The “state of the art” of the diffraction analysis of crystallite sizeand lattice strain1

Eric J. Mittemeijer* and Udo Welzel

Max Planck Institute for Metals Research, Heisenbergstraße 3, 70569 Stuttgart, Germany

Received June 1, 2008; accepted July 4, 2008

X-ray powder diffraction / Line-profile analysis /Crystallite size / Microstrain / Coherency of diffraction

Abstract. This paper addresses both old, but “renovated”methods and new methods for diffraction line-profileanalysis. Classical and even extremely simple single-linemethods for separating “size” and “strain” broadeningeffects have merit for characterization of the material im-perfectness, but it is generally very difficult to interpretthe data obtained in terms of microstructure parameters asused in materials science. Developments of recent years,focusing on distinct anisotropic line-broadening effects, asdue to the type, orientation and distribution of dislocationsand minute compositional variation, will be touched upon.The most promising development may be the synthesis ofline profiles on the basis of a microstructure model andapplication of the (kinematical) diffraction theory withoutany further assumption, which contrasts with the othermethods. This approach can in principle be applied in sin-gle-line and multiple-line variants and also in analyses ofthe whole diffraction pattern. The advantage is the directevaluation of microstructure parameters as used in materi-als science. The challenge is to develop microstructuremodels which are flexible enough to be applicable inmore than one case . . .

1. Introduction

Diffraction lines of crystalline materials contain a wealthof microstructural information: The amount and distribu-tion of the phases in the material, compositional inhomo-geneity, the crystallite size and shape distributions, thecrystallographic orientation distribution function, the con-centrations and distributions of crystal defects such as va-cancies, dislocations, stacking and twin faults, and, notleast, lattice distortions due to mechanical stresses, etc.(see, for example, Ref. [1] and references therein). In

many cases such information is not easily and statisticallyassured accessible by methods other than diffraction.

The analysis of diffraction-line broadening, the topic ofthis paper, evolved already shortly after the discovery ofdiffraction of X-rays by crystals by Friedrich, Knippingand von Laue (1912): Scherrer (1918) found that thebreadth of a diffraction line is related to the finite size ofthe diffracting crystals. Considering that, as follows fromdifferentiating Bragg’s law, lattice-parameter fluctuationsare also exhibited by diffraction-line broadening, Dehlin-ger and Kochendorfer, already as early as 1939, realizedthat a separation of the diffraction-line broadening in size-and strain-related contributions can, in principle, beachieved provided that the diffraction angle dependence ofthe line breadth is known [2].

Unfortunately, straightforward extraction of quantitativeinformation on size and strain from the shape (“width”) datais normally impossible. The least of the problems met isprobably the elimination of instrumental broadening effects,for which more or less reliable approaches exist whichdepart either from recordings of broadening by standardspecimens, or, more recently, from calculation of the instru-mental broadening on the basis of the known instrumental/geometrical details of the diffraction experiment.

Fundamental problems are the unravelling of the var-ious contributions to the observed, broadened diffractionlines due to the various types of microstructural details, ascrystallite size, lattice (micro)strain, planar faults (not con-sidered in this paper; cf. e.g. Refs. [3, 4]), etc., and theirinterpretation in terms of parameters that are used in mate-rials science, as dislocation densities, faulting probabilitiesand crystallite size.

On the one hand, more and more advanced methods toextract microstructural parameters from the profile para-meters of a single or multiple diffraction lines using moreor less realistic, general assumptions on the material im-perfection/line shape are developed: line-profile decompo-sition. On the other hand, a recent, powerful, yet virginalapproach appears to be line-profile synthesis, where themicrostructural parameters are determined by fitting lineprofiles, calculated on the basis of a model for the micro-structure specific for the material investigated, to measuredprofiles (i.e. no line-shape assumptions are employed).

