技術報告

先進セラミックス研究センター年報（2014）．Vol．3，33-38

Implementation of the Williamson–Hall and Halder–Wagner Methods
into RIETAN-FP
Fujio Izumi*, **, Takuji Ikeda****Guest Professor of Advanced Ceramics Research Center, Nagoya Institute of Technology
10-6-29, Asahigaoka, Tajimi, Gifu 507-0071, JAPAN**National Institute for Materials Science
1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, JAPAN***National Institute of Advanced Industrial Science and Technology
4-2-1 Nigatake, Miyagino-ku, Sendai, Miyagi 983-8551, JAPAN
Abstract
Microstructural evaluation is important to understand physical and chemical properties of polycrystalline materials, particularlyin the field of nanotechnology. The latest version of a multi-purpose pattern-fitting system, RIETAN-FP, offers new features ofWilliamson–Hall (WH) and Halder–Wagner (HW) methods to determine crystallite sizes and microstrains from integral breadths,
β, evaluated after the Rietveld or Le Bail analysis of X-ray and neutron powder diffraction data. Contributions of instrumental
broadening are subtracted with analytical results of instrumental standards showing negligible sample broadening. Gnuplot isused for the graphical representation of linear relationships: βcosθ vs. sinθ in the WH plot and (β/tanθ)2 vs. β/(tanθ sinθ) in the HW
one. Deviations from the linear relationships can easily be recognized by the resulting graphs. With these two methods, the mean
volume-weighted size,〈D〉
v , of microcrystalline CeO2 was determined at 29.96 nm (WH) and 28.92 nm (HW), which are in good
agreement with values reported for the round-robin sample. The present new features concerning microstructural characterizationmust deliver added value to users of RIETAN-FP in both academic institutions and industries.
Key-words
Crystallite size, Microstrain, Powder diffraction, Williamson–Hall plot, Halder–Wagner plot, RIETAN-FP

1. Information extractable from powder dif-quantities (a), (d), (e), and (g) are obtained by pattern de-fraction data by whole-pattern fitting

composition, i.e., Le Bail analysis [4] plus hybrid pattern

The whole-pattern fitting of X-ray and neutron powder

decomposition in RIETAN-FP. Further, iterations of maxi-

diffraction data is powerful means of characterization to

mum-entropy method (MEM) analyses by Dysnomia [5]obtain the following structural, microstructural, and compo-

and while-pattern fitting by RIETAN-FP make it possible

sitional information:

to determine distributions of electron and scattering-length(a) Lattice parameters,

densities, (h) and (i), in the unit cell [6].
Rietveld analysis can also be applied to the quantitative(b) Crystal-structure parameters,(c) Magnetic moment (neutron diffraction),analysis of a mixture of crystalline materials (f); this feature(d) Crystallite sizes [1],is very important in the characterization of reaction products(e) Microstrains [1],and quality control in industries. The content of amorphous(f) Mass fractions (multi-phase samples).

materials can also be determined by addition of an internal(g) Observed integrated intensities, │Fo│2 , for ab initio

standard material.
Thanks to the above wealth of functions, RIETAN-FP has

structure solution,(h) Electron densities (X-ray diffraction),

been contributing to many studies in a variety of fields such(i) Densities of coherent-scattering lengths, bc (neutron dif-

as physics, chemistry, materials science, and earth science.
The present report deals with microstructural characteri-fraction),
A multi-purpose pattern-fitting program, RIETAN-FP [2],zation, (d) and (e), with RIETAN-FP v2.6 or later, helping its

is capable of evaluating all the above physical quantities.

users to understand details in procedures to determine crys-

Quantities (a)–(f) result from Rietveld refinement [3] whiletallite sizes and microstrains after Rietveld or Le Bail analy-－ 33 －

Implementation of the Williamson–Hall and Halder–Wagner Methods into RIETAN-FP

sis. This new version provides us with powerful and conven-（5）

ient means of investigating microstructures from isotropic
profile broadening by X-ray and neutron powder diffraction.

where U i and Yi are contributions of the instrument to U and

Crystallite-size broadening can be used to determine theY , respectively. U i and Y i can be estimated by Rietveld anal-

crystallite size of less than 1 μm in materials. On the otherysis from diffraction data of a sample whose crystallinity ishand, microstrains are caused by a distribution of both ten-high enough to show negligible profile broadening due to thesile and compressive forces, which causes broadening ofsample.

diffraction profiles about the original position. Crystallite-

This simple method, however, suffers from possible errors

size broadening follows a 1/cosθ function while strain-

due to strong correlations among profile parameters, U , V, W ,induced profile broadening has a tanθ dependence [1],X , and Y . In addition, no dependence of profile broadening

which allows us to separate these two effects in diffraction

on a function containing θ is graphically illustrated, which isdata over a wide 2θ range. In what follows, three different

unfavorable for finding variations in the data and anisotropic

methods of microstructural analysis from powder diffrac-broadening of diffraction profiles.

tion data will be described.

