The "state of the art" of diffraction analysis of crystallite size and lattice strain.

552 Z. Kristallogr. 223 (2008) 552–560 / DOI 10.1524/zkri.2008.1213
# by Oldenbourg Wissenschaftsverlag, Munchen
¨

The “state of the art” of the diffraction analysis of crystallite size
1
and lattice strain

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 / many cases such information is not easily and statistically
Crystallite size / Microstrain / Coherency of diffraction assured accessible by methods other than diffraction.
The analysis of diffraction-line broadening, the topic of
Abstract. This paper addresses both old, but “renovated” this paper, evolved already shortly after the discovery of
methods and new methods for diffraction line-profile diffraction of X-rays by crystals by Friedrich, Knipping
analysis. Classical and even extremely simple single-line and von Laue (1912): Scherrer (1918) found that the
methods for separating “size” and “strain” broadening breadth of a diffraction line is related to the finite size of
effects have merit for characterization of the material im- the diffracting crystals. Considering that, as follows from
perfectness, but it is generally very difficult to interpret differentiating Bragg’s law, lattice-parameter fluctuations
the data obtained in terms of microstructure parameters as are also exhibited by diffraction-line broadening, Dehlin-
used in materials science. Developments of recent years, ger and Kochendorfer, already as early as 1939, realized
¨
focusing on distinct anisotropic line-broadening effects, as that a separation of the diffraction-line broadening in size-
due to the type, orientation and distribution of dislocations and strain-related contributions can, in principle, be
and minute compositional variation, will be touched upon. achieved provided that the diffraction angle dependence of
The most promising development may be the synthesis of the line breadth is known [2].
line profiles on the basis of a microstructure model and Unfortunately, straightforward extraction of quantitative
application of the (kinematical) diffraction theory without information on size and strain from the shape (“width”) data
any further assumption, which contrasts with the other is normally impossible. The least of the problems met is
methods. This approach can in principle be applied in sin- probably the elimination of instrumental broadening effects,
gle-line and multiple-line variants and also in analyses of for which more or less reliable approaches exist which
the whole diffraction pattern. The advantage is the direct depart either from recordings of broadening by standard
evaluation of microstructure parameters as used in materi- specimens, or, more recently, from calculation of the instru-
als science. The challenge is to develop microstructure mental broadening on the basis of the known instrumental/
models which are flexible enough to be applicable in geometrical details of the diffraction experiment.
more than one case . . . Fundamental problems are the unravelling of the var-
ious contributions to the observed, broadened diffraction
lines due to the various types of microstructural details, as
crystallite size, lattice (micro)strain, planar faults (not con-
1. Introduction
sidered in this paper; cf. e.g. Refs. [3, 4]), etc., and their
interpretation in terms of parameters that are used in mate-
Diffraction lines of crystalline materials contain a wealth rials science, as dislocation densities, faulting probabilities
of microstructural information: The amount and distribu- and crystallite size.
tion of the phases in the material, compositional inhomo- On the one hand, more and more advanced methods to
geneity, the crystallite size and shape distributions, the extract microstructural parameters from the profile para-
crystallographic orientation distribution function, the con- meters of a single or multiple diffraction lines using more
centrations and distributions of crystal defects such as va- or less realistic, general assumptions on the material im-
cancies, dislocations, stacking and twin faults, and, not perfection/line shape are developed: line-profile decompo-
least, lattice distortions due to mechanical stresses, etc. sition. On the other hand, a recent, powerful, yet virginal
(see, for example, Ref. [1] and references therein). In approach appears to be line-profile synthesis, where the
microstructural parameters are determined by fitting line
1
Presented as keynote lecture at the International Conference on profiles, calculated on the basis of a model for the micro-
the Diffraction Analysis of the Microstructure of Materials (“Size- structure specific for the material investigated, to measured
Strain V”; Garmisch-Partenkirchen, Germany, October 7–9, 2007).
Full Proceedings available at www.zkristallogr.de (Z. Kristallogr. profiles (i.e. no line-shape assumptions are employed).
Suppl. 27 (2008); open access). The length of the diffraction vector (and the correlation
* Correspondence author (e-mail: e.j.mittemeijer@mf.mpg.de) of the positions of the scattering atoms) is crucial for the

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

occurrence of incoherency of diffraction, and as a conse- beam diffractometer is insensitive to defocusing errors.
quence, apart from extreme cases, the so-called crystallite- This implies that the specimen can be tilted and rotated in
size values depend on the reflection considered. Thereby, a parallel-beam diffractometer, as is required for stress and
and this is not often realized, classical methods for line- texture measurements and investigations of the inhomo-
profile analysis, but also recent developments, where all geneity and anisotropy of the microstructure [18], without
reflections in the entire diffraction pattern are simulta- changing the (extent of) instrumental broadening. This
neously analysed, under simple assumptions as a size (invariance of) instrumental broadening has been investi-
broadening independent of the length of the diffraction gated both for diffractometers based on X-ray (polycapil-
vector, can become invalidated. lary) lenses [12, 17] and X-ray mirrors [18, 19].