The length of the diffraction vector (and the correlationof the positions of the scattering atoms) is crucial for the

552 Z. Kristallogr. 223 (2008) 552–560 / DOI 10.1524/zkri.2008.1213

# by Oldenbourg Wissenschaftsverlag, Munchen

1 Presented as keynote lecture at the International Conference onthe Diffraction Analysis of the Microstructure of Materials (“Size-Strain V”; Garmisch-Partenkirchen, Germany, October 7–9, 2007).Full Proceedings available at www.zkristallogr.de (Z. Kristallogr.Suppl. 27 (2008); open access).

* Correspondence author (e-mail: e.j.mittemeijer@mf.mpg.de)

occurrence of incoherency of diffraction, and as a conse-quence, apart from extreme cases, the so-called crystallite-size values depend on the reflection considered. Thereby,and this is not often realized, classical methods for line-profile analysis, but also recent developments, where allreflections in the entire diffraction pattern are simulta-neously analysed, under simple assumptions as a sizebroadening independent of the length of the diffractionvector, can become invalidated.

2. Correction for instrumental broadening

The measurement apparatus, usually a diffractometer orsome type of camera, generally brings about a significantintrinsic, instrumental broadening of the diffraction lines.Two approaches can be considered to determine instru-mental line broadening: On the one hand, a specimen withnegligible structural line broadening can be investigated;the observed broadening is then (taken as) the instrumen-tal broadening. On the other hand, the broadening can becalculated provided that sufficiently accurate models/meth-ods exist. The determination of the instrumental broaden-ing is dealt with in Section 2.1.

For analysing the line broadening measured for a parti-cular specimen under investigation, a correction for instru-mental broadening has to be performed. This is dealt within Section 2.2.

2.1 Determination of the instrumental profile

The selection of an appropriate reference specimen forcharacterising instrumental diffraction-line broadening re-quires careful consideration. In passing it is noted herethat the standard reference material SRM 660 distributedby the National Institute of Standard and Technology exhi-bits a small, but in some cases, even for a laboratory pow-der diffractometer, non-negligible structural diffraction-linebroadening obscuring the genuine instrumental line broad-ening [5, 6]. A newer batch SRM 660a should lift thisproblem (cf. also Ref. [7]).

For a calculation of the instrumental diffraction-linebroadening, the fundamental parameter [8, 9] and ray-tra-cing [10–12] approaches are usually adopted. Whereas theformer presupposes that the aberrations due to different in-strumental aberrations can each be quantified by analyticalfunctions and can be treated independently, the latter is timeconsuming. An approach overcoming both drawbacks, byconsidering different aberrations simultaneously and provid-ing calculation efficiency, has been proposed recently forlaboratory Bragg-Brentano powder diffractometers [13]. Acomparison of methods for modelling the effect of axial di-vergence in laboratory powder diffraction arrived at the con-clusion that a computationally simplified approximationbased on Edgeworth series can be employed [14]. Specialattention has also been paid to instrumental diffraction-linebroadening of synchtrotron-based diffractometers based oncollimating [6] and focusing [15] optics.

In recent years diffractometers operating in parallel-beam geometry mode have become available also for la-boratory measurements [12, 16–19]. As the parallel beamgeometry does not rely on a focusing condition, a parallel-

beam diffractometer is insensitive to defocusing errors.This implies that the specimen can be tilted and rotated ina parallel-beam diffractometer, as is required for stress andtexture measurements and investigations of the inhomo-geneity and anisotropy of the microstructure [18], withoutchanging the (extent of) instrumental broadening. This(invariance of) instrumental broadening has been investi-gated both for diffractometers based on X-ray (polycapil-lary) lenses [12, 17] and X-ray mirrors [18, 19].

2.2 Subtraction/incorporationof the instrumental broadening

Depending on the strategy of analysis of diffraction-linebroadening, line-profile decomposition versus line-profilesynthesis (cf. Section 1), instrumental diffraction-linebroadening has to be taken into account differently.Whereas the former approach requires a subtraction of theinstrumental from the measured broadening, the latter ap-proach requires an ‘addition’ of the instrumental to the(calculated/modelled) specimen broadening. This is usual-ly achieved by a deconvolution method and a convolutionmethod, respectively.

