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J. Appl. Cryst. (2003). 36, 53±64 Peter F. Peterson et al. Pair distribution function 53
research papers
Journal of
Applied
Crystallography
ISSN 0021-8898
Received 15 March 2002
Accepted 10 October 2002
#2003 International Union of Crystallography
Printed in Great Britain ± all rights reserved
Improved measures of quality for the atomic pair
distribution function
Peter F. Peterson,
a
² Emil S. Boz
Ïin,
a
Thomas Proffen
b
and Simon J. L. Billinge
a
*
a
Department of Physics and Astronomy and Center for Fundamental Materials Research, Michigan
State University, East Lansing, MI 48824-1116, USA, and
b
Los Alamos National Laboratory,
LANSCE-12, Mailstop H805, Los Alamos, NM 87544, USA. Correspondence e-mail:
billinge@pa.msu.edu
The introduction of neutron spallation-source instruments, such as the General
Materials Diffractometer (GEM) at ISIS, allows measurement of pair
distribution function (PDF) data at signi®cantly higher rates than previously
possible. As a result of the increased rate, a single experiment can produce over
a hundred individual runs. Manual processing of all these data using traditional
methods becomes inconvenient and inef®cient. This article presents quality
criteria that help produce automated direct Fourier transformed PDFs of quality
similar to hand-processed data, and compares optimization methods.
1. Introduction
Different techniques for structure determination have devel-
oped over the years. X-ray and neutron diffraction studies are
useful since they allow structure measurements on the
a
Êngstro
Èm length scale. The technique discussed in this article is
the atomic pair distribution function (PDF) method. The PDF
is widely used to study glasses and amorphous materials
(Wagner, 1978; Warren, 1990), but more recently it has been
used to study the structure of crystalline materials (Egami,
1990, 1998). An important aspect of every structural technique
is to obtain the best results possible from a given measure-
ment. This can be achieved either by improving the
measurement method or by processing the data better; we will
be discussing the latter.
The PDF, G(r), is obtained from the experimentally deter-
mined total-scattering structure function, S(Q), by a Sine
Fourier transform
Gr 2
Z
1
0
QSQÿ1sinQrdQ
4rrÿ0;1
where ris the distance between two atoms and Qis the
momentum transfer. (r), de®ned as (Faber & Ziman, 1965)
r 1
4r2X
ij
bibj
hbi2rÿrij;2
is the microscopic pair density, and
0
is the average number
density of the sample. The sum is taken over all atoms in the
sample and r
ij
=|r
i
ÿr
j
| is the distance separating atoms iand j.
It gives the probability of ®nding two atoms separated by a
distance r, weighted by the scattering lengths, and averaged
over all pairs of atoms in the sample.
In a real experiment, a number of corrections to the
intensity data must be carried out in order to obtain the
structure function, S(Q), normalized to the total-scattering
cross section of the sample. In principle, the data corrections
are well known and well understood (Klug & Alexander, 1974;
Wagner, 1978; Soper et al., 1989; Hannon et al., 1990; Warren,
1990; Billinge & Egami, 1993; Wright et al., 1995) and the data
analysis can be carried out with no adjustable parameters. In
practice, a number of approximations must be made in
calculating these corrections and certain parameters are not
known with high accuracy. Using these approximations results
in a corrected normalized S(Q) that contains distortions. The
distortions are usually dealt with in somewhat arbitrary ways,
as described below. Fortunately, the structural information in
the PDF is rather robust with respect to these distortions.
Inadequacies in the corrections tend to result in very long
wavelength distortions of S(Q), giving rise to nonphysical
features in G(r) at low values of atomic separation, r, too small
for real atomic separations. The distortions do not affect the
data except insofar as ripples from these features propagate
into the high-rregion. Frequently, an expert eye is needed to
minimize distortions of S(Q) so that the physics can be
studied. Nonetheless, it is clearly of interest to ®nd more
quantitative criteria for assessing the quality of a PDF and to
minimize these distortions and obtain the most accurate PDFs
possible. This becomes more important as new instruments,
such as the General Materials Diffractometer (GEM) at ISIS,
come online, and increased data acquisition rates necessitate
an automated data processing method. Though the data
corrections to obtain S(Q) from X-ray data are different from
the neutron case described above and discussed in detail in the
paper below, the S(Q) function before direct Fourier trans-
formation is important in both cases. The quality criteria and
² Current address: Intense Pulsed Neutron Source, Building 360, 9700 South
Cass Ave., Argonne, IL 60439-4814, USA.
research papers
54 Peter F. Peterson et al. Pair distribution function J. Appl. Cryst. (2003). 36, 53±64
optimization procedures described here are applicable to both
X-ray and neutron-derived S(Q) functions.