2.2 Subtraction/incorporation
2. Correction for instrumental broadening of the instrumental broadening
The measurement apparatus, usually a diffractometer or Depending on the strategy of analysis of diffraction-line
some type of camera, generally brings about a significant broadening, line-profile decomposition versus line-profile
intrinsic, instrumental broadening of the diffraction lines. synthesis (cf. Section 1), instrumental diffraction-line
Two approaches can be considered to determine instru- broadening has to be taken into account differently.
mental line broadening: On the one hand, a specimen with Whereas the former approach requires a subtraction of the
negligible structural line broadening can be investigated; instrumental from the measured broadening, the latter ap-
the observed broadening is then (taken as) the instrumen- proach requires an ‘addition’ of the instrumental to the
tal broadening. On the other hand, the broadening can be (calculated/modelled) specimen broadening. This is usual-
calculated provided that sufficiently accurate models/meth- ly achieved by a deconvolution method and a convolution
ods exist. The determination of the instrumental broaden- method, respectively.
ing is dealt with in Section 2.1. Various approximate strategies for a correction of
For analysing the line broadening measured for a parti- breadth parameters (as full widths at half maximum and
cular specimen under investigation, a correction for instru- integral breadths) by “simple subtraction” of the corre-
mental broadening has to be performed. This is dealt with sponding breadth parameters for the instrumental broaden-
in Section 2.2. ing have been developed (cf. Refs. [20, 21] and references
therein). Since the pioneering work of Stokes [22] invol-
2.1 Determination of the instrumental profile ving a rigorous deconvolution of the measured broadening
The selection of an appropriate reference specimen for with the instrumental broadening in Fourier space, no ma-
characterising instrumental diffraction-line broadening re- jor progress in deconvolutive methods has been made until
quires careful consideration. In passing it is noted here recently: a novel method for deconvolution has been pro-
that the standard reference material SRM 660 distributed posed both for laboratory [23] and synchrotron [24] pow-
by the National Institute of Standard and Technology exhi- der diffractometers. This method combines scale transfor-
bits a small, but in some cases, even for a laboratory pow- mation, data interpolation and fast Fourier transformation
der diffractometer, non-negligible structural diffraction-line and permits a rigorous subtraction, in steps, of broadening
broadening obscuring the genuine instrumental line broad- contributions due to axial divergence, flat specimen aberra-
ening [5, 6]. A newer batch SRM 660a should lift this tion, specimen transparency and the wavelength distribu-
problem (cf. also Ref. [7]). tion of the X-ray source, provided that the individual in-
For a calculation of the instrumental diffraction-line strumental aberrations can be parameterized each by only
broadening, the fundamental parameter [8, 9] and ray-tra- one diffraction-angle dependent width parameter.
cing [10–12] approaches are usually adopted. Whereas the
former presupposes that the aberrations due to different in- 3. Size-strain broadening
strumental aberrations can each be quantified by analytical
functions and can be treated independently, the latter is time 3.1 Simple approaches
consuming. An approach overcoming both drawbacks, by
considering different aberrations simultaneously and provid- If data of high quality are unavailable (e.g. in the analysis of
ing calculation efficiency, has been proposed recently for in-situ, non-ambient measurements) or an application does
laboratory Bragg-Brentano powder diffractometers [13]. A not merit the expenditure of time and effort required for
comparison of methods for modelling the effect of axial di- advanced line-profile analysis/synthesis methods (as whole
vergence in laboratory powder diffraction arrived at the con- powder pattern modelling), a simple analysis of integral
clusion that a computationally simplified approximation breadths may be appropriate for obtaining semi-quantitative
based on Edgeworth series can be employed [14]. Special estimates of crystallite size and microstrain. Two basic ap-
attention has also been paid to instrumental diffraction-line proaches for the separation of size and strain broadening on
broadening of synchtrotron-based diffractometers based on the basis of integral breadths can be adopted:
collimating [6] and focusing [15] optics. (i) Single-line method [25]. A diffraction line is con-
In recent years diffractometers operating in parallel- ceived as a convolution of a Gaussian and a Lorentzian
beam geometry mode have become available also for la- (also called Cauchy) profile (i.e. as a Voigt function),
boratory measurements [12, 16–19]. As the parallel beam where the Gaussian component is due to microstrain and
geometry does not rely on a focusing condition, a parallel- the Lorentzian component is due to finite crystallite size.

