preferred orientation factors

calculated for symmetric reflection and capillary transmission
methods.

(16)

2.4 Preferred orientation factor for asymmetricreflection mode

The above formula corresponds to the application of
the Chebyshev quadrature to the integral formulamodified by variable transformation (Appendix B).
Tables 1 – 3 compare the values obtained by finiteterm numerical integrals applying the two different
formulas, the formula (I) given by eq. (10) and theformula (II) given by eqs. (11)–(16).
As shown in Tables 1 and 2, the formula (II)converges to the correct value by increasing the numberof sampling points (N) more rapidly than the formula (I),particularly in the cases 1 << r. They also confirm thevalidity of the analytical formula given by eq. (7)derived for the capillary transmission mode.
Furthermore, numerical instability of the formula (I) for
the nearly symmetric reflection case (), as seen inTable 3, seems to be reduced by application of theformula (II). It is expected that the formula (II) is
always more efficient and stable than the formula (I).
The formula (II) becomes singular just in the case of

The formula of the preferred orientation factor for theasymmetric reflection mode may numerically becalculated straightforwardly by eqs. (2)–(4), assuggested by Dollase [2]. And it is generally
recommended that sampling points for numericalintegral should be evenly spaced, when the integrand isa periodic function [3]. Then the formula for numericalintegral would be given by the following equation,(10)
is given by
where the definition of the function
eqs. (1), (3) and (4).
However, when the degree of preferred orientation isstrong (1 << r), the integrand in eq. (2) behaves almost
singularly for the variable in the case:―9―軌道放射Ｘ線粉末回折における選択配向の影響

Table 1. Numerical values of preferred orientation factors for

Table 3. Numerical values of preferred orientation factors for

Δ = 90º, α = 30º, and r = 0.5 and 2, calculated by finite-term

Δ = 0.01º, α = 30º, and r = 0.5 and 2, calculated by thenumerical integrals (N = 1, 2, 4, 8, 16) based on the formulas (I)formulas (I) and (II). The solutions forand (II). The exact solutions calculated by eq. (7) are 0.42668

calculated by eq. (5) are 1.75425 and 0.18102 for r = 0.5 and 2,and 1.38810 for r = 0.5 and 2, respectively.

respectively.

Δ = 0º and α = 30º

Formula

(I)

(II)

(I)

(II)

Formular

0.5

0.5

22

r

0.5

0.5

22

N=10.51200

0.40000

0.620221.70561

N=11.75324

1.754260.181070.18102

N=20.42064

0.42708

1.10165

1.40836

N=21.75353

1.754260.181050.18102

N=40.42665

0.42668

1.36347

1.38850

N=41.75359

1.754260.181050.18102

N=80.42668

0.42668

1.38798

1.38810N=81.75360

1.754260.181050.18102N = 16

0.42668

0.42668

1.388101.38810

(I)

(II)

(I)

(II)

Table 2. Numerical values of preferred orientation factors for
Δ = 90º, α = 30º, and r = 0.25 and 4, calculated by the formulas(I) and (II). The exact solutions calculated by eq. (7) are
0.15508 and 1.33115 for r = 0.25 and 4, respectively.
Formula

(I)

(II)

(I)

(II)

r0.25

0.25

44

N=10.19096

0.143960.11670

1.95471

N=20.15221

0.155270.30258

1.42285

N=40.15506

0.155080.73815

1.34181N=80.15508

0.155081.20927

1.33164

N = 16

0.15508

0.15508

1.328021.33116

exactly symmetric reflection, Δ = 0, but it can easily betreated by eq. (5) as an exceptional case in a practicalcomputer application.
Figure 2 shows the dependence of the preferredorientation factor (POF) on the diffraction angle 2θ,
expected for the asymmetric reflection mode with thefixed incident glancing angle of Ω. Note that the POF
have maximum or minimum at the diffraction angle 2θ
of just twice the incident glancing angle Ω, 2θ = 2Ω,while the POF is expected to be constant andindependent of the measured diffraction angle 2θ in thesymmetric reflection mode. It is likely that the misuseof the symmetric formula will cause systematic errors onevaluation of atomic displacement parameters.
3. Discussion

The author would like to emphasize that this article is
not intended to insist that the March-Dollase model
should always be used as the model for preferredorientation. The validity of a preferred orientation
model should be examined by well-controlledexperiments. The author has already found such
experimental data that cannot be well modeled by theFig. 2. An example of preferred orientation factors for an

asymmetric reflection mode measurement.

March-Dollase pole density profile, and the study on
extension or modification of the March-Dollase
preferred orientation model is currently in progress.
4. Conclusion

An analytical expression of the preferred orientation
factor for the capillary transmission mode and an
efficient numerical algorithm to evaluate the preferredorientation factor for the asymmetric reflection mode in
powder diffraction measurements have been derived. Itis shown that the formulas of the preferred orientation
factors for the capillary transmission and asymmetricreflection modes are definitely different from that for theconventional symmetric reflection mode. It is
recommended to Rietveld programmers to implement
the correct formulas for preferred orientation factor,
applicable to the data collected with synchrotron powder
diffraction measurements.

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井田隆

Appendix A. Computational method for
elliptic integral(B6)

The Legendre complete elliptic integral of the second
kind is calculated by(A1)which is equivalent towhere the functions RF( x,y,z ) and R J( x,y,z,p ) are theCarlsonʼs elliptic integral of the first and second kinds,
respectively, defined by(B7)
This transformation leads the following relations,

(A2)

(A3)
(B8)
An efficient algorithm for evaluation of the Carlson ʼselliptic integral is known [4].

(B9)Appendix B. Transformation of variable innumerical integral

As shown in sec. 2.1, the general formula of thepreferred orientation factor is given by the followingequations,However, the above transformation from to introduces
other singularities at the edge points,and.
It is found that another transformation from togiven by

(B1)

(B10)

(B2)
(B3)
It is suggested by the relation :(B4)

can remove the singularities at the edge points, keeping
the smooth behavior of the integrand almost unchanged.
The second transformation of the variable given by eq.
(B10) is equivalent to the method known as the GaussChebyshev quadrature [4].
Referencesthat the quasi-singularity of the integrandeq. (B1) atin

[1] H. M. Rietveld, J. Appl. Cryst. 2, 65–71 (1969).
[2] W. A. Dollase, J. Appl. Cryst. 19, 267–272 (1986).
[3] M. Iri, Y. Fujino, “Suchi-Keisan no Joshiki,” Chapter 7,(B5)Section 4, Kyoritsu Shuppan (1985) [in Japanese].
[4] W. H. Press, S. A. Teukolsky, W. T. Vetterling, “Numerical

can be removed by the transformation :Recipes,” Sections 5.8, 6.12, 12.4, Cambridge UniversityPress (1986).

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