A number of indirect methods have been proposed for the
Fourier transform, such as the use of the Reverse Monte Carlo
method (Zetterstrom & McGreevy, 2000) and Bayesian
methods (Terwilliger, 1994). Here we consider how to obtain
the best S(Q) possible for direct Fourier transform. Also, in
most experiments multiple diffraction patterns are combined
to form a single S(Q). The process of combining multiple
spectra, from different scans or banks, is a sizable topic on its
own and will be dealt with in a separate article.
The S(Q)andG(r) functions exhibit certain known prop-
erties that can be made use of to assess and optimize the data
corrections. With real data, the easiest problem to detect is
when S(Q) does not asymptote to unity as Q!1. In prac-
tice, the data are adjusted to obtain the right asymptote.
However, it is not a priori clear whether the correction should
be to add a constant, multiply by a factor, or apply some other
correction. This is because it is often not clear which data
distortion, or distortions, are primarily to blame for the
problem. Some data distortions are additive, such as back-
ground, empty can, multiple scattering and incoherent scat-
tering subtractions, and others are multiplicative, such as
normalization for ¯ux and number of atoms in the illuminated
sample volume, and absorption corrections.
A number of different approaches are often taken at this
point. The most common is to make the sample density a
parameter and vary it until the asymptotic behavior of S(Q)is
correct [S(Q)!1asQ!1]. This practice is somewhat
arbitrary, since in many cases this will not be the limiting factor
in the accuracy of the corrections as the sample density is easy
to determine with reasonable accuracy. It should be noted that
varying the sample density applies a predominantly multi-
plicative correction (it strongly affects the sample normal-
ization and the absorption correction) to the data with a small
additive part (from the multiple-scattering correction and an
incorrect background subtraction due to the incorrect
absorption correction). Another commonly varied parameter
is the effective beam width. The beam size is known from the
collimation of the instrument, but the beam may not be
homogeneous (Soper et al., 1989; Hannon et al., 1990).
Therefore, the effective beam width will be different from the
physical beam size due to varying intensity across the beam
pro®le. This beam size is a predominantly multiplicative
correction due to ¯ux normalization; however, it will also have
an additive component due to differently evaluated multiple-
scattering corrections. Less commonly used parameters are
sample height or effective beam height.
Since the choice of parameters to vary is mostly arbitrary, it
is interesting to see whether taking a completely arbitrary
approach of simply multiplying the data by a constant and/or
adding a constant results in PDFs of equally good quality.
Here we compare a number of different approaches for data
normalization. Since our primary interest is crystalline mate-
rials, we have chosen to study a neutron data set from pure
germanium. We systematically apply both a multiplicative and
an additive correction to the processed S(Q), obtain the PDF
by direct Fourier transform, and analyse the results using a
number of PDF quality criteria de®ned below. The PDF is very
sensitive to the asymptotic behavior of S(Q) so we con®ne our
interest to data where limQ!1hSQi = 1.00 (2). Since
germanium is crystalline with a well de®ned structure, by
modeling we can easily determine when the data are properly
normalized. It is then possible to compare different approa-
ches that satisfy these criteria; for example, varying a multi-
plicative constant, , and an additive constant, , varying the
sample density
eff
and , and so on. The resulting PDFs are
then compared using the quality factors. Finally, we suggest a
protocol for automatically obtaining the best PDF given an
initial experimentally derived S(Q). While the analysis is only
carried out for a single data set, the methods described below
can be used to automate the analysis of several data sets.
As discussed above, the corrections to the data are incom-
plete. This is not only due to the simpli®cations made to the
data corrections but also because characterization runs do not
yield the complete picture of the physical setup. Multiple
scattering events when an incident neutron or X-ray scatters
off the sample, then the sample environment, before entering
a detector, cannot be fully subtracted using characterization
runs since the sample is not present. These inadequacies in the
measurement and processing result in a total-scattering
structure function, S0(Q), that is different from the true S(Q).