554 E. J. Mittemeijer and U. Welzel

Determination of the crystallite size (volume-weighed do- crystallite-size distribution and vice versa if a crystallite
main size in the direction parallel to the diffraction vector) shape is adopted; see e.g. Ref. [33]). In principle the col-
D and of a measure for the width of the microstrain distri- umn-length distribution is given by the second derivative
bution e is possible making use of the equations of the Fourier transform of the only size-broadened profile
l [34, 35]. However, reliable determination of the column-
bL ¼ ; ð1Þ length distribution on this basis suffers from problems due
D cos q to background subtraction and truncation [20]. In particu-
bG ¼ 4e tan q ð2Þ lar, the obtained size distributions can be highly unreliable
where l is the wavelength, 2q is the Bragg angle of reflec- in the presence of general strain broadening, which, in the
tion, bL is the integral breadth of the Lorentzian compo- line-profile decomposition approach, has to be separated
nent and bG is the integral breadth of the Gaussian com- from the size broadening on the basis of usually unvali-
ponent (for details, see Ref. [20] and references therein). dated assumptions (corresponding results, for example ob-
For the case of a Gaussian microstrain distribution it is tained in Ref. [36], should be mistrusted) [37].
possible to calculate the local root-mean-square strain An alternative approach departs from the presupposi-
h"2 i1=2 from e: h"2 i1=2 ¼ ð2=pÞ1=2 e [20, 26]. tion of a certain type of column-length or crystallite size
0 0
(ii) Williamson-Hall (WH) method [27]. Assuming that distribution. For the description of monomodal distribu-
the size and strain profile components are Lorentzian pro- tions, the Gamma- and lognormal distributions have been
files, the corresponding integral breadths are linearly addi- proposed:
tive to obtain the total integral breadth in reciprocal space Gamma distribution [20]:
b* ¼ ðb cosqÞ=l (cf. Eqs. (1) and (2): 1 r
pðnÞ ¼ n exp ðÀunt Þ ð4Þ
1 C
b* ¼ þ 2ed * ; ð3Þ
D where n denotes column length or crystallite size, C is a
where d * ¼ ð2 sin qÞ=l. A plot of b* versus d * should re- normalisation constant and r, u and t are adjustable para-
sult in a straight line and the values for size and strain can meters (note that usually, t is (unnecessarily) taken as
then be obtained directly from the intercept and the slope one).
of the straight line, respectively. Equation (3) presents one Log-normal distribution:
specific expression for a ubiquitously adopted (but non-tri-
pðnÞ ¼ ðð2pÞ1=2 sÞÀ1 exp ðÀðln n=no Þ2 =ð2s 2 ÞÞ=n ð5Þ
vial, see what follows in Section 3.5) assumption that size
broadening does not depend on the length of the diffraction where no , the median, and s, the variance, are the adjusta-
vector whereas strain broadening does. Other variants of ble parameters (cf., for example, Refs. [38, 39]). It has
the WH method exist [e.g. adopting Gaussian shaped func- been found that in particular highly deformed metals often
tions, taking into account anisotropic line broadening (see exhibit log-normal column length/size distributions (e.g.
Section 3.3.3), as due to dislocations etc.], but all are based Refs. [38–40]).
on the assumption of specific profile shapes. Recently, the determination of column-length/crystal-
Integral-breadth methods have been used in various also lite-size distributions without a prior assumption about the
recent studies with the supposition that the results have a type of distribution has been attempted on the basis of
quantitative meaning (e.g. Refs. [5, 28–30]; for a critical whole powder pattern modelling. An approach involving
overview of such methods, see also Ref. [21]). Results quan- histograms with “tuned” bin width and adjustable bin
titatively consistent with results obtained by more advanced height, but assuming a spherical crystallite shape, has been
methods can be obtained, in particular and obviously for proposed in Ref. [41] (see Fig. 1).
cases where one source of line broadening prevails [31, 32]. An approximate solution for obtaining the crystallite-
size distribution together with information on crystallite
3.2 Column length/crystallite size distribution shape has been proposed in Refs. [42, 43]. In the latter ap-
proach a microstrain distribution with a homogeneous strain
The column length will generally exhibit a distribution in each crystallite has been presupposed (this can be a se-
(the column-length distribution can be calculated from the vere limitation; cf. Section 3.5 and see next paragraph).

Fig. 1. Whole powder pattern modelling: Crystallite size, D (diameter of the sphere (¼ crystallite)), distributions, pðDÞ, of nanocrystalline ceria
powders calcinated for 1 hour at different temperatures (increasing from the left to the right). The full histogram is the result of the analysis
without 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 not the distance between the defects (projected onto the dif-
assuming a distribution function) for determining the col- fraction vector), the probability function for the amplitude
umn-length distribution (and, possibly, the crystallite-size of the component strain fields and a function describing
distribution on the basis of an additional assumption on the average shape (width) of the component strain fields.
crystallite shape) both require that broadening from In the simplest case for application of the strain-field mod-
sources other than finite size (as microstrains) is marginal el, the Fourier coefficient for the only strain broadened
or absent: the required assumptions to separate the size profile Ad ðLÞ is described by only three parameters (cf.
broadening from the other broadening components renders Eq. (7) of van Berkum et al. [47]): (i) the mean projected
a subsequent determination of column-length or crystallite- (onto the diffraction vector) defect distance hsi, (ii) the
size distributions unreliable (corresponding results, e.g. as root-mean-square strain he2 i, and (iii) the width of the
o
published in Ref. [36] (see first paragraph of this section) (Lorentz shaped) component strain fields, w. A component
or Ref. [42] (see above paragraph) should be mistrusted). representing a possible size broadening can simply be in-
cluded [39]. For applications of this strain-field model to
3.3 Microstrain broadening ball-milled metal powders, see Refs. [39, 47, 48].
Methods departing from specific microstructural models
Whereas the fundamentals of size broadening are well es- have been developed for analysing line broadening due to
tablished and in a mature state already since the 1950s, inclusions in a crystalline matrix [49] and due to disloca-
[34, 35] analysis of strain broadening is a field of currently tions. In the following the focus is on dislocation line
strong activity, where both methods imposing assumptions broadening.
on the kinematical diffraction theory and methods depart- The pioneering work in this field is due to Krivoglaz
ing from a microstructural model are developed. and Ryaboshapka [50] and Wilkens [51]. Krivoglaz and
Ryaboshapka considered sets of statistically random distri-
3.3.1 Methods imposing assumptions butions of non-interacting (edge or) screw dislocations.
on the kinematical diffraction theory Wilkens demonstrated that a random distribution of dislo-
cations (in a set) is unrealistic and introduced the concept
An overview of the methods based on specific assump- of the restrictedly random dislocation arrangement. To this
tions about the strain distributions in materials without re- end, the degree of correlation in the dislocation distribu-
ferring to a specific microstructural model is provided by tion of a set was described by the so-called cut-off radius
Table 1 [27, 35, 44–46]. Re , which can be considered as the radius of a cylinder
A quantitative evaluation of size and strain parameters within which the dislocation arrangement is random: No
derived from broadened line profiles requires thorough con- elastic interaction of the various dislocations sets in the
sideration of the underlying assumptions in the methods crystal is considered to occur.
used. A comparative application of the different methods to The strain Fourier coefficients Ad ðLÞ can be approxi-
an imperfect material is not straightforward because the as- mated by [52]:
sumptions are incompatible (e.g. Gaussian strain distribu-
tions in the Warren-Averbach method versus small strain gra- Ad ðLÞ ¼ exp ½ÀðcLÞp Š ; ð6Þ
dients in the alternative method) and the resulting parameters d
where c characterizes the width of A ðLÞ and the exponent
are not defined in the same way (e.g. volume-versus area-
p takes values between 1 (Lorentzian line profile) and 2
weighted crystallite size) [39]; see also Fig. 1 in Ref. [46].
(Gaussian line profile). The shape parameter M,
3.3.2 Methods departing from a microstructural model M ¼ Re ðrÞ1=2 ; ð7Þ
A flexible general method based on a microstructural can be calculated from p (cf. Eq. (2.19) of Vermeulen et al.
model without referring to a particular type of defect is [52]). c is related to the square-root of the dislocation density
the strain-field model proposed by Van Berkum et al. [47]. r. Note that dislocation line broadening is usually anisotro-
In this approach, the strain field is composed of a super- pic, i.e. it depends on the hkl reflection (i.e. it depends on the
position of the (component) strain fields of individual de- orientation and length of the diffraction vector; cf. Section
fects. The strain fields of the lattice defects are described 3.3.3). This can be rationalized by the so-called dislocation-
statistically by three functions: the probability function for contrast factor, which is contained in c in Eq. (6).

