Full text of DOI:10.1007/s100510070073

Eur. Phys. J. B 18, 23–30 (2000)
THE EUROPEAN
PHYSICAL JOURNAL B
EDP Sciences
Societ a` Italiana di Fisica
Springer-Verlag 2000
ꢀ
c
Whole diffraction pattern-fitting of polycrystalline fcc materials
based on microstructure
a
P. Scardi , M. Leoni, and Y.H. Dong
Dipartimento di Ingegneria dei Materiali, Universit `a di Trento, 38050 Mesiano (TN), Italy
Received 16 May 2000
Abstract. A method is proposed for modelling the complete diffraction pattern of fcc polycrystalline ma-
terials. The algorithm permits a simultaneous refinement of several parameters related to microstructure
and lattice defects responsible for line broadening effects. Linear (dislocations) and planar (stacking faults)
defects are considered in detail, together with the effect of size and shape of coherent scattering domains
(
crystallites). Experimentally observed profiles are modelled by Voigt functions, whose parameters are
connected with those describing the dislocation field (dislocation density, outer cut-off radius, average con-
trast factor), twin and deformation fault probabilities, and domain size, also considering the effect of a
symmetrical instrumental profile. Domain shape is assumed spherical, with a lognormal distribution of di-
ameters; however, the approach can be generalised to different shapes and size distributions. The proposed
algorithm can be extended to other crystalline structures, and can be used within the Rietveld method or
as a Whole Powder Pattern Fitting (WPPF), as in the present work.
PACS. 61.72.Dd Experimental determination of defects by diffraction and scattering – 61.72.Lk Linear
defects: dislocations, disclinations – 61.72.Nn Stacking faults and other planar or extended defects
1
Introduction
particularly effective to study lattice defects and cell pa-
1
rameters in single-phase and multiphase materials [5] .
Most methods proposed so far are based on the mod-
elling of experimental data with analytical functions, like
Voigt (convolution of a Gaussian (G) and a Lorentzian (or
Cauchy) (C) curve [11]), pseudo-Voigt (pV ) (weighted av-
erage of G and C, with mixing factor η: (1−η)G+ηC [12])
or Pearson VII (a modified C curve [13]), just to cite the
most common ones [14]. Using an analytical function to
model a diffraction profile is certainly an arbitrary choice;
however, this approach has become increasingly popular
for practical reasons, as it permits a fast and efficient
extraction of information from the experimental pattern,
and is a simple way to study broad and overlapped diffrac-
tion profiles. Experience has also shown that, frequently,
analytical functions reproduce quite well real diffraction
In the recent years, analytical methods for the processing
of diffraction data from polycrystalline materials devel-
oped in the direction of whole powder pattern fitting, i.e.,
simultaneous analysis of an extended portion of the ex-
perimental pattern containing several diffraction profiles.
This new paradigm in the powder diffraction research led
to considerable benefits in two main areas: (a) structure
solution and refinement from powder data and (b) study
of the microstructure and lattice defects in polycrystalline
materials. While (a) is of direct interest in crystallography
and structural studies [1], (b) is a promising development
in materials science [2,3], and is the direct object of the
present study.
WPPF methods are used to determine (i) phase per- data (e.g., see [15]). Therefore this approach can be con-
centages in powder mixtures [4], (ii) lattice parame- sidered as partially legitimate; however, limits and pos-
ters [5], (iii) nature and concentration of line and plane sible artefacts due to the use of analytical functions to
defects [2,3], (iv) crystalline domain size and shape [3,6] fit experimental data (which limit possible shapes of the
and (v) texture [7]. Many existing WPPF computer pro- modelled profile) should always be borne in mind.
grams are based on the Rietveld algorithm [4,8], even if
structural parameters (mainly, atomic positions and ther-
mal factors) may be known and therefore considered as
constants. It is also possible to use a WPPF without
modelling the diffracted intensity by means of a struc-
tural model, but as a fitting parameter. This approach is
Several approaches have been proposed to extract in-
formation (i–v) after WPPF, using the refined profile data
(
like position, width and shape) [5]; we can consider these
as a posteriori techniques, whose reliability is strongly
1
Some authors attribute to Pawley [9,10] the introduction
of WPPF, even if considerable changes have been made to the
original approach; for instance, see [1,5].