In general, the measured S0(Q) can be written in terms of the
true S(Q)as
S0QQSQQ;3
where (Q) and (Q) are dimensionless functions. The
simplest approximation is to take (Q)and(Q) as being
independent of momentum transfer, Q, when equation (3)
becomes
S0QSQ: 4
Then the PDF associated with S0(Q) can be written in terms of
S(Q)as
G0r 2
Z
1
0
QS0Qÿ1sinQrdQ
2
Z
1
0
QSQÿ1sinQrdQ:5
In addition to and , the other experimental effect is a ®nite
measurement range. The PDF then becomes
G0r 2
Z
Qmax
Qmin
QSQÿ1sinQrdQ
2
Z
Qmax
Qmin
QSQÿÿ1sinQrdQ
2
Z
Qmax
Qmin
QSQÿ1sinQrdQ
ÿ12
Z
Qmax
Qmin
QsinQrdQ
Gcrÿ1r;6
where G
c
(r) is the ideal G(r) convoluted with a termination
function and (r) is de®ned in equation (6). The term (r)is
the result of an improper high-Qasymptote of the S(Q)and
gives rise to low-rripples often seen in experimental PDFs.
Both G
c
(r)and(r) are discussed in more detail in Appendix
A. For compactness, in the rest of the article G(r) will refer to
the measured PDF approximated by equation (6), G0(r).
2. Quality measures
2.1. Definition of quality measures
Quantifying the difference between the measured and ideal
PDF is of great interest when the PDF is unknown. G(r) has
certain known properties that can be used to assess its quality.
Here we list several criteria used for determining an optimal
PDF that is closest to the real structural PDF. The quality
criteria presented here are derived and symbols are de®ned in
Appendix B.
In the case where the crystallographic structure of the
sample is being studied there are well established methods of
determining the quality of data. Programs such as PDFFIT
(Proffen & Billinge, 1999) and GSAS (Larson & Von Dreele,
1994; Toby, 2001) allow a structural model to be re®ned and
then ®nd how well the model and data agree. Usually this is
used to determine how correct a model is; however, if the
structure is already well known then these programs can be
used to test the quality of measured data. In PDFFIT two
criteria that tell us about data quality are the weighted pro®le
agreement factor, R
wp
, and the scale factor, N
m
.R
wp
is de®ned
as (Proffen & Billinge, 1999)
Rwp PiwriGobsriÿNmGcalc ri
2
PiwriGobsri
2
()
1=2
;7
where G
obs
(r) and G
calc
(r) are the measured and model PDFs,
respectively, and w(r) is the weighting factor. This de®nition is
used in standard crystal structure re®nements. Note that the
de®nition of R
wp
, equation (7), is valid as it stands and
provides a useful way of optimizing a model and of comparing
the goodness-of-®t of different models (Peterson et al., 2001).
However, because points in the PDF are statistically corre-
lated, R
wp
does not have the same statistical signi®cance as
crystallographic R
wp
functions. For example, it is not possible
to obtain a reliable
2
value from it. The scale factor, N
m
, is the
factor by which the model must be multiplied to give good
agreement with the data. We therefore de®ne the factor N
d
=
N=1/N
m
as the factor by which the data must be multiplied to
result in a properly scaled G(r)(N
m
= 1). These two criteria,
and others resulting from such ®tting, require signi®cant
knowledge of the material a priori. In general they will not be
used to determine data quality but instead model quality. For
this reason they are not discussed further. The remaining
criteria presented will require knowledge of the chemical
composition and average number density,
0
.
Using only the information about the general behavior of
S(Q) and the PDF, there are three criteria that S(Q) should
conform to. The ®rst is the equality (Appendix B)
Sint Z
1
0
Q2SQÿ1dQÿ220:8
Secondly, the high-Qportion of S(Q) should approach unity.
This is seen in
Savg lim
Q!1hSQi 1;9
where the angle brackets indicate an average over a range in
Q. In this article the average is taken over 24 A
Ê
ÿ1
<Q<Q
max
=40A
Ê
ÿ1
for the synthetic data in x2.2, and 15 A
Ê
ÿ1
<Q<Q
max
=25A
Ê
ÿ1
for the measured data presented in x3.1. This range,
0.6Q
max
<Q<Q
max
, was determined empirically. Thirdly, one
can also look at the total dispersion of S(Q) between the low-
Qand high-Qasymptotes to ®nd (Appendix B)
Sdisp lim
Q!1 SQÿ lim
Q!0SQ
1hb2iÿhbi2
hbi2;10
where bis the neutron scattering length and the averages are
over isotopes and elements. An analog of equation (10) exists
for X-rays where the neutron scattering lengths are replaced
with X-ray form factors evaluated at Q=0A
Ê
ÿ1
,f(0). The
methods for calculating S
avg
and S
disp
are both done by aver-
aging S(Q) over a range. While the principles behind S
avg
and
S
disp
are sound, a better method of calculating their values
should be determined. The method of determining S
avg
and
S
disp
could be improved, for example, through the use of
Bayesian statistics (David & Sivia, 2001), though this is
beyond the scope of this article.