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

Method Assumptions Size Strain

Williamson-Hall conventional Lorentz shaped peak Volume-weighted Maximum strain, e related to local
plot [27, 44] 1949, 1953 profiles for size- and strain- column length mean squared strain he02i
broadened profile for Gaussian strain distributions
Warren-Averbach [35, 45] Gaussian strain distribution Area-weighted column length Mean squared strain, he2(L)i,
1950, 1952 or small strains related to Ad(L)
Alternative method [46] 1994 Small strain gradients Area-weighted column length No analytical relation between
and broad size distribution he2(L)i and the strain Fourier
coefficients


Dislocation densities and configurations have been in- tropic diffraction-line broadening may be categorized as
vestigated in thin films and plastically deformed materials follows:
(see, for example, Refs. [39, 54, 55]). In addition to the (i) Only small (negligible) microstrain gradients within
dislocation density and the cut-off radius, the fractions of crystallites. In this case, the increase of line broadening
screw and edge dislocations can be determined. For a re- with increasing length of the diffraction vector, for a given
cent review on dislocation line broadening, see Ref. [56]. set of diffracting lattice planes, is proportional to tan q (cf.
also Section 3.5) [47, 57]. Phenomenological models for
3.3.3 Anisotropic microstrain-like diffraction-line this type of anisotropic microstrain diffraction-line broaden-
broadening ing have been developed and implemented in Rietveld-re-
finement programs (see, for example, Ref. [58]). A model
The occurrence of anisotropic diffraction-line broadening
case for this type of line broadening is a (hypothetical) iso-
(i.e. the diffraction-line broadening depends non-monoto-
tropic microstress distribution which, in combination with
nously on the hkl reflection when plotted versus 2q) is a
single-crystal elastic anisotropy, results in an anisotropic
quite general phenomenon which has attracted consider-
microstrain distribution [57].
able attention both in phenomenological and microstruc-
This approach is likely to overestimate the anisotropy of
ture-based modelling of diffraction-line broadening. Aniso-
diffraction-line broadening, as an isotropic microstress distri-
bution gives rise to geometrically incompatible strains in dif-
ferently oriented crystallites. The real grain interaction in a
polycrystalline material is more likely to be between isotro-
pic stress and isotropic strain distributions. Another recently
considered source of anisotropic (microstrain-like) line
broadening are composition fluctuations in a non-cubic ma-

a

a

b

Fig. 2. (a) Full width at half-maximum (FWHM) of the reflections of
a e-FeN0.433 powder and LaB6 (used for the determination of the in-
strumental line broadening) measured using a Bragg-Brentano dif-
b
fractometer with Co Ka radiation. The apparent ‘scatter’ of the line
widths of the powder is due to compositional inhomogeneities. Fig. Fig. 3. (a) The FWHM (w*) and the integral breaths (B*) as a func-
f
2a and b have been taken from Ref. [59]. (b). The anisotropy of the tion of the reciprocal space coordinate d* in the classical Williamson-
microstrain-like broadening observed from a e-FeN0.433 powder. The Hall plot in the case of Nb ball milled for one day. The indices of the
direction dependence of the FWHM, BfD2 q;hkl , as a function of the reflections have also been indicated in the figure. Note the pronounc-
angle of the diffraction vector relative to the c axis for the hexagonal edly anisotropic nature of line broadening. (b) The modified William-
crystals system. The separate points indicate the experimental data; son-Hall plot of the same data as in Fig. 3a. As a function of d *C 1/2
the solid line represents the curve obtained by fitting a model for line (where C is the dislocation contrast factor). The indices of the reflec-
broadening due to compositional fluctuations to the experimental tions have also been indicated in the figure. Fig. 3a and b have been
data. A compositional fluctuation of e-FeN0.433Æ0.008 is obtained. taken from Ref. [62].