a
e-mail: paolo.scardi@ing.unitn.it

24
The European Physical Journal B
limited by the quality of the WPPF itself. The statisti- will be introduced later. A general approach for asymmet-
cal weight of the intensity, which is an important piece of ric profiles involves an additional sine term in (1) [21]. In
information, is not directly related to the estimated errors the following, however, we will only consider cosine terms,
of structural and defect parameters. In spite of that, this so we will refer to them as Fourier coefficients, omitting
approach is probably the most used at present, because is the adjective “cosine”.
quite straightforward and can be used in principle for any
The Fourier coefficients, A(L), carry the information
crystalline phase without a knowledge of structural data. on peak shape and width and can be written as:
A different procedure has been recently proposed for
IP
S,F
D
A(L) = T (L)A (L)A (L).
(2)
fcc materials by some of the authors [3]. The line pro-
file parameters connected with domain size, faulting and
lattice distortions (due to dislocations) where directly in-
cluded in a least square algorithm for WPPF based on
the use of pV profile functions. Applications to different
inorganic materials was successful, and provided an ef-
fective means to account for line broadening anisotropy,
i.e., changes in peak shape and width for different (hkl),
additional to the effect of the reflection order on lattice
distortions. The main limit was in the oversimplified size
broadening model, which was rather arbitrary and inac-
curate to fit high angle reflections. In the present work
we propose an extension of the cited approach, based on
much more general assumptions regarding domain size ef-
fects, and a fast and effective least squares algorithm that
can easily be incorporated in WPPF or Rietveld programs.
The three terms in (2) represent the effect of instru-
IP
mental profile (T (L)), crystallite size and stacking faults
S,F
D
(A
(L)), and lattice distortions (A (L)). Expressions
for instrumental, size/faulting and distortion terms can
be written according to suitable models; in this work they
are based on the following hypotheses:
• Studied materials have fcc structure and no appre-
ciable texture is present; extension to bcc and hcp is
straightforward. The proposed approach, in principle,
can be further extended to any symmetry, provided
that suitable relations for size/faulting and distortion
terms in (2) are available.
• Crystalline domains are spherical, with a lognormal
distribution of diameters. This assumption is quite
general, and seems to be appropriate in several cases,
including materials as different as finely dispersed ce-
ramics and highly deformed metals (see [6] and Refs.
therein). The corresponding size-broadening effect is
isotropic (i.e., independent of (hkl)), and can be mod-
elled in terms of lognormal mean (γ) and variance (ω).
The model can be extended to consider different crys-
tallite shapes and size distributions.
The contribution of faulting is written according to
Warren’s theory [17]. Modelling is limited to low fault-
ing probabilities, and an average profile is calculated
for each family of planes {hkl}. This is an approxi-
mation, since faulting should be treated by separating
the diffraction signal from a {hkl} family into profile
sub-components with different hkl. However, the com-
plexity of this operation is beyond the limits of the
proposed model.
2
Methodology
2
.1 Introductory remarks
Diffraction profiles from polycrystalline materials can be
written as Fourier integrals, whose coefficients are con-
nected with the various possible sources of line broaden-
ing, which typically include an instrumental component
and sample-dependent physical effects [16]. For a given
set of planes with Miller indices {hkl}, the diffracted in-
tensity is conveniently represented in the reciprocal lat-
tice, introducing the modulus of the diffraction vector,
•
∗
d = 2 sin θ/λ, where θ is the diffraction angle and λ is
the wavelength. If profiles are symmetrical, the diffracted
intensity can be expressed by a cosine Fourier integral
(
analogous expressions should be written for all the spec-
tral components of the radiation employed),
2
.2 Instrumental broadening
Z
∞
∗
∗
∗
∗
The instrumental profile (IP) is assumed to be symmetri-
cal, condition easily obtained by using suitable diffraction
optics [22]; in general, since we are mostly concerned with
broadened profiles, where the instrument contribution is
not critical, the assumption of a symmetrical IP is appro-
priate. Under these conditions, the IP can be analytically
modelled for a wide portion of the diffraction pattern by
means of Voigt or pseudo-Voigt curves [22,23]. If we con-
sider a pV , the Fourier transform can be written as:
I(d ) = k(d )
A(L) cos [2πL(d − d − δ)] dL (1)
B
0
∗
∗
where dB is the value of d in Bragg condition, in ab-
sence of lattice defects, and δ is the shift due to fault-
∗
ing. k(d ) accounts for all effects that are constant for a
∗
given peak, or are known functions of d (e.g., diffraction
2
geometry, Lorentz-polarisation factor, multiplicity, |F| ,
etc. [17]). The variable (L) in the integral is the Fourier
length, which is a length in the real space along the di-
∗
IP
pV
2
pV,G
2
rection of the diffraction vector (L = n/d , where n is an
T
(L) = (1 − k) exp(−πβ
L ) + k exp(−2βpV,CL)
integer).