Similar to the previous criteria of S(Q), the PDF can be
looked at without prior structural knowledge. The real-space
analogy of equation (8) is (Appendix B)
Gint Z
1
0
rGrdrÿ1:11
The largest effect of distortions in S(Q) is to introduce ripples
in G(r) at low r. It is therefore reasonable to adjust and in
J. Appl. Cryst. (2003). 36, 53±64 Peter F. Peterson et al. Pair distribution function 55
research papers
research papers
56 Peter F. Peterson et al. Pair distribution function J. Appl. Cryst. (2003). 36, 53±64
such a way as to minimize these ripples. We propose the
following criterion, G
low
, to accomplish this:
Glow Rrlow
0rGr4r2fit2dr
Rrlow
04r2fit2dr;12
where
®t
is the average number density determined by ®tting
the low-rregion of the PDF and r
low
is before the ®rst peak.
For the data presented here, r
low
of 2 A
Êwas used. G
low
is
designed as a robust criterion for automatically estimating (i.e.
with no user input) the magnitude of ripples in the unphysical
low-rregion of the PDF below the ®rst atom-pair peak. The
exact form of G
low
is justi®ed in Appendix B. While many
criteria presented in this section are de®ned for an in®nite
range, for real data they are evaluated over a ®nite range. In
the next section these criteria will be further explored using
synthetic data.
2.2. Testing the quality measures
In order further to understand the quality measures
presented in the previous section, they will be tested against
synthetic PDFs of varying quality. All of the test PDFs were
generated from the same initial PDF. The initial PDF was
created by calculating G(r) from the known structure of
germanium (at 10 K) using PDFFIT. Instrumental parameters
in PDFFIT were chosen to be appropriate for the Glass,
Liquid and Amorphous Materials Diffractometer (GLAD) at
the Intense Pulsed Neutron Source (IPNS), so comparisons
can be made with measured data (the instrument resolution,
Q
, is 0.0657 A
Ê
ÿ1
) (Proffen & Billinge, 1999). This PDF was
Fourier transformed to produce an ideal S(Q) using
QSQÿ1
Z
130 A
0A
GrsinQrdr:13
At r=60A
Ê, the PDF has already reached its asymptotic value
of zero, with this instrument resolution, as seen in Fig. 1(a).
The S(Q) produced by this method can be Fourier trans-
formed back from 0 to 100 A
Ê
ÿ1
to reproduce the initial PDF. A
higher instrument resolution could be created by calculating
to larger rvalues. For example, to synthesize GEM data (
Q
'
0.035 A
Ê
ÿ1
), where the PDF reaches its asymptotic value near
160 A
Ê, the integral in equation (13) would need to be eval-
uated to a signi®cantly higher upper limit.
The utility of the quality factors can be seen by looking at
the synthetic data for different values of Q
max
representing
ideal and measured data. The different quality factors calcu-
lated for Q
max
values of 100 and 40 A
Ê
ÿ1
(r
max
= 100 A
Ê) are
shown in Table 1. In the table, the reader will quickly notice
that the quality factors in equations (8) and (11) vary signi®-
cantly from their theoretical values, especially as Q
max
is
reduced to 40 A
Ê
ÿ1
, even in the current case where there are no
distortions to the data. This is because of the instrument
resolution,
Q
. In real space, the instrument resolution
dampens the peaks, removing weight from the PDF. This
dampening is the reason for the PDF reaching its high-r
behavior at the small distance of 60 A
Ê. This was seen as S
int
and G
int
varied with the instrument resolution. Since the peaks
are missing weight, G
int
and S
int
are not their ideal values. For
this reason, they will not be considered further. Besides S
int
and G
int
, the values of the other criteria are not more than 3%
different from their theoretical values. This also gives a
measure of the minimum signi®cant uncertainty in the quality
criteria, 3%.
Measured spectra tend to have errors which are both
random and systematic in nature. Random noise originates
from measurement statistics. There are many sources of
systematic errors, from bad detectors to an unstable source.