terial (see Fig. 2 for an example) [59]. For a recent general a so-called columnar microstructure occurs, where the film
treatment on anisotropic microstrain broadening due to a consists of e.g. columnar-shaped grains separated by grain
field-tensor (rank 0, pertaining to composition variation; rank boundaries oriented more or less perpendicularly to the
2, pertaining to stress/strain distributions), cf. Ref. [60]. layer surface. For such a thin film, the crystallite size is an
(ii) No assumption about microstrain gradients: Adop- anisotropic quantity: the crystallite size along the surface
tion of a microstructural model. In this case, the depen- normal is much larger than the crystallite size in the plane
dence of strain broadening on the length of the diffraction of the film. Thus, macroscopically anisotropic size broad-
vector follows from the microstructural model. The re- ening occurs (see Fig. 4a for an example) [18]. Anisotro-
cently most frequently studied case is dislocation line pic size broadening can be accompanied by anisotropic
broadening, for which anisotropic line broadening is due strain broadening (see Fig. 4b) [18], which can also occur
to the orientation of the diffraction vector with respect to due to unequal densities of defects (as dislocations on dis-
slip systems and the anisotropy of elastic constants (cf. tinct slip systems) along different directions in the speci-
Section 3.3.2 and Fig. 3; e.g. Refs. [56, 61–63]). men [52].
The analysis of macroscopically anisotropic diffraction-
3.4 Macroscopic anisotropy line broadening is considerably simplified experimentally
by the use of parallel-beam diffractometers, because instru-
Massive and polycrystalline specimens generally exhibit mental aberrations occurring for focusing diffractometers
an anisotropic microstructure. Consider, as an example, (i.e. ‘defocusing’) upon changing the orientation of the
thin films deposited by physical vapour deposition: Often diffraction vector (from e.g. the specimen surface normal
direction, 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 takes
values between 0 and 2p with equal probability. In this
case, the total diffracted intensity can be taken as the sum
of intensities scattered by the individual crystallites sepa-
rately. This naturally leads to the usually adopted concept
of size broadening due to the finite size of individual crys-
tallites and strain broadening related to the relative displa-

a cement of atoms within one grain. A more general ap-
proach is to consider the whole irradiated volume of a
polycrystal 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 for
strain fields associated with lattice defects (cf. also Section
3.3.2) [47]. As the phase difference of scattered waves
originating from different scatterers (atoms) is the scalar
product of the diffraction vector and the position (differ-
ence) vector of the scatterers, both the character of the
strain fields in a specimen and the length of the diffraction
vector are decisive for diffraction-line broadening. It has
been demonstrated that for general strain broadening, the
order-dependence of the diffraction-line width is complex,

b i.e. neither order-independent broadening (traditionally
termed ‘size broadening’) nor broadening proportional to
Fig. 4. (a) The crystallite, grain sizes of a 250 nm thick Ti3Al layer the length of the diffraction vector (traditionally termed
as viewed along different hhkli* directions, i.e. as function of the ‘strain broadening’) occurs (see Fig. 5a). Two limiting
tilting angle w. The schematic figure represents the rectangular Ti3Al
grains (with a height of 50 nm and a width of 6 nm) in the Ti3Al
cases have been identified:
layer, the crystallite size of the rectangular grain DhhkliÃ measured (i) For infinitely broad component strain fields of lat-
along the hhkli* direction is 50 nm/cos w for 0 w 6.89 ; 6 nm/ tice defects (w=hsi ! 1; cf. Section 3.3.2), the broaden-
sin w for 6.89 w 90 , as shown by the solid and dashed lines. ing is proportional to the length of the diffraction vector
(b) The microstrains and the grain boundary fraction FGB of the d * (cf. Eq. (3)):
Ti3Al 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 a b ¼ ð2pÞ1=2 d *he2 i1=2 ; ð8Þ
geometrical constant and equals 1. The results suggest that FGB and
the microstrain behave similarly as function of w. Fig. 4a and b have where he2 i is the mean squared strain. This is the well-
been taken from Ref. [18]. known strain broadening for a specimen with a constant


and constant at large lengths; see Fig. 5b) has recently
been experimentally confirmed on the basis of measured
line broadening of nanocrystalline thin films by Rafaja
et al. (see Fig. 5b) [64]: i.e. a loss of coherency with in-
creasing length of the diffraction vector at small lengths of
d * leading to incoherent diffraction of the individual crys-
tallites at larger length d*. For another study of the effect
´
of 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].

a 4. Concluding remarks and perspectives

(1) Whereas unprejudiced individual peak-profile analysis
(still) allows the most severe microstructural model
testing, a simultaneous analysis of all reflections, sub-
ject to more severe (e.g. profile-shape) constraints, is
required if overlap of diffraction lines occurs. Distinc-
tion of both evaluation approaches can be expected to
diminish 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 diffraction