(3)
Instrumental features as well as some types of lattice
defects can produce asymmetric profiles [17–20]; in this where k = ηβC/β, and β, βC and βG are integral
work we deal with the asymmetry due to twin faults, which breadths (peak area/maximum intensity) of the pV and

P. Scardi et al.: Whole diffraction pattern-fitting of fcc polycrystalline materials
25
its Lorentzian and Gaussian components, respectively. By and
modelling the experimental pattern of a suitable profile
standard [23], one can obtain a parametric description of
1
/2
Z = (1 − 3α − 2β)
.
(11)
IP
the IP and its transform, TpV (L) [3].
P
The
(where m is the multiplicity) is intended as a sum
m
over all the (hkl) components belonging to a {hkl} plane
family. As can be seen from (9) and (10), the number of
independent components is less than m; in fact, only the
value of L0 and the selection rules of (10) are important
2
.3 Effect of faulting and domain size
Planar defects, which are assumed to lay in the (111)
planes, can be described in terms of deformation and
twin faulting probabilities (α and β, respectively). The
(
maximum number of independent components is four).
Faulting is also responsible for a hkl-dependent shift in
S
F
2
size (A ) and faulting (A ) coefficients can then be sepa-
the diffraction peak centroid proportional to α :
rated as:
√
X
1
3α L0
2
4π h
0
S,F
S
F
hkl
∗
d
B
A
(L) = A (L)A (L; α, β).
(4)
δhkl
=
σL0
.
(12)
m
m
Size coefficients can be calculated from the column length
distribution, p(L). If we assume to divide each spherical In conclusion, the contribution of faulting to line broaden-
grain in columns of cells along the direction of the diffrac- ing and peak position can be written in terms of faulting
tion vector, p(L) is the distribution of distances between probabilities (α and β), and is different for the various
cells inside columns [17]. Such a distribution can be calcu- (hkl) reflections.
lated for spheres whose diameters are distributed accord-
ing to a lognormal (p(D)):
ꢀ
ꢁ
2
.4 Lattice distortions due to dislocations
2
1
(ln D − γ)
p(D) =
√
exp −
·
(5)
2
Dω 2π
2ω
D
The calculation of the last term in (2) (A (L)) requires
some assumption on the nature of the defects producing
the lattice distortion. Dislocations are frequently respon-
sible for lattice distortion (usually referred to as micros-
train) in a wide variety of materials. The corresponding
line broadening effect, which is anisotropic, can be de-
scribed in terms of dislocation density (ρ) and outer cut-
off radius (Re) (or similar characteristic distances [27]);
To obtain p(L), with need to consider the distribution for
a single sphere, ps(L; D) = 2L/D [24], weighted over the
2
lognormal distribution (5):
Z
∞
p(L) ∝
ps(L; D)g(D)dD.
(6)
|
L|
Solving the integral (6), and after suitable normalisation, following Wilkens [28,29] and van Berkum [30], the fol-
the column length distribution is
lowing expression can be written for the distortion Fourier
coefficients:
ꢂ
ꢃ
ln L − γ
2
ꢀ
ꢁ
p(L) = L exp(−2γ − 2ω )erfc
√
·
(7)
1
2
2
2
ω
D
2
∗
2
∗
A (L) = exp − π|b| Chklρd L f (L/Re)
(13)
hkl
The size Fourier coefficients, according to Bertaut [17,25],
can be calculated as:
where b is the Burgers vector (the slip system in fcc struc-
√
Z
∞
1
hLi
tures is {111}h110i, for which |b| = a0/ 2 , where a0 is the
s
0
0
0
A (L) =
(L − L)p(L )dL
(8)
∗
lattice parameter) and f (L/Re) is a known function [30].