To overcome such drawbacks of the methodology adopted
in GSAS [7], the Williamson–Hall [9] and Halder–Wagner

2. Determination of crystallite sizes and[10,11] methods have recently been added to the latest version

microstrains from profile parameters

of RIETAN-FP. With these two features, D and ε can easily

With RIETAN-FP, crystallite sizes and microstrains havebe determined from integral breadths, β, defined as (peakbeen estimated in the same manner as GSAS [7] so far.

area)/(peak intensity) after Rietveld or Le Bail analysis.

That is, profile parameters in the pseudo-Voigt function ofDiffraction data measured with characteristic X-rays suchThompson, Cox, and Hastings [8] are refined by the Riet-

as Cu Kα radiation are good enough to get reliable D and εveld or Le Bail method from powder diffraction data of anvalues. Because the graphing of the two kinds of plots with

instrumental standard and an analysis sample to determine

free software, gnuplot [12], is supported in RIETAN-FP,

crystallite sizes and microstrains.

additional information is obtainable on the characterization

In the pseudo-Voigt function of Thompson et al. [8], theof powders by use of profile broadening, for example,

full-width at half maximum (FWHM), H G, for the Gaussiananisotropic one depending on directions of scattering vectors.

component is represented by(1)

3. Williamson–Hall method

On the other hand, the FWHM of the Lorentzian one, H L , isCrystallites whose size, D, is less than ca. 1 μm exhibit
profile broadening. The integral breadth (in radians), βD , due

computed by(2)

to the effect of small crystallites is related to D via the so-where θ is the Bragg angle, and U , V , W , X , and Y are(6)called Scherrer equation [13],profile parameters to be refined by a method of nonlinear
least squares. In both GSAS and RIETAN-FP, the unit of U , V,corresponding to Eq. (3).

and W in Eq. (1) is (degrees) whereas that of X and Y in Eq.

The effect of isotropic microstrain, ε, on profile broadening(2) is degrees.

can be derived by differentiating Bragg’s law,

2

Let K be the shape factor (dimensionless), λ the wavelength of the X-ray or neutron beam, r a constant (=π/180=(7)
with respect to θ (in radians):

0.0174533）to convert degrees into radians. Then, the crys-(8)tallite size, D , is determined by(3)

That is, profile broadening due to microstrain, ε = Δd/d (d:lattice-plane spacing), is proportional to tanθ. Stokes andwith the unit of D being equal to that of λ. K depends on

Wilson [14] pointed out that the integral breadth, βε, arising

assumptions made during the derivations. Microstrains cor-from isotropic microstrain is related to the integral breadth of

responding to the Gaussian and Lorentzian components are,the strain distribution, , by(9)respectively, computed by(4)Comparison between Eq. (6) and Eq. (9) shows that thedependence of βD on θ is quite different from that of β ε .

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Fujio Izumi, Takuji Ikeda

Williamson and Hall [9] introduced a simple approximation(14)

that the integral breadth, β, due to both of small crystallite
sizes and microstrains is simply the sum of Lorentzianwithcomponent, βD , and the Gaussian one, βε:

（15）

(10)
Thus, Eq. (6) and Eq. (9) are combined together to yield

(16)
(11)

where C is the proportional constant to convert

into themicrostrain ε. The value of C, which depends on the as-on the assumptions that the Lorentzian and Gaussiancomponents of β* are solely due to the size and strain effects, respectively. Equation (14) can be rewritten as

sumptions made concerning the nature of the inhomogene-(17)

ous strain, lies between 4 and 5 [15], with C=4 corresponding to the maximum (upper limit) of strain [16,17].

on the basis of direct space. Inside RIETAN-FP, anotherIn part of previous work, C was carelessly set at 2, which

equationprobably arose out of confusingwith ε. Multiplying both(18)

sides of Eq. (11) by cosθ, we obtain(12)

equivalent to Eq. (17) is used because both of Eq. (12) andEq. (18) contain sinθ and βcosθ. Equation (17) has a form of a