Various approximate strategies for a correction ofbreadth parameters (as full widths at half maximum andintegral breadths) by “simple subtraction” of the corre-sponding breadth parameters for the instrumental broaden-ing have been developed (cf. Refs. [20, 21] and referencestherein). Since the pioneering work of Stokes [22] invol-ving a rigorous deconvolution of the measured broadeningwith the instrumental broadening in Fourier space, no ma-jor progress in deconvolutive methods has been made untilrecently: a novel method for deconvolution has been pro-posed both for laboratory [23] and synchrotron [24] pow-der diffractometers. This method combines scale transfor-mation, data interpolation and fast Fourier transformationand permits a rigorous subtraction, in steps, of broadeningcontributions due to axial divergence, flat specimen aberra-tion, specimen transparency and the wavelength distribu-tion of the X-ray source, provided that the individual in-strumental aberrations can be parameterized each by onlyone diffraction-angle dependent width parameter.

3. Size-strain broadening

3.1 Simple approaches

If data of high quality are unavailable (e.g. in the analysis ofin-situ, non-ambient measurements) or an application doesnot merit the expenditure of time and effort required foradvanced line-profile analysis/synthesis methods (as wholepowder pattern modelling), a simple analysis of integralbreadths may be appropriate for obtaining semi-quantitativeestimates of crystallite size and microstrain. Two basic ap-proaches for the separation of size and strain broadening onthe basis of integral breadths can be adopted:

(i) Single-line method [25]. A diffraction line is con-ceived as a convolution of a Gaussian and a Lorentzian(also called Cauchy) profile (i.e. as a Voigt function),where the Gaussian component is due to microstrain andthe Lorentzian component is due to finite crystallite size.

The “state of the art” of the diffraction analysis of crystallite size and lattice strain 553

Determination of the crystallite size (volume-weighed do-main size in the direction parallel to the diffraction vector)D and of a measure for the width of the microstrain distri-bution e is possible making use of the equations

bL ¼l

D cos q; ð1Þ

bG ¼ 4e tan q ð2Þwhere l is the wavelength, 2q is the Bragg angle of reflec-tion, bL is the integral breadth of the Lorentzian compo-nent and bG is the integral breadth of the Gaussian com-ponent (for details, see Ref. [20] and references therein).For the case of a Gaussian microstrain distribution it ispossible to calculate the local root-mean-square strainh"2

0i1=2 from e: h"2

0i1=2 ¼ ð2=pÞ1=2 e [20, 26].

(ii) Williamson-Hall (WH) method [27]. Assuming thatthe size and strain profile components are Lorentzian pro-files, the corresponding integral breadths are linearly addi-tive to obtain the total integral breadth in reciprocal spaceb* ¼ ðb cosqÞ=l (cf. Eqs. (1) and (2):

b* ¼ 1

Dþ 2ed* ; ð3Þ

where d* ¼ ð2 sin qÞ=l. A plot of b* versus d* should re-sult in a straight line and the values for size and strain canthen be obtained directly from the intercept and the slopeof the straight line, respectively. Equation (3) presents onespecific expression for a ubiquitously adopted (but non-tri-vial, see what follows in Section 3.5) assumption that sizebroadening does not depend on the length of the diffractionvector whereas strain broadening does. Other variants ofthe WH method exist [e.g. adopting Gaussian shaped func-tions, taking into account anisotropic line broadening (seeSection 3.3.3), as due to dislocations etc.], but all are basedon the assumption of specific profile shapes.

Integral-breadth methods have been used in various alsorecent studies with the supposition that the results have aquantitative meaning (e.g. Refs. [5, 28–30]; for a criticaloverview of such methods, see also Ref. [21]). Results quan-titatively consistent with results obtained by more advancedmethods can be obtained, in particular and obviously forcases where one source of line broadening prevails [31, 32].