L
Following the theory of Wilkens, a higher order term can
where hLi is the mean column length [17]. The integra- be added to the exponent in (13), whose leading term is
s
2
4
tion of (8) gives an analytical expression for A (L; γ, ω),
4
2
∗
4
of the type: g(L )ρ C
d
; the expression of g(L ) can
be worked out from references [28–30]. The weight of the
higher order term is considerably less than that propor-
tional to L f in (13), and its importance is appreciated
only for high diffraction angles (high d ) and around the
hkl hkl
which according to our hypotheses is independent of (hkl);
therefore, size Fourier coefficients for all diffraction profiles
depend on γ and ω only.
2
∗
According to Warren [17], the faulting coefficients can
be written as:
∗
peak top, where large L are important.
L0
Chkl is the average contrast factor, which accounts for
(9) the anisotropy of the dislocation strain field that is respon-
sible for the anisotropic line broadening effect (actually,
X
∗
1
m
LdB 2 σL0
F
hkl
h
A
(L) =
Z
0
m
2
0
2
2
2
2
where L0/h = (h + k + l)/(h + k + l ),
Expressions given by Warren were obtained for low fault-

−
ing probabilities; for higher probabilities suitable relations re-
cently derived by Velterop et al. [26] should be used. On the
other hand, the present approach is not suitable for high fault-
ing probabilities, since, as discussed previously (see Sect. 2.1)
profile sub-components should be calculated separately.

+1 for L = 3N + 1

0
σL =
0
for L = 3N
N = 0, ꢀ1, ꢀ2, ... (10)
0
0


1
for L0 = 3N − 1

26
The European Physical Journal B
according to the different dislocation models, part of the βV,C for each diffraction profile, in terms of the fitting pa-
dependence on (hkl) is carried by Re [28]; in the present rameters connected to size/faulting and distortion effects:
work we consider Re as a fitting parameter independent α, β, γ, ω, ρ, R , (Ch00) and q.
e
of (hkl) [29]). As originally proposed several years ago by
Stokes and Wilson [31], line broadening anisotropy can be
introduced by assuming a linear dependence of the mi-
crostrain on the orientational parameter H (also referred
to as Γ [32]); this corresponds to write the contrast factor
for the cubic system as [3,33]:
As a weight function in (16) we can use wL = A(L);
in this way we can give more weight to the information
carried by the tails of profiles (low L values) with respect
to the peak top, which is consistent with the statistical
weight in the experimental pattern. Formally, the upper
limit in the summation of (16) is infinity; from a prac-
tical point of view, the calculation can be stopped when
ꢀ
ꢁ
2
2
2 2
2 2
h k + h l + k l
−3
A(L) falls below a given threshold (e.g. 10 ). Most im-
Chkl = Ch00(1 + qH) = Ch00 1 + q
2
2
2 2
(
h + k + l )
portantly, we note that (16) can be solved analytically,
which means in a very fast and efficient way in terms of
computation time. This also provides an efficient way to
test the validity of a given model for line broadening, ex-
pressed by (2).
(14)
where Ch00 and q can be calculated from the elastic con-
stants of the material (cij or sij ) for different types of
dislocations [34]. As shown recently by Ung ´a r et al. [33],
Ch00 and q for screw and edge dislocations can be con-
veniently expressed in a parametric form as a function of
the elastic constants. If the elastic constants are known, 2.6 Profile asymmetry due to twin faults
D
the average contrast factor can be calculated, and A (L)
depends on ρ and Re only. Additionally, q can be refined
Finally, we should consider that twin faults are cause of
in order to adjust the modelling result for the appropriate
screw/edge character of the dislocations [3].
asymmetry in some profiles. As indicated above, a rigorous
model would require the introduction of sine terms in (1).