b
vector in either the reciprocal lattice or the specimen
frame of reference, is a prerequisite.
Fig. 5. (a) Application of the strain-field model: Relative integral (3) Preference for microstructure model-based line-profile
breadths of line profiles in reciprocal space for different relative synthesis methods over line-profile decomposition
widths w, of the component strain fields. The subscript ‘r’ indicates methods is due to the application of flawed micro-
that the respective quantity has been normalized by the mean pro-
jected (on the diffraction vector) defect distance. Taken from Ref. structure/diffraction models in line-profile decomposi-
[47]. (b) Diffraction line broadening measured for a nanocrystalline tion methods.
Ti0.38Al0.62N thin film (points) and the numerical simulation (solid (4) Is a general, practically applicable microstructure/dif-
line) taking into account the partial coherence of neighbouring nano- fraction model possible?
sized crystallites. Taken from Ref. [64]. Note the similarity with re-
sults shown in Fig. 5a for, e.g., wr ¼ 0:03.
References
lattice spacing d within each crystallite and a Gaussian [1] Mittemeijer, E. J.; Scardi, P. (Eds.): Diffraction Analysis of the
spacing distribution over the crystallites, where: Microstructure of Materials. Springer, Germany 2004.
[2] Dehlinger, U.; Kochendorfer, A.: Linienverbreiterung von ver-
¨
he2 i ¼ ðhd2 i À hdi2 Þ=hdi2 : ð9Þ formten Metallen. Z. Kristallogr. 101 (1939) 134–148.
[3] Estevez-Rams, E.; Penton Madrigal, A.; Scardi, P.; Leoni, M.:
This proportionality of line broadening (e.g. integral Powder diffraction characterization of stacking disorder. Z. Kris-
tallogr. Suppl. 26 (2007) 99–104.
breadth) and length of the diffraction vector (see Fig. 5a) [4] Leoni, M.: Diffraction analysis of layer disorder. Z. Kristallogr.
has often been used for strain broadening in general (cf. 223 (2008) 561–568.
Eqs. (2) and (3); Section 3.3.1). [5] Pratapa, S.; O’Connor, B.; Hunter, B.: A comparative study of
(ii) For infinitely narrow component strain fields single-line and Rietveld strain–size evaluation procedures using
MgO ceramics. J. Appl. Cryst. 35 (2002) 155–162.
(w=hsi ! 0, as corresponds to small-angle grain bound-
[6] Masson, O.; Dooryhee, E.; Fitch, A. N.: Instrument line-profile
´
aries in polycrystalline material; cf. Section 3.3.2), it fol- synthesis in high-resolution synchrotron powder diffraction. J.
lows that Appl. Cryst. 36 (2003) 286–294.
[7] Balzar, D.; Audebrand, N.; Daymond, M. R.; Fitch, A; Hewat,
b / ðd *Þ2 ð10Þ A.; Langford, J. I.; Le Bail, A.; Louer, D.; Masson, O.; McCo-
¨
wan, C. N.; Popa, N. C.; Stephens, P. W.; Toby, B. H.: Size-
for small lengths of the diffraction vector. For larger lengths, strain line-broadening analysis of the ceria Round-Robin sam-
a constant line width b is obtained (‘classical’ size broaden- ple. J. Appl. Cryst. 37 (2004) 911–924.
[8] Cheary, R. W.; Coelho, A. A.: A fundamental parameters approach
ing). A model microstructure producing this type of broaden- to X-ray line profile fitting. J. Appl. Cryst. 25 (1992) 109–121.
ing consists of domains of undistorted material shifted with [9] Kern, A.; Coelho, A. A.; Cheary, R. W.: Convolution based pro-
respect to each other. For increasing length of the diffraction file fitting. In: Diffraction Analysis of the Microstructure of Ma-
vector, the phase differences (modulo 2p) are almost uni- terials (Eds. E. J. Mittemeijer, P. Scardi) pp. 17–50. Springer,
Germany 2004.
formly distributed, thus incoherency of diffraction occurs. [10] www.bgmn.de
This prediction of this type of diffraction-line broaden- [11] Bergmann, J.; Friedel, P.; Kleeberg, R.: BGMN – A new funda-
ing (increasing at small lengths of the diffraction vector mental parameters based Rietfeld program for laboratory X-ray