3.2 Column length/crystallite size distribution

The column length will generally exhibit a distribution(the column-length distribution can be calculated from the

crystallite-size distribution and vice versa if a crystalliteshape is adopted; see e.g. Ref. [33]). In principle the col-umn-length distribution is given by the second derivativeof the Fourier transform of the only size-broadened profile[34, 35]. However, reliable determination of the column-length distribution on this basis suffers from problems dueto background subtraction and truncation [20]. In particu-lar, the obtained size distributions can be highly unreliablein the presence of general strain broadening, which, in theline-profile decomposition approach, has to be separatedfrom the size broadening on the basis of usually unvali-dated assumptions (corresponding results, for example ob-tained in Ref. [36], should be mistrusted) [37].

An alternative approach departs from the presupposi-tion of a certain type of column-length or crystallite sizedistribution. For the description of monomodal distribu-tions, the Gamma- and lognormal distributions have beenproposed:

Gamma distribution [20]:

pðnÞ ¼ 1

Cnrexp ð�untÞ ð4Þ

where n denotes column length or crystallite size, C is anormalisation constant and r, u and t are adjustable para-meters (note that usually, t is (unnecessarily) taken asone).

Log-normal distribution:

pðnÞ ¼ ðð2pÞ1=2 sÞ�1 exp ð�ðln n=noÞ2=ð2s2ÞÞ=n ð5Þwhere no, the median, and s, the variance, are the adjusta-ble parameters (cf., for example, Refs. [38, 39]). It hasbeen found that in particular highly deformed metals oftenexhibit log-normal column length/size distributions (e.g.Refs. [38–40]).

Recently, the determination of column-length/crystal-lite-size distributions without a prior assumption about thetype of distribution has been attempted on the basis ofwhole powder pattern modelling. An approach involvinghistograms with “tuned” bin width and adjustable binheight, but assuming a spherical crystallite shape, has beenproposed in Ref. [41] (see Fig. 1).

An approximate solution for obtaining the crystallite-size distribution together with information on crystalliteshape has been proposed in Refs. [42, 43]. In the latter ap-proach a microstrain distribution with a homogeneous strainin each crystallite has been presupposed (this can be a se-vere limitation; cf. Section 3.5 and see next paragraph).

554 E. J. Mittemeijer and U. Welzel

Fig. 1. Whole powder pattern modelling: Crystallite size, D (diameter of the sphere (¼ crystallite)), distributions, pðDÞ, of nanocrystalline ceriapowders calcinated for 1 hour at different temperatures (increasing from the left to the right). The full histogram is the result of the analysiswithout prior assumption on the crystallite size distributions, whereas the line is the result of the analysis restricted to a log-normal size distribu-tion. Taken from Ref. [41].

The two categories of approaches (i.e. whether or notassuming a distribution function) for determining the col-umn-length distribution (and, possibly, the crystallite-sizedistribution on the basis of an additional assumption oncrystallite shape) both require that broadening fromsources other than finite size (as microstrains) is marginalor absent: the required assumptions to separate the sizebroadening from the other broadening components rendersa subsequent determination of column-length or crystallite-size distributions unreliable (corresponding results, e.g. aspublished in Ref. [36] (see first paragraph of this section)or Ref. [42] (see above paragraph) should be mistrusted).

3.3 Microstrain broadening

Whereas the fundamentals of size broadening are well es-tablished and in a mature state already since the 1950s,[34, 35] analysis of strain broadening is a field of currentlystrong activity, where both methods imposing assumptionson the kinematical diffraction theory and methods depart-ing from a microstructural model are developed.

3.3.1 Methods imposing assumptionson the kinematical diffraction theory

An overview of the methods based on specific assump-tions about the strain distributions in materials without re-ferring to a specific microstructural model is provided byTable 1 [27, 35, 44–46].