In the proposed approach we adopt a simplified model,
always in the limit of low faulting probabilities, which is
essentially the same as that used by Scardi and Leoni [3],
based on the use of split profile function; the basic differ-
ence is that we now use split Voigt curves (Voigtians with
different integral widths for low and high angle tails (βlow
and βhigh, respectively)) instead of split pV curves [3].
2
.5 WPPF algorithm
From the discussion above it is possible to write an ana-
lytical expression for A(L) in (2). In order to model the
diffraction peaks in the experimental pattern, we assume
profiles to be Voigtians. This is clearly an arbitrary hy-
pothesis, since experimental profiles can in principle have
any shape; however, much of the work in the recent litera- mate of asymmetry can be obtained considering that
ture has shown that Voigt curves are appropriate to model the difference between βhigh and βlow is proportional to
many observed patterns from polycrystalline samples [15].
As we will see in the following, the advantage of this hy-
pothesis is the possibility to develop a rather simple and
fast algorithm to model the line broadening effects of (2).
In principle, however, the present approach could also be
used with a different analytical profile function (e.g., a pV
or a Pearson VII).
According to the Warren theory, a reasonable esti-
P
L0
m |L0|
β
σL0 . Also in this case, limitations expressed in
Section 2.1 and footnote 2 apply. Details on the algorithm
to include asymmetry with split profile functions are re-
ported in the cited references [3,21].
The Fourier transform of a Voigtian is a simple two-
parameter expression:
3
Experimental
2
V,G
2
TV (L) = exp(−πβ
L − 2βV,CL)
(15)
X-ray diffraction (XRD) patterns were collected using a
Rigaku PMG-VH diffractometer, adopting the conven-
tional Bragg-Brentano geometry. Goniometer optics in-
cluded a graphite curved-crystal analyser in the diffracted
where βV,G and βV,C are the integral breadths of the
Gaussian and Lorentzian components (that should not be
confused with the corresponding parameters for a pV func-
tion [11,15]).
For each (hkl) diffraction profile, the basic equation
for the whole powder pattern fitting can be written as:
◦
◦
beam and narrow slits (Soller = 2 ; DS = 1/2 ; Soller
◦
=
2 ; RS = 0.15 mm, in the order, from source to
secondary circle (graphite analyser) and scintillation
counter).
X
2
The IP was measured over a wide angular range (25–
wL [TV (L) − A(L)] = min.,
(16)
◦
1
50 ) from the experimental pattern of a suitable KCl
L
profile standard [22,23], and no asymmetry was observed
where wL is a weight. In practice, a least squares algo- within the accuracy of pV curve fitting. The procedure
rithm is used to fit the Fourier transform of a Voigtian also permitted an optimisation of the observed ratio and
profile to (2); the fitting of the experimental data is there- angular distance between the spectral components of the
fore carried out by optimising the values of βV,G and Cu Kα radiation.

P. Scardi et al.: Whole diffraction pattern-fitting of fcc polycrystalline materials
27
Table 1. Results of WPPF: lattice parameter (a ), lognormal
mean (γ), variance (ω) and mean grain diameter (hDi); aver-
0
1
1
1
4000
2000
0000
(111)
(a)
age contrast factor along for (h00) (Ch00) and slope in (12)
(
q); Wilkens’ parameter (µ) and dislocation density (ρ); de-
formation fault (α) and twin fault (β) probabilities; statistical
quality indices of fitting: R , Rexp, GoF [4].
8000
6000
4000
2000
0
w
Ni powder
Li, Mn spinel [35]
◦
(200)
As received 24h ball milling treated at 800 C
(
220) (311)
a0 (nm)
0.35228(1)
4.10(2)
0.10(3)
61(2)
0.35242(1)
2.95(2)
0.824544(5)
2.0(5)
(420)
(
222)
(400)
γ
ω
0.39(1)
0.99(6)
40
60
80
100
120
140
hDi (nm)
20.1(5)
a
0.266
12(1)
a
0.3
2
θ
(degrees)
a
C
q
h00
0.266
a
10000
(b)
1.81
1.88(3)
1.5(1)
10.8(5)
0.68(2)
0
2.55(4)
0.9(2)
0.22(2)
0
µ
∼2
0.58(1)
0
15
−2
ρ(×10 ) m
α (%)
1
000
β (%)
0
0
R
wp (%)
exp (%)
9.40
6.29
1.49
7.34
13.90
11.99
1.16
R
6.45
100
GoF
1.14
a
fixed value, calculated for 50% screw - 50% edge disloca-
tions [3,33].