sources, it’s use in quantitative analysis and structure investiga- [35] Warren, B. E.; Averbach, B. L.: The effect of cold-work distor-
tions. CPD Newsletter 20 (1998) 5–8. tion on X-ray patterns. J. Appl. Phys. 21 (1950) 595–599.
[12] Leoni, M.; Welzel U.; Scardi, P.: Polycapillary optics for materi- [36] Garin, J. L. ; Mannheim, R.; Soto, M. A.: Particle size and mi-
als science studies: Instrumental effects and their correction. J. crostrain measurement in ADI alloys. Powder Diffraction 17
Res. Natl. Inst. Stand. Technol. 109 (2004) 27–48. (2002) 119–124.
[13] Zuev, A. D.: Calculation of the instrumental function in X-ray [37] Kaszkur, Z.; Mierzwa, B.; Pielaszek, J.: Ab initio test of the
powder diffraction. J. Appl. Cryst. 39 (2006) 304–314. Warren-Averbach analysis on model palladium nanocrystals. J.
[14] Prince, E.; Toby, B. H.: A comparison of methods for modeling Appl. Cryst. 38 (2005) 266–273.
the effect of axial divergence in powder diffraction. J. Appl. [38] Scardi, P.; Leoni, M.: Diffraction line profiles from polydisperse
Cryst. 38 (2005) 804–807. crystalline systems. Acta Cryst. A57 (2001) 604–613.
[15] Gozzo, F.; De Caro, L.; Giannini, C.; Guagliardi, A.; Schmitt, [39] Lucks, I.; Lamparter, P.; Mittemeijer, E. J.: An evaluation of
B.; Prodi, A.: The instrumental resolution function of synchro- methods of diffraction-line broadening analysis applied to ball-
tron radiation powder diffractometers in the presence of focusing milled molybdenum. J. Appl. Cryst. 37 (2004) 300–311.
optics. J. Appl. Cryst. 39 (2006) 347–357. [40] Ungar, T.; Borbely, A.; Goren-Muginstein, G. R.; Berger, S.; Rosen,
´ ´
[16] Xiao Q.-F.; Kennedy R. J.; Ryan T. W.; York B. R.: Multifiber A. R.: Particle-size, size distribution and dislocations in nanocrys-
polycapillary collimator for X-ray powder diffraction. Mat. Sci. talline tungsten-carbide. Nanostruct. Mater. 11 (1999) 103–113.
Forum 278–281 (1998) 236–241. [41] Leoni, M.; Scardi, P.: Nanocrystalline domain size distributions
[17] Scardi, P.; Setti, S.; Leoni, M.: Multicapillary optics for materi- from powder diffraction data. J. Appl. Cryst. 37 (2004) 629–
als science studies. Mat. Sci. Forum 321–324 (2000) 162–167. 634.
[18] Welzel, U.; Mittemeijer, E. J.: The analysis of homogeneously [42] Kojdecki, M. A.; Bastida, J.; Pardo, P.; Amoros, P.: Crystalline
´
and inhomogeneously anisotropic microstructures by X-ray dif- microstructure of sepiolite influenced by grinding. J. Appl.
fraction. Powder Diffraction 20 (2005) 376–392. Cryst. 38 (2005) 888–899.
[19] Wohlschlogel, M.; Schulli, T. U.; Lantz, B.; Welzel, U.: Appli-
¨ ¨ [43] Kojdecki, M. A.; Ruiz de Sola, E.; Serrano, F. J.; Delgado-Pi-
cation of a single-reflection collimating multilayer optic for X- nar, E.; Reventos, M. M.; Esteve, V. J.; Amigo, J. M.; Alarcon,
´ ´ ´
ray diffraction experiments employing parallel-beam geometry. J.: Microstructural evolution of mullites produced from single-
J. Appl. Cryst. 41 (2008) 124–138. phase gels. J. Appl. Cryst. 40 (2007) 260–276.
[20] Delhez, R.; de Kejser, Th. H.; Mittemeijer, E. J.: Determination of [44] Hall, W. H.: X-ray line broadening in metals. Proc. Phys. Soc.
crystallite size and lattice distortions through X-ray diffraction London 62 (1949) 741–743.
line profile analysis. Fresenius Z. Anal. Chem. 312 (1982) 1–16. [45] Warren, B. E.; Averbach, B. L.:The separation of cold-work dis-
[21] Scardi, P. ; Leoni, M. ; Delhez, R.: Line broadening analysis tortion and particle size broadening in X-ray patterns. J. Appl.
using integral breadth methods: A critical review. J. Appl. Cryst. Phys. 23 (1952) 497.
37 (2004) 381–390. [46] van Berkum, J. G. M.; Vermeulen, A. C.; Delhez, R.; de Keij-
[22] Stokes, A. R.: A numerical Fourier-analysis method for the cor- ser, Th.; Mittemeijer, E. J.: Applicabilities of the Warren-Aver-
rection of widths and shapes of lines on X-ray powder photo- bach analysis and an alternative analysis for separation of size
graphs. Proc. Phys. Soc. 61 (1948) 382–391. and strain broadening. J. Appl. Cryst. 27 (1994) 345–357.
[23] Ida, T. ; Toraya, H.: Deconvolution of the instrumental functions [47] van Berkum, J. G. M.; Delhez, R.; de Keijser, Th.; Mittemeijer,
in powder X-ray diffractometry. J. Appl. Cryst. 35 (2002) 58–68. E. J.: Diffraction-line broadening due to strain fields in materi-
[24] Ida, T.; Hibino, H.; H. Toraya, H.: Deconvolution of instrumen- als; Fundamental aspects and methods of analysis. Acta Cryst.
tal aberrations for synchrotron powder X-ray diffractometry. J. A52 (1996) 730–747.
Appl. Cryst. 36 (2003) 181–187. [48] Lucks, I.; Lamparter, P.; Mittemeijer, E. J.: Diffraction-line pro-
[25] de Keijser, Th. H.; Langford, J. I.; Mittemeijer, E. J.; Vogels, A. file analysis – A simple way to characterize ball-milled Mo?
B. P.: Use of the Voigt function in a single-line method for the Mat. Sci. Forum 378–3 (2001) 451–456.
analysis of X-ray diffraction line broadening. J. Appl. Cryst. 15 [49] van Berkum, J. G. M.; Delhez, R.; de Keijser, Th.; Mittemeijer,
(1982) 308–314. E. J.: Characterization of deformation fields around misfitting
[26] Langford, J. I.; Delhez, R.; de Keijser, Th. H.; Mittemeijer, E. Inclusions in solids by means of diffraction line broadening.
J.: Profile analysis for microcrystalline properties by the Fourier Phys. Stat. Sol. (A) 134 (1992) 335–350.
and other methods. Aust. J. Phys. 41 (1988) 173–187. [50] Krivoglaz, M. A.; Ryaboshapka, K. P.: The effect of dislocations
[27] Williamson, G. K.; Hall, W. H.: X-ray line broadening from on line profiles. Phys. Met. Metallogr. 15 (1963) 18–27.
filed aluminium and wolfram. Acta Metall. 1 (1953) 22–31. [51] Wilkens, M.: The Determination of density and distribution of
[28] Kapoor, K.; Lahiri, D.; Rao, S. V. R.; Sanyal, T.; Kashyap, B. dislocations in deformed single crystals from broadened X-ray
P.: X-ray diffraction line profile analysis for defect study in Zr- diffraction profiles. Phys. Stat. Sol. (A) 2 (1970) 359–370.
2.5% Nb material. Bull. Mater. Sci. 27 (2004) 59–67. [52] Vermeulen, A. C.; Delhez, R.; de Keijser, Th. H.; Mittemeijer,
[29] Zhang, Y. W.; Yang, Y.; Jin, S.; Liao, C. S.; Yan, C. H.: Doping E. J.: Changes in the densities of dislocations on distinct slip
effect on the grain size and microstrain in the sol-gel-derived systems during stress relaxation in thin aluminium layers: The
rare earth stabilized zirconia nanocrystalline thin films. J. Mater. interpretation of X-ray diffraction line broadening and line shift.
Sci. Letters 21 (2002) 943–946. J. Appl. Phys. 77 (1995) 5026–5049.
[30] Mukherjee, P.; Sarkar, A.; Barat, P.; Banyopadhyay, S. K.; Sen, [53] Ungar, T.; Borbely, A.: The effect of dislocation contrast on x-
´ ´
P.; Chattopadhyay, S. K.; Chatterjee, P.; Chatterjee, S. K.; Mitra, ray line broadening: A new approach to line profile analysis.
M. K.: Deformation characteristics of rolled zirconium alloys: A Appl. Phys. Lett. 69 (1996) 3173–3175.
study by X-ray diffraction line profile analysis. Acta Mat. 52 [54] Scardi, P.; Leoni, M.: Whole powder pattern modelling. Acta
(2004) 5687–5696. Cryst. A58 (2002) 190–200.
[31] Louer, D. ; Bataille, T. ; Roisnel, T. ; Rodriguez-Carvajal, J. : A
¨ [55] Dragomir-Cernatescu, I.; Gheorghe, M.; Thadhani, N.; Snyder,
study of nanocrystalline yttrium oxide from diffraction-line-pro- R. L.: Dislocation densities and character evolution in copper
file analysis: Comparison of methods and crystallite growth. deformed by rolling under liquid nitrogen from X-ray peak pro-
Powder Diffraction 17 (2002) 262–269. file analysis. Powder Diffraction 20 (2005) 109–111.
[32] Audebrand, N. ; Bourgel, C. ; Louer, D.: Ex-oxalate magnesium
¨ [56] Kuzel, R.: Kinematical diffraction by distorted crystals – dislo-
oxide, a strain-free nanopowder studied with diffraction line pro- cation X-ray line broadening. Z. Kristallogr. 222 (2007) 136–
file analysis. Powder Diffraction 21 (2006) 190-199. 149.
[33] Krill, C. E. ; Birringer, R.: Estimating grain-size distributions in [57] Stokes, A. R.; Wilson, A. J. C.: The diffraction of X-rays by
nanocrystalline materials from X-ray diffraction profile analysis. distorted crystal aggregates – I. Proc. Phys. Soc. London 56
Phil. Mag. A 77 (1998) 621–640. (1944) 174–181.
[34] Bertaut, M.: Etude aux rayons-X de la repartition des dimen- [58] Stephens, P. W.: Phenomenological model of anisotropic peak
sions des crystallites dans une poudre crystalline. Comptes Re- broadening in powder diffraction. J. Appl. Cryst. 32 (1999)
ndus 228 (1949) 492-494. 281–289.