A quantitative evaluation of size and strain parametersderived from broadened line profiles requires thorough con-sideration of the underlying assumptions in the methodsused. A comparative application of the different methods toan imperfect material is not straightforward because the as-sumptions are incompatible (e.g. Gaussian strain distribu-tions in the Warren-Averbach method versus small strain gra-dients in the alternative method) and the resulting parametersare not defined in the same way (e.g. volume-versus area-weighted crystallite size) [39]; see also Fig. 1 in Ref. [46].

3.3.2 Methods departing from a microstructural model

A flexible general method based on a microstructuralmodel without referring to a particular type of defect isthe strain-field model proposed by Van Berkum et al. [47].In this approach, the strain field is composed of a super-position of the (component) strain fields of individual de-fects. The strain fields of the lattice defects are describedstatistically by three functions: the probability function for

the distance between the defects (projected onto the dif-fraction vector), the probability function for the amplitudeof the component strain fields and a function describingthe average shape (width) of the component strain fields.In the simplest case for application of the strain-field mod-el, the Fourier coefficient for the only strain broadenedprofile AdðLÞ is described by only three parameters (cf.Eq. (7) of van Berkum et al. [47]): (i) the mean projected(onto the diffraction vector) defect distance hsi, (ii) theroot-mean-square strain he2

oi, and (iii) the width of the(Lorentz shaped) component strain fields, w. A componentrepresenting a possible size broadening can simply be in-cluded [39]. For applications of this strain-field model toball-milled metal powders, see Refs. [39, 47, 48].

Methods departing from specific microstructural modelshave been developed for analysing line broadening due toinclusions in a crystalline matrix [49] and due to disloca-tions. In the following the focus is on dislocation linebroadening.

The pioneering work in this field is due to Krivoglazand Ryaboshapka [50] and Wilkens [51]. Krivoglaz andRyaboshapka considered sets of statistically random distri-butions of non-interacting (edge or) screw dislocations.Wilkens demonstrated that a random distribution of dislo-cations (in a set) is unrealistic and introduced the conceptof the restrictedly random dislocation arrangement. To thisend, the degree of correlation in the dislocation distribu-tion of a set was described by the so-called cut-off radiusRe, which can be considered as the radius of a cylinderwithin which the dislocation arrangement is random: Noelastic interaction of the various dislocations sets in thecrystal is considered to occur.

The strain Fourier coefficients AdðLÞ can be approxi-mated by [52]:

AdðLÞ ¼ exp ½�ðcLÞp� ; ð6Þwhere c characterizes the width of AdðLÞ and the exponentp takes values between 1 (Lorentzian line profile) and 2(Gaussian line profile). The shape parameter M,

M ¼ ReðrÞ1=2 ; ð7Þcan be calculated from p (cf. Eq. (2.19) of Vermeulen et al.[52]). c is related to the square-root of the dislocation densityr. Note that dislocation line broadening is usually anisotro-pic, i.e. it depends on the hkl reflection (i.e. it depends on theorientation and length of the diffraction vector; cf. Section3.3.3). This can be rationalized by the so-called dislocation-contrast factor, which is contained in c in Eq. (6).

The “state of the art” of the diffraction analysis of crystallite size and lattice strain 555

Table 1. Summary of basic assumptions made in line profile decomposition methods and the type of size and strain data obtained [39]. AdðLÞ isthe strain (‘distortion’) Fourier coefficient of a line profile, L is the correlation distance perpendicular to the diffracting planes.