40
60
80
100
120
140
0
0
0
0
0
.08
.06
.04
.02
.00
2
θ
(degrees)
Fig. 1. Results of Whole Powder Pattern Fitting for a Ni sam-
ple ball milled for 24 hours. Experimental (circle) and mod-
elled (line) patterns are reported together with their difference
(residual, below) in linear (a) and logarithmic (b) scale. Arrows
indicate the position of strong lines of the native oxide (NiO).
p)(dD
4
Results and discussion
The method described so far was applied to two different
inorganic polycrystalline materials. The first example is a
Ni powder ball milled for 24 hours. Figure 1a shows the
experimental and modelled patterns with their difference
0
20
40
60
80
100
D (nm)
(
residual). The quality of the modelling in the tail region
Fig. 2. Distribution of diameters of spherical grains for ball
milled (line-circle) and as received (line) Ni powder.
can be better appreciated in the log scale plot (Fig. 1b),
where it is also possible to note the little contribution
from Ni oxide (NiO, Bunsenite), which was included in
the modelling algorithm. Refined parameters for the ball on line broadening and asymmetry due to faulting should
milled Ni sample are reported in Table 1.
be justified.
It is interesting to compare these results with those
We included the Wilkens’ parameter (µ = Re√
ρ),
which can be related to the distribution of dislocations for the starting powder, which was composed of spherical
and their interaction [29]; the refined value is comparable particles with diameters of the order of several microns.
to that from previous studies on similar materials [3,33], As shown in Table 1, the concentration of defects is much
within the limits of validity of (13) (cf. Wilkens [29]) .
lower in the as received powder, toward the limits of sen-
Line broadening anisotropy is quite evident, and ac- sitivity of the method; for this reason q was not refined,
cording to our results is mostly due to a high dislocation and Wilkens’ parameter is affected by a large error. The
density; from the refined value of q, we can judge that the main source of line broadening in the as received powder
dislocation character is not far from 50% edge - 50% screw. is due to crystallite size; quite obviously metal particles
Profiles are reasonably symmetric, and consequently twin were not single-crystals (a result confirmed by a prelim-
faults are absent, whereas a low fraction of deformation inary TEM investigation). Figure 2 shows a comparison
faults is present; under these conditions, approximations between the lognormal distribution of sphere diameters for

28
The European Physical Journal B
1.0
0.8
0.6
0.4
0.2
0.0
0
-
5
-
-
-
10
15
20
()L]
V
ln[T
-25
-
30
35
(
a)
0
10
20
30
40
-
0
.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
L (nm)
2
(d*)
Fig. 3. Cosine Fourier coefficients for the (111) (
◦
) and
(
200) ( ) profiles of Figure 1, calculated according to the
size/faulting/distortion model of (2) (A(L), symbols) and for
the corresponding Voigt curves (T (L), line) according to (15).
V
the two studied samples. Before ball milling the average
diameter is peaked about 60 nm, with a quite narrow dis-
tribution: under these conditions (ω ≈ 0, see Tab. 1) the
distribution is not far from a Gaussian. After ball milling
the distribution shifts to smaller size and shows a higher
variance, but still preserves a tail toward the values of the
starting distribution. Finally we can note the increase in
lattice parameter due to lattice defect incorporation after
ball milling.
Figure 3 shows the Fourier coefficients (2) compared
with those of the refined Voigtian profiles (given by (15))
for the (111) and (200) reflections. The agreement between
the size/faulting/distortion model and Voigtian transform
is good; in addition, a comparison between the trends
for (111) and (200) clearly demonstrates the strong line
broadening anisotropy. This last feature is better appreci-
ated in the so-called Warren-Averbach plot [17], where
the logarithm of the Fourier coefficients (corrected for
Fig. 4. Conventional (a) and modified (b) Warren-Averbach
plot for the ball milled Ni sample of Figure 1 (see text): L =
1
0 ( ), L = 20 ( ) and L = 30 ( ) nm.