Systematic errors are normally dealt with by determining
where they come from, ®xing the problem and remeasuring.
Frequently, it is not possible to remeasure a data set, or the
systematic error is subtle enough not to be noticed. In these
cases one can either disregard the data or get an idea of their
quality and try to understand the underlying physics, knowing
Figure 1
Synthetic 10 K germanium PDF: (a) as calculated by PDFFIT and
resulting S(Q); (b) calculated by Fourier transform. Insets are a reduced
vertical scale of the functions. The synthetic data were calculated using
parameters to mimic a GLAD measurement.
Table 1
Values of quality criteria for unspoiled data with Q
max
of 100 and 40 A
Ê
ÿ1
.
Theory 100 A
Ê
ÿ1
% difference 40 A
Ê
ÿ1
% difference
S
avg
1.0000 1.0002 0.02 1.0003 0.03
S
disp
1.0000 0.9814 1.86 0.9774 2.26
S
int
ÿ0.8725 ÿ0.8614 1.27 ÿ1.5843 81.59
G
int
ÿ1.0000 ÿ0.7292 27.08 ÿ0.7286 27.14
G
low
0.0000 0.0000 ± 0.0000 ±
®t
0.0442 0.0439 0.68 0.0439 0.68
that the data do have a systematic error and knowing its effect
on the data. The following examples all started from the
synthetic data set described above with known errors intro-
duced into S(Q). The S(Q) functions were Fourier trans-
formed using a Q
max
of 40 A
Ê
ÿ1
, obtainable at spallation-source
neutron and synchrotron X-ray instruments.
The effect of random noise is handled in two ways: by
adding constant noise and noise that increases with Q. Noise is
introduced by adding a random number between 0.5 dS,
where dS is shown in the insets to Fig. 2, along with the
resulting S(Q) functions and PDFs. The effect of random noise
on the criteria can be seen in Table 2. As seen in Fig. 2, the
effect of random noise is most easily seen in the regions where
there are no PDF peaks. Being average criteria, the values of
S
avg
and S
disp
do not appreciably change, within the prescribed
3%; the value of G
low
does.
One might expect the effect of systematic errors on data to
be more dramatic than random noise. Since systematic errors
carry information, their effect is more complicated than
random noise. Three types of systematic errors will be
presented here. While the source of the errors is not
mentioned, they are typical of errors that have been seen in
real data.
The most common systematic error is a scaling error. This
comes from the fact that diffraction data are inherently arbi-
trarily scaled. Therefore, this type of systematic error will
always be encountered. Many authors have described various
techniques for ®nding either an absolute scale factor (Kartha,
1953; Krogh-Moe, 1956; Norman, 1957; Kaszkur, 1990;
Cumbrera et al., 1995; Leadbetter & Wright, 1972) or a rela-
tive scale factor (Hannon et al., 1990; Soper et al., 1989; Louca
& Egami, 1999) to compare data sets. When S(Q) is scaled,
this changes the asymptotic behavior. To ensure proper
asymptotic behavior we set =1ÿ; hence
S0QSQ1ÿ:14
The effect of scaling can be seen in Fig. 3 and Table 3. As
expected from the analytic result, equation (6), G0(r)=G
c
(r)
and G(r) remains undistorted but changes its scale. By de®-
nition, S
avg
does not change for the three cases within
reasonable accuracy, while it is also noticed that G
low
is scale
invariant as well. The other two criteria, S
disp
and
®t
, do vary
with scale, as seen in the second half of Table 3. In principle,
therefore, S
disp
and
®t
could be used to determine the scale of
the data. S
disp
only requires knowledge of the sample chemical
composition and
®t
only the average sample number density.
This result shows that the `quality' of the PDF [i.e. spurious
ripples and distortions of G(r)] does not change with scale
factor, provided that the asymptotic behavior of S(Q)is
satis®ed. Obtaining the correct absolute scale factor, or rela-
tive scale factor between data sets, is important when carrying
out model-independent analyses of data such as peak inte-
grations or peak height analyses. However, when ®tting
models to data, provided that the model PDF can be scaled, it
is not necessary to satisfy both S
avg
and Nindependently.
More interesting and widely encountered in real data are Q-
dependent additive and multiplicative distortions. We now
consider a slowly oscillating additive sine wave that might
originate from an imperfectly corrected background (Fig. 4a).