[59] Leineweber, A.; Mittemeijer, E. J.: Diffraction line broadening [62] Dragomir, I. C.; Ungar, T.: The dislocations contrast factors of
´
due to lattice-parameter variations caused by a spatially varying cubic crystals in the Zener constant range between zero and
scalar variable: Its orientation dependence caused by locally unity. Powder Diffraction 17 (2002) 104–111.
varying nitrogen content in epsilon-FeN0.433. J. Appl. Cryst. 37 [63] Leoni, M.; Martinez-Garcia, J.; Scardi, P.: Dislocation effects in
(2004) 123–135. powder diffraction. J. Appl. Cryst. 40 (2007) 719–724.
[60] Leineweber, A.: Anisotropic microstrain broadening due to field- [64] Rafaja, D.; Klemm, V.; Schreiber, G.; Knapp, M.; Kuzel, R.:
tensor distributions. J. Appl. Cryst. 40 (2007) 362–370. Interference phenomena observed by X-ray diffraction in nano-
[61] Ungar, T.; Gubicza, J.; Ribarik, G.; Borbely, A.: Crystallite size
´ ´ ´ crystalline thin films. J. Appl. Cryst. 37 (2004) 613–620.
distribution and dislocation structure determined by diffraction [65] Ribarik, G.; Audebrand, N.; Palancher, H.; Ungar, T.; Louer, D.:
´ ´ ¨
profile analysis: Principles and practical application to cubic and Dislocation densities and crystallite size distribution in nanocrys-
hexagonal crystals. J. Appl. Cryst. 34 (2001) 298–310. talline ball-milled fluorides MF2 (M ¼ Ca, Sr, Ba and Cd), deter-
mined by X-ray diffraction line-profile analysis. J. Appl. Cryst.
38 (2005) 912–926.

Zeitschrift für Kristallographie
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International journal for structural, physical,
Z. Kristallogr. Suppl. 27 (2008)
and chemical aspects of crystalline materials
Proceedings of the ”5th Size Strain“ Conference
ZEITSCHRIFT FÜR
KRISTALLOGRAPHIE
(Diffraction Analysis of the Microstructure of Materials)
held in October 2007 in Garmisch-Partenkirchen, Germany
16. Jahrestagung
Editors: Eric J. Mittemeijer, Paolo Scardi, Andreas
der Deutschen Gesellschaft für Kristallographie
f

Erlangen, 3.– 6. März 2008
Leineweber and Udo Welzel
Referate
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Z. Kristallogr. Suppl. 28 (2008)
Supplement Issue No. 28 2008
Abstract of the 16th Annual Meeting
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The "state of the art" of diffraction analysis of crystallite size and lattice strain.

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The "state of the art" of diffraction analysis of crystallite size and lattice strain.