Equation (12) is regarded as a straight line, y=ax+b. A plot ofstraight line, y=ax+b, in a similar manner as Eq. (12). In they = βcosθ against x = sinθ is referred to as the Williamson–

Halder–Wagner (HW) plot, y = (β/tanθ)2 is plotted against x

Hall (WH) plot since Williamson and Hall [9] proposed this= β/(tanθ sinθ). Then, the slope and y intercept of the resulting

methodology in 1953. However, this designation is somewhat

straight line afford Kλ/D and 16ε2 , respectively. As described

unfair because Hall was the first to report the idea in 1949 [18].

in the previous section, the value of K = 4/3 [20] is believed

The slope of the straight line is Cε while its y intercept is Kλ/D.

to be valid on the definition of the crystallite size as theEquation (12) holds true for isotropic line broadening.

volume-weighted average one for spherical crystallites.

If both of crystallite-size and microstrain profiles are

Despite the approximations and assumptions made on theGaussian, then the plot is convex downward, having the same

derivation of Eq. (17), the HW plot has a great advantage thatterminal slope at a high angle as the Lorentzian case anddata for reflections at low and intermediates angles are given

intercepting the y axis at Kλ/D [19].

more weight than those at higher diffraction angles, which

The WH plot is a very useful diagnostic tool for learning

are often less reliable. Further, Eq. (17) does not contain thethe kind(s) of profile broadening and determining approxi-

constant C unlike Eq. (12), which is another advantage of themate values of D and ε.

HW plot over the WH one.

Regrettably, various K values such as 0.89, 0.9, 0.94, and 1.0
have been arbitrarily used in the literature without any regard

5. Procedures of calculating crystallite sizes andto the definition of the crystallite size. Ida et al. [20] derived

microstrains in RIETAN-FP

the K value of 4/3 for the mean volume-weighted average

In what follows, ‘hoge’ is the metasyntactic variable thatsize,〈D〉
v , defined as the ratio of the mean forth power of D

should be replaced by a string (usually a sample name); hoge.

divided by the mean cube of D,

ins is a user input file, hoge.int is an intensity data file, hoge.

(13)
in the case of spherical crystallites. Note that the K value of

lst is a standard-output file, hoge.gpd is a gnuplot data file,

4/3 is valid on the representation of profile broadening not byof gnuplot.

and hoge.plot is a gnuplot script file consisting of commands

the FWHM but by the integral breadth, β, in the same way as
Eq. (12).

5.1 An instrumental standard
After Rietveld or Le Bail analysis with a pair of files,
4. Halder–Wagner method

hoge.ins and hoge.int, has been finished, sinθ and β(instr)

For the determination of D and ε, Halder and Wagner [10,11]

cosθ, where β(instr) denotes the integral breadth due to the

proposed an alternative equation containing the integral

instrument, are estimated for all the reflections observed inbreadth, β*, of the reciprocal lattice point and the lattice-

the whole diffraction pattern and output to the gnuplot data

plane spacing, d*, for the reciprocal cell:

file, hoge.gpd. Profile broadening arising from the instrument－ 35 －

Implementation of the Williamson–Hall and Halder–Wagner Methods into RIETAN-FP

includes spectral distribution of X-ray and neutron beams,

The power, n, is input by the user in hoge.ins. In happy anddeviations from the ideal geometry, axial divergence, sample

special situations, n is 1 for Lorentzian instrumental andtransparency, etc. In all profile functions, G(Δ2θ）used insample broadening and 2 for Gaussian instrumental and

RIETAN-FP, the peak area of each reflection is normalized

sample broadening [19]. In the case of the intermediatein such a way thatcharacter (Voigtian instrumental and sample broadening), n(19)

is expected to lie between 1 and 2. Equations (12) and (17)are then calculated from a set of sinθ and β(sample)cosθ

Therefore, β is simply equal to the reciprocal of the peakpairs, which is followed by linear regression analysis to yield

intensity.