•
2
∗
the IP component) is plotted against d . As shown in
Figure 4a, there is a large scatter of the data caused by
the anisotropic line broadening due to dislocations [3,19].
The effectiveness of the model (and (13) in particular) can
be judged from the linearity of the trend in Figure 4b,
where the same Fourier coefficients are plotted as a func-
served in cases of practical interest, when broad profiles
are measured.
As a further example we considered a Li,Mn spinel
sample. The studied system, obtained from a crystalli-
◦
2
sation (800 C) of an amorphous gel [35], is a finely
∗
tion of d Chkl.
dispersed ceramic powder for which the assumption of
equiaxial grains with a lognormal size distribution is ex-
It is worth underlining that the results of Figure 4b
were obtained by the proposed WPPF method, i.e., by pected to be appropriate [6]. Previous studies on similar
a simultaneous refinement of the size/faulting/distortion samples [3,36,37] indicated the presence of a marked
line broadening anisotropy, reasonably attributed to
dislocations.
parameters directly performed on the observed data. This
is a major improvement in Line Profile Analysis (LPA),
where conventional approaches mostly rely on a posteri-
Figure 5 shows the results of WPPF; defects parame-
ori methods (cf. Sect. 1), that can be biased by the model ters and quality indices are reported in Table 1. In this
used to extract profile information from raw data. For ex- case 35 profiles were modelled simultaneously with an
ample, limits of conventional LPA methods are evident excellent agreement between observed and modelled pat-
in the presence of peak overlapping, which is always ob- terns. The log scale plot of Figure 5b highlights the quality

P. Scardi et al.: Whole diffraction pattern-fitting of fcc polycrystalline materials
29
(b)
1000
100
30
40
50
60
70
80
90
100 110 120 130
2
θ (degrees)
Fig. 5. Results of WPPF for a Li,Mn spinel sample. Graphical output in linear (a) and logarithmic (b) scale.
of peak tail fitting, whereas the inset of Figure 5a shows jor fault in Warren’s theory, concerning the average over
a detail of the line broadening anisotropy effect in the low m (see (9, 12) and footnote 2), produces significant devi-
2
θ range.
The results discussed so far demonstrate the effective-
ations from the correct result when faulting is the main
source of broadening, a situation that is likely to happen
in a few cases [26]; examples illustrated in this work seems
not to fall in this category. In any case, splitting each re-
flection of a {hkl} plane family into sub-components with
different width, asymmetry and shift is a necessary devel-
opment [21].
ness of the proposed method, and the quality and detail of
information attainable. The devised algorithm is simple,
and leads to a convergence in a few least squares iterations.
Limits in the modelling that need to be improved concern
faulting; however, possible inadequacies of the present ap-
proach are likely to be reduced when size and distortion
Another important issue concerns avoiding the use of
effects are prevalent sources of line broadening. The ma- analytical functions to model the pattern. We have already

30
The European Physical Journal B
underlined that this is an arbitrary hypothesis, whose
consequences on the reliability of the results are not eas-
ily controlled, and are certainly sample-dependent. For
instance, the limitation of using Voigt profile functions
is apparent from both curves in Figure 3 for small L: a
small but non negligible discrepancy can be observed. As
recently stated by Langford et al. [6], in finely dispersed
systems (like ceramic powders and deformed metals) there
may well be an appreciable systematic error in hDi due to
deviation of the Voigt profile shape from the profile func-
tion produced by a real system; in our case, deviations are
with respect to (5).
7. M. J ¨a rvinen, in Defect and Microstructure Analysis by
Diffraction, edited by R.L. Snyder, J. Fiala, H.-J. Bunge
(Oxford University Press, Oxford, 1999), pp. 556–569; V.
Honkim ¨a ki, P. Suortti, ibidem, pp. 41–58.
8. H.M. Rietveld, J. Appl. Cryst. 2, 65 (1969).
9. H. Toraya, in The Rietveld Method, edited by R.A. Young
(
Oxford University Press, Oxford, 1993), pp. 254–275.
0. G.S. Pawley, J. Appl. Cryst. 14, 357 (1981).
1. J.I. Langford, J. Appl. Cryst. 11, 10 (1978).
2. G.K. Wertheim, M.A. Butler, K.W. West, D.N.E.
Buchanan, Rev. Sci. Instrum. 11, 1369 (1974).