Method Assumptions Size Strain

Williamson-Hall conventionalplot [27, 44] 1949, 1953

Lorentz shaped peakprofiles for size- and strain-broadened profile

Volume-weightedcolumn length

Maximum strain, e related to localmean squared strain he0

2ifor Gaussian strain distributions

Warren-Averbach [35, 45]1950, 1952

Gaussian strain distributionor small strains

Area-weighted column length Mean squared strain, he2(L)i,related to Ad(L)

Alternative method [46] 1994 Small strain gradientsand broad size distribution

Area-weighted column length No analytical relation betweenhe2(L)i and the strain Fouriercoefficients

Dislocation densities and configurations have been in-vestigated in thin films and plastically deformed materials(see, for example, Refs. [39, 54, 55]). In addition to thedislocation density and the cut-off radius, the fractions ofscrew and edge dislocations can be determined. For a re-cent review on dislocation line broadening, see Ref. [56].

3.3.3 Anisotropic microstrain-like diffraction-linebroadening

The occurrence of anisotropic diffraction-line broadening(i.e. the diffraction-line broadening depends non-monoto-nously on the hkl reflection when plotted versus 2q) is aquite general phenomenon which has attracted consider-able attention both in phenomenological and microstruc-ture-based modelling of diffraction-line broadening. Aniso-

tropic diffraction-line broadening may be categorized asfollows:

(i) Only small (negligible) microstrain gradients withincrystallites. In this case, the increase of line broadeningwith increasing length of the diffraction vector, for a givenset of diffracting lattice planes, is proportional to tan q (cf.also Section 3.5) [47, 57]. Phenomenological models forthis type of anisotropic microstrain diffraction-line broaden-ing have been developed and implemented in Rietveld-re-finement programs (see, for example, Ref. [58]). A modelcase for this type of line broadening is a (hypothetical) iso-tropic microstress distribution which, in combination withsingle-crystal elastic anisotropy, results in an anisotropicmicrostrain distribution [57].

This approach is likely to overestimate the anisotropy ofdiffraction-line broadening, as an isotropic microstress distri-bution gives rise to geometrically incompatible strains in dif-ferently oriented crystallites. The real grain interaction in apolycrystalline material is more likely to be between isotro-pic stress and isotropic strain distributions. Another recentlyconsidered source of anisotropic (microstrain-like) linebroadening are composition fluctuations in a non-cubic ma-

556 E. J. Mittemeijer and U. Welzel

a�

b�

Fig. 2. (a) Full width at half-maximum (FWHM) of the reflections ofa e-FeN0.433 powder and LaB6 (used for the determination of the in-strumental line broadening) measured using a Bragg-Brentano dif-fractometer with Co Ka radiation. The apparent ‘scatter’ of the linewidths of the powder is due to compositional inhomogeneities. Fig.2a and b have been taken from Ref. [59]. (b). The anisotropy of themicrostrain-like broadening observed from a e-FeN0.433 powder. Thedirection dependence of the FWHM, Bf

D2 q;hkl, as a function of theangle of the diffraction vector relative to the c axis for the hexagonalcrystals system. The separate points indicate the experimental data;the solid line represents the curve obtained by fitting a model for linebroadening due to compositional fluctuations to the experimentaldata. A compositional fluctuation of e-FeN0.433�0.008 is obtained.

a�

b�

Fig. 3. (a) The FWHM (w*) and the integral breaths (Bf*) as a func-tion of the reciprocal space coordinate d* in the classical Williamson-Hall plot in the case of Nb ball milled for one day. The indices of thereflections have also been indicated in the figure. Note the pronounc-edly anisotropic nature of line broadening. (b) The modified William-son-Hall plot of the same data as in Fig. 3a. As a function of d*C1/2

(where C is the dislocation contrast factor). The indices of the reflec-tions have also been indicated in the figure. Fig. 3a and b have beentaken from Ref. [62].

terial (see Fig. 2 for an example) [59]. For a recent generaltreatment on anisotropic microstrain broadening due to afield-tensor (rank 0, pertaining to composition variation; rank2, pertaining to stress/strain distributions), cf. Ref. [60].