D and ε. Of course, graphing of the plot with gnuplot is also

Equations (12) and (17) are calculated from sinθ and β(instr)

possible in this case because both sinθ and β(sample)cosθ forcosθ, which is followed by linear regression analysis to

all the reflections in the whole 2θ range are output to hoge.

obtain D and ε from the slope and y intercept. The D and ε

gpd together with the corresponding script file, hoge.plt, forvalues, which are output to hoge.lst, are expected to give an

gnuplot. Deviations from the linear relationships, i.e., Eq. (12)approximately straight line, which can easily be checked by

and Eq. (17), because of anisotropic profile broadening candrawing a graph by gnuplot with a pair of files, hoge.gpd andeasily be recognized by the resulting graph.

hoge.plt, created by RIETAN-FP. Commands in hoge.plt may
be freely modified by the user to change the appearance of

6. Determination of a crystallite size withthe graph.

diffraction data of CeO2

An instrumental standard having the same crystal struc-

The features of the WH and HW methods in RIETAN-FP

ture and chemical composition is preferred, of course. Un-v2.6 were tested with two sets of X-ray powder diffraction

less such a sample of high crystallinity is available, a stand-data for CeO 2: a broadened sample and an instrumental

ard reference material such as NIST SRM 640 (Si) and SRM

standard used for the first size/strain round robin [17]. These

660 (LaB6) with a similar linear attenuation coefficient, μ,

two intensity data (a “common” instrumental setup withmay be used as a substitute. Such a caution is demanded

Cu Kα radiation) were downloaded from a Web page of theparticularly when a sample of a small μ value is mounted onround robin [21].

a flat-plate holder in an X-ray powder diffractometer with

Nanocrystalline CeO 2 exhibiting profile broadening due

the Bragg–Brentano geometry where the effect of sample

to the crystallite-size effect was produced by the thermaltransparency is appreciable.

treatment of hydrated ceria at 923 K for 45 h [22]. Substantial strain is expected to be absent in this sample. The in-

5.2 A sample showing broadened diffraction profiles

strumental standard was prepared by annealing commer-

The hoge.gpd file for the instrument standard sample iscially available CeO2 at 1573 K for 3 h in air.

renamed instrument.gpd and copied into a folder where aWith RIETAN-FP v2.6, the two intensity data were ana-

series of files, e.g., hoge.ins and hoge.int, for an analysis

lyzed by the Rietveld method to determine the crystallite size

sample are contained. If needed, one or more lines storingand microstrain of the annealed sample of CeO2 according tohkl, sinθ, βcosθ, etc. for another instrument standard may

the procedures described in Sect. 5. The n value in Eq. (20)be inserted into instrument.gpd to complement reflection

was set at 2.

data in a low-angle region. Then, Rietveld or Le Bail analy-

Figure 1 illustrates the HW plot drawn from the two sets

sis is carried out to obtain sinθ and β(obs)cosθ. The sinθof the Rietveld refinement. A distinct linear relationship isand β(instr)cosθ values are input from instrument.gpd andfound between (β/tanθ) 2 and β/(tanθ sinθ). The y intercept

interpolated by spline interpolation to give β(instr)cosθ cor-

is as small as 9.443×10 –6, corresponding to a very small mi-

responding to the present sample. Such a procedure is very

crostrain of 0.07682 % in this sample annealed at the highconvenient, requiring only instrument.gpd.

temperature. The WH plot also showed that βcosθ bears

Strictly speaking, the observed profile is the convolution ofa linear relationship to sinθ. The upper limit of ε was esti-

profiles due to the instrument and the sample. To evaluate

mated at 0.4957 % from the slope of the resultant straight line

the approximate contribution of the broadening, β(sample),with C = 4. This strain value is comparable to the average

originating in the sample, the following approximation

one, 0.4(9) %, in the round robin [21].
The crystallite sizes determined by the WH and HWformula is used:(20)

methods were 22.47K nm and 21.69K nm, respectively. K
depends on the definition of the crystallite size. If the K－ 36 －

Fujio Izumi, Takuji Ikeda

Acknowledgments
We thank professor Takashi Ida of Nagoya Institute ofTechnology for fruitful discussion on the analysis of crystallite sizes and microstrains and Dr. Manabu Inukai of Nagoya Institute of Technology for letting us know of the HW
method.
References[1] R. Jenkins, R. L. Snyder, “Introduction to X-ray Powder
Fig. 1 The Halder–Wagner plot for the round-robin sample ofnanocrystalline CeO2. The straight line was obtained by linear regression analysis. Note that (β/tanθ) 2 is multiplied by
1000.