1
1
1
13. M.M. Hall, V.G. Veeraraghavan, H. Rubin, P.G. Winchell,
J. Appl. Cryst. 10, 66 (1977).
1
1
4. R.A. Young, D.B. Wiles, J. Appl. Cryst. 15, 430 (1982).
5. J.I. Langford, in Defect and Microstructure Analysis by
Diffraction, edited by R.L. Snyder, J. Fiala, H.-J. Bunge
5
Conclusion
(
Oxford University Press, Oxford, 1999), pp. 59–81.
A new method of whole powder pattern fitting based on 16. H.P. Klug, L.E. Alexander, X-ray Diffraction Procedures,
2nd edn. (John Wiley and Sons, New York, 1974).
the use of Voigt functions has been described and tested
on two different cases of study. The algorithm proved to be 17. B.E. Warren, X-ray Diffraction (Addison Wesley, New
York, 1969).
8. R.L. Snyder, in The Rietveld Method, edited by R.A.
Young (Oxford University Press, Oxford, 1993), pp. 111–
131.
9. T. Ung ´a r, in Defect and Microstructure Analysis by
Diffraction, edited by R.L. Snyder, J. Fiala, H.-J. Bunge
fast and effective in reproducing the experimental pattern
1
1
2
of fcc materials, and can in principle be adapted to other
structures within WPPF or Rietveld programs.
The major advancement with respect to previous ap-
proaches is that all the main components of sample-related
line broadening are considered on the basis of well-defined
physical models:
(
0. A.C. Nunes, D. Lin, J. Appl. Cryst. 28, 274 (1995).
Oxford University Press, Oxford, 1999), pp. 165–199.
•
size broadening is connected with the column length 21. P. Scardi, M. Leoni, Y.H. Dong, Mat. Sci. Forum (2000)
distribution in a system of spherical grains, whose di-
ameters are distributed according to a lognormal (in
principle, different grain shapes and distributions can
be considered);
planar defects are dealt with in terms of probability
of finding twin and stacking faults in the (111) planes,
with the assumption of low defect concentrations;
lattice distortions are attributed to dislocations, whose
anisotropic strain field is considered in terms of the
average contrast factor, dislocation density and outer
cut-off radius.
in press; P. Scardi, M. Leoni, Y.H. Dong (in preparation).
2. P. Scardi, L. Lutterotti, P. Maistrelli, Powder Diffraction
2
2
9
, 180 (1994).
3. M. Leoni, P. Scardi, J.I. Langford, Powder Diffraction 13,
10 (1998).
2
•
•
24. W.L. Smith, J. Appl. Cryst. 9, 187 (1976).
25. F. Bertaut, C.R. Acad. Sci. Paris 228, 492 (1949).
26. L. Velterop, R. Delhez, Th.H. de Keijser, E.J. Mittemeijer,
D. Reefman, J. Appl. Cryst. 33, 296 (2000).
27. M.A. Krivoglaz, O.V. Martynenko, K.P. Ryaboshapka,
Phys. Met. Metall. 55, 1 (1983).
2
8. M. Wilkens, in Fundamental Aspects of Dislocation The-
ory, edited by J.A. Simmons, R. de Wit, R. Bullough (Nat.
Bur. Stand., (US) Spec. Publ. No 317, Washington, DC.
USA, 1970), Vol. II, pp. 1195–1221.
The authors wish to thank J.I. Langford for helpful discus-
sion and suggestions, and S. Setti for assistance in laboratory
measurements.
2
3
9. M. Wilkens, Phys. Stat. Sol. (a) 2, 359 (1970).
0. J.G.M. van Berkum, Ph.D. thesis, Delft University of Tech-
nology, 1994.
31. A.R. Stokes, A.J.C. Wilson, Proc. Phys. Soc. London 56,
174 (1944).
3
2. V. Hauk, Structural and Residual stress Analysis by Non-
destructive Methods (Elsevier, Amsterdam, 1997); I.C.
Noyan, J.B. Cohen, Residual Stress (Springer-Verlag, New
York, 1987).
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