(ii) No assumption about microstrain gradients: Adop-tion of a microstructural model. In this case, the depen-dence of strain broadening on the length of the diffractionvector follows from the microstructural model. The re-cently most frequently studied case is dislocation linebroadening, for which anisotropic line broadening is dueto the orientation of the diffraction vector with respect toslip systems and the anisotropy of elastic constants (cf.Section 3.3.2 and Fig. 3; e.g. Refs. [56, 61–63]).

3.4 Macroscopic anisotropy

Massive and polycrystalline specimens generally exhibitan anisotropic microstructure. Consider, as an example,thin films deposited by physical vapour deposition: Often

a so-called columnar microstructure occurs, where the filmconsists of e.g. columnar-shaped grains separated by grainboundaries oriented more or less perpendicularly to thelayer surface. For such a thin film, the crystallite size is ananisotropic quantity: the crystallite size along the surfacenormal is much larger than the crystallite size in the planeof the film. Thus, macroscopically anisotropic size broad-ening occurs (see Fig. 4a for an example) [18]. Anisotro-pic size broadening can be accompanied by anisotropicstrain broadening (see Fig. 4b) [18], which can also occurdue to unequal densities of defects (as dislocations on dis-tinct slip systems) along different directions in the speci-men [52].

The analysis of macroscopically anisotropic diffraction-line broadening is considerably simplified experimentallyby the use of parallel-beam diffractometers, because instru-mental aberrations occurring for focusing diffractometers(i.e. ‘defocusing’) upon changing the orientation of thediffraction vector (from e.g. the specimen surface normaldirection, for the case of Bragg-Brentano diffractometers)can be avoided [18].

3.5 Crystallite size and coherency of diffraction

For most polycrystalline specimens, the phase difference(reduced modulo 2p) of a wave scattered by one crystal-lite and the wave scattered by a second crystallite takesvalues between 0 and 2p with equal probability. In thiscase, the total diffracted intensity can be taken as the sumof intensities scattered by the individual crystallites sepa-rately. This naturally leads to the usually adopted conceptof size broadening due to the finite size of individual crys-tallites and strain broadening related to the relative displa-cement of atoms within one grain. A more general ap-proach is to consider the whole irradiated volume of apolycrystal as a coherently scattering domain. Such an ap-proach has been followed by van Berkum et al. for analys-ing strain broadening on the basis of a flexible model forstrain fields associated with lattice defects (cf. also Section3.3.2) [47]. As the phase difference of scattered wavesoriginating from different scatterers (atoms) is the scalarproduct of the diffraction vector and the position (differ-ence) vector of the scatterers, both the character of thestrain fields in a specimen and the length of the diffractionvector are decisive for diffraction-line broadening. It hasbeen demonstrated that for general strain broadening, theorder-dependence of the diffraction-line width is complex,i.e. neither order-independent broadening (traditionallytermed ‘size broadening’) nor broadening proportional tothe length of the diffraction vector (traditionally termed‘strain broadening’) occurs (see Fig. 5a). Two limitingcases have been identified:

(i) For infinitely broad component strain fields of lat-tice defects (w=hsi ! 1; cf. Section 3.3.2), the broaden-ing is proportional to the length of the diffraction vectord* (cf. Eq. (3)):

b ¼ ð2pÞ1=2 d*he2i1=2 ; ð8Þ

where he2i is the mean squared strain. This is the well-known strain broadening for a specimen with a constant

The “state of the art” of the diffraction analysis of crystallite size and lattice strain 557

a�

b�

Fig. 4. (a) The crystallite, grain sizes of a 250 nm thick Ti3Al layeras viewed along different hhkli* directions, i.e. as function of thetilting angle w. The schematic figure represents the rectangular Ti3Algrains (with a height of 50 nm and a width of 6 nm) in the Ti3Allayer, the crystallite size of the rectangular grain Dhhkli� measuredalong the hhkli* direction is 50 nm/cos w for 0� < w < 6.89�; 6 nm/sin w for 6.89� < w < 90�, as shown by the solid and dashed lines.(b) The microstrains and the grain boundary fraction FGB of theTi3Al layer as viewed along different hhkli* directions, i.e. as func-tion of the tilting angle w. FGB ¼ gD=D, where D is a constant rela-tive to the grain boundary thickness, and equals 1.0 nm here; g is ageometrical constant and equals 1. The results suggest that FGB andthe microstrain behave similarly as function of w. Fig. 4a and b havebeen taken from Ref. [18].