Diffractometry,” John Wiley & Sons, Inc., New York
(1996) pp. 89–91.
[2] F. Izumi, K. Momma, Solid State Phenom. 130 (2007)
15–20.
[3] “The Rietveld Method,” ed. by R. A. Young, Oxford
University Press (1995).
[4] A. Le Bail, H. Duroi, J. L. Fourquet, Mater. Res. Bull. 23

value of 4/3 for the mean volume-weighted size of sphericalcrystallites [20] is adopted,〈D〉
v is calculated at 29.96 nm
in the WH method and 28.92 nm in the HW one. These two(1988) 447–452.
[5] K. Momma, T. Ikeda, A.A. Belik, F. Izumi, Powder
Diffr. 28 (2013) 184–193.

values slightly less than 30 nm fall within ranges of〈D〉
v,[6] F. Izumi, Solid State Ionics 172 (2004) 1–6.

(32±11) nm, reported by Balzar et al. [17] and are compa-[7] A. C. Larson, R. B. Von Dreele, “General Structure

rable to〈D〉
v determined by a fundamental-parameter ap-

Analysis System (GSAS),” Report LAUR 86-748, Los

proach [23] (see Fig. 4 in Ref. 17).

Alamos National Laboratory, Los Alamos (2004) pp.
162–164.

7. Conclusion[8] P. Thompson, D. E. Cox, J. B. Hastings, J. Appl. Crys-

The WH and HW plots, which have recently been implemented into RIETAN-FP v2.6 or later, are the most
rapid diagnostic tools for determining the kind(s) of profile
broadening present in samples analyzed by the Rietveld or Le
Bail method [19]. Both of them present the following valuable
information about microstructures of various polycrystalline
materials:

tallogr. 20 (1987) 79–83.
[9] G. K. Williamson, W. H. Hall, Acta Metall. 1 (1953)
22–31.
[10] N. C. Halder, C. N. J. Wagner, Acta Crystallogr. 20(1966) 312–313.
[11] N. C. Halder, C. N. J. Wagner, Adv. X-Ray Anal. 9 (1966)
91–102.

(a) Clear discrimination between crystallite-size and microstrain effects,[12] http://gnuplot.sourceforge.net/[13] P. Scherrer, Nachr. Ges. Wiss. Göttingen, Math.-Phys.

(b) An estimate of the crystallite size, D,

Klasse 2 (1918) 96–100.

(c) An estimate of the microstrain, ε,[14] A. R. Stokes, A. J. C. Wilson, Proc. Cambridge Philos.

(d) Clear distinction between isotropic (monotonic curve)

Soc. 40 (1944) 197–198.

and anisotropic (scatter) broadening by graphical repre-[15] http://pd.chem.ucl.ac.uk/pdnn/peaks/size.htm

sentation with gnuplot.

[16] A. R. Stokes, A. J. C. Wilson, Proc. Phys. Soc. (London,

For example, (b) and (c) are useful to characterize nano-

U. K.) 56 (1944) 174–181.

crystalline materials showing marked surface effects while[17] D. Balzar, N. Audebrand, M. R. Daymond, A. Fitch, A.

(c) helps us to estimate the degree of inhomogeneous distri-

Hewat, J. I. Langford, A. Le Bail, D. Louër, O. Masson, C.

bution of solute atoms in solid solutions. With only numeri-

N. McCowan, N. C. Popa, P. W. Stephens, B. H. Toby, J.

cal data, anisotropic broadening cannot be well perceived

Appl. Crystallogr. 37 (2004) 911–924.

contrary to (d). We are confident that the present new features of microstructural characterization in RIETAN-FP deliver added value to its users in both academic institutions
and industries.

[18] W. H. Hall, Proc. Phys. Soc. A (London, U. K.) 62 (1949)
741–743.
[19] E. H. Kisi, C. J. Howard, “Applications of Neutron
Powder Diffraction,” Oxford University Press, New York－ 37 －

Implementation of the Williamson–Hall and Halder–Wagner Methods into RIETAN-FP(2008) Chap. 9.
[20] T. Ida, S. Shimazaki, H. Hibino, H. Toraya, J. Appl.
Crystallogr. 36 (2003) 1107–1115.
[21] http://www.ccp14.ac.uk/ccp/web-mirrors/balzar/div853/
balzar/s-s_rr.htm[22] N. Audebrand, J.-P. Auffrédic, D. Louër, Chem. Mater.
12 (2000) 1791–1799.
[23] R. W. Cheary, A. Coelho, J. Appl. Crystallogr. 25 (1992)
109–121.

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