lattice spacing d within each crystallite and a Gaussianspacing distribution over the crystallites, where:

he2i ¼ ðhd2i � hdi2Þ=hdi2 : ð9Þ

This proportionality of line broadening (e.g. integralbreadth) and length of the diffraction vector (see Fig. 5a)has often been used for strain broadening in general (cf.Eqs. (2) and (3); Section 3.3.1).

(ii) For infinitely narrow component strain fields(w=hsi ! 0, as corresponds to small-angle grain bound-aries in polycrystalline material; cf. Section 3.3.2), it fol-lows that

b / ðd*Þ2 ð10Þ

for small lengths of the diffraction vector. For larger lengths,a constant line width b is obtained (‘classical’ size broaden-ing). A model microstructure producing this type of broaden-ing consists of domains of undistorted material shifted withrespect to each other. For increasing length of the diffractionvector, the phase differences (modulo 2p) are almost uni-formly distributed, thus incoherency of diffraction occurs.

This prediction of this type of diffraction-line broaden-ing (increasing at small lengths of the diffraction vector

and constant at large lengths; see Fig. 5b) has recentlybeen experimentally confirmed on the basis of measuredline broadening of nanocrystalline thin films by Rafajaet al. (see Fig. 5b) [64]: i.e. a loss of coherency with in-creasing length of the diffraction vector at small lengths ofd* leading to incoherent diffraction of the individual crys-tallites at larger length d*. For another study of the effectof partial coherence on size broadening, see Ribarik et al.[65] For an application of the strain-field model incorpor-ating coherency effects, see Lucks et al. [39].

4. Concluding remarks and perspectives

(1) Whereas unprejudiced individual peak-profile analysis(still) allows the most severe microstructural modeltesting, a simultaneous analysis of all reflections, sub-ject to more severe (e.g. profile-shape) constraints, isrequired if overlap of diffraction lines occurs. Distinc-tion of both evaluation approaches can be expected todiminish as more sound microstructure-based diffrac-tion models are developed.

(2) For sensitive crystal-imperfection analysis, an evalua-tion of anisotropic diffraction-line broadening, with re-spect to the (orientation and) length of the diffractionvector in either the reciprocal lattice or the specimenframe of reference, is a prerequisite.

(3) Preference for microstructure model-based line-profilesynthesis methods over line-profile decompositionmethods is due to the application of flawed micro-structure/diffraction models in line-profile decomposi-tion methods.

(4) Is a general, practically applicable microstructure/dif-fraction model possible?

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558 E. J. Mittemeijer and U. Welzel

a�

b�

Fig. 5. (a) Application of the strain-field model: Relative integralbreadths of line profiles in reciprocal space for different relativewidths w, of the component strain fields. The subscript ‘r’ indicatesthat the respective quantity has been normalized by the mean pro-jected (on the diffraction vector) defect distance. Taken from Ref.[47]. (b) Diffraction line broadening measured for a nanocrystallineTi0.38Al0.62N thin film (points) and the numerical simulation (solidline) taking into account the partial coherence of neighbouring nano-sized crystallites. Taken from Ref. [64]. Note the similarity with re-sults shown in Fig. 5a for, e.g., wr ¼ 0:03.

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Z. Kristallogr. Suppl. 27 (2008)Proceedings of the ”5th Size Strain“ Conference (Diffraction Analysis of the Microstructure of Materials) held in October 2007 in Garmisch-Partenkirchen, GermanyEditors: Eric J. Mittemeijer, Paolo Scardi, Andreas Leineweber and Udo Welzel

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