As before, we want S0(Q) to have the right asymptotic form,
S
avg
= 1, so the constant is changed in such a way that this is
satis®ed at our chosen Q
max
. This is a common situation in real
experimental PDFs: an unknown slowly oscillating additive
J. Appl. Cryst. (2003). 36, 53±64 Peter F. Peterson et al. Pair distribution function 57
research papers
Figure 2
Reduced total-scattering structure function (left), form of noise dS added
to S(Q) (inset), and associated PDFs (right). From top to bottom the
synthetic data are pure, constant-noise added, Q-dependent-noise added.
The abscissae in the insets are Q(A
Ê
ÿ1
).
Table 2
Values of quality criteria for different amounts of noise.
R
wp
is calculated between the test data and the ideal PDF.
Theory None Constant Q-dependent
S
avg
1.0000 1.0001 0.9999 1.0005
S
disp
1.0000 0.9772 0.9770 0.9774
G
low
0.0000 0.0000 0.0152 0.0847
®t
0.0442 0.0439 0.0440 0.0446
R
wp
0.0174 0.0487 0.0492
Figure 3
S(Q) (left) and associated PDFs (right). From top to bottom the scale, ,
is 0.5, 1.0 and 2.0. Both S(Q) and G(r) are offset for clarity.
research papers
58 Peter F. Peterson et al. Pair distribution function J. Appl. Cryst. (2003). 36, 53±64
correction is arbitrarily corrected with a (mostly) multi-
plicative correction. The exact form of S0(Q) used here is
S0QSQ0:1 sin 2
100 Q
:15
The choices of the amplitude (0.1) and wavelength (100 A
Ê
ÿ1
)
were made to produce S0(Q) similar to what might be
encountered in real measurements. The reduced total-scat-
tering structure function is shown in Fig. 4(a) and the asso-
ciated PDF can be seen in Fig. 5(a), with the quality factors
being listed in Table 4, column (a). From equation (6) we
expect to see the PDF scaled and low-rripples due to (r).
While there is little noticeable change in S(Q), the effect on
the PDF is quite large. This type of systematic error shows the
behavior often seen in PDF data of large spurious peaks near
r=0A
Êwith small oscillations extending into the physical
portion of the PDF. S
avg
is within a reasonable range of its
ideal value, while G
low
varies signi®cantly. This is expected
because G
low
is a quanti®cation of the low-rnoise in the
PDF.
The next type of systematic error to be discussed here is a
slowly varying multiplicative factor. This type of systematic
error might result from an improper absorption correction.
The exact form of the function used here is
S0QSQexp ÿQ
150 A
Êÿ1
2
"#()
:16
The reduced total-scattering structure function is in Fig. 5(b)
and the associated PDF can be seen in Fig. 5(b), with the
quality factors being listed in Table 4 column (b). In this case
the value of S
disp
is less than one because this systematic error
suppresses the high-Qintensity. There is also a clear effect on
the low-rportion of the PDF.
For completeness, the fourth spoiling of the synthetic data
was achieved by introducing both Q-dependent random noise
Table 3
Values of quality criteria for various scale factors.
= 0.5 = 1.0 = 2.0
S
avg
1.0005 1.0003 0.9999
S
disp
0.4894 0.9774 1.9557
G
low
0.0000 0.0000 0.0000
®t
0.0219 0.0439 0.0877
S
disp
/0.9788 0.9774 0.9778
®t
/0.0438 0.0439 0.0439
Figure 4
Q[S(Q)ÿ1] for three types of systematic errors: (a) additive long-
wavelength sine oscillation, (b) Gaussian multiplicative function, and (c)
additive sine oscillation with Q-dependent random noise. Below each
structure function is the difference between the data with and without
errors. The insets are the same data plotted from 0 to 100 A
Ê
ÿ1
.
Figure 5
G(r) for three types of errors: (a) long-wavelength sine oscillation, (b)
step function, and (c) sine oscillation with Q-dependent random noise.
Below each PDF is the difference between the data with and without
errors.
Table 4
Values of quality criteria for various systematic errors.
R
wp
is calculated between the test data and the appropriately scaled ideal PDF.
The three columns are shown in Figs. 4 and 5 as (a), (b) and (c), respectively.
Theory (a)(b)(c)
± 0.97 1.07 0.97
S
avg
1.0000 1.0124 1.0218 1.0129
G
low
0.0000 0.0194 0.0543 0.0625
R
wp
ÿ0.0175 0.0217 0.0442
and a sine wave oscillation. The results of this synthetic data
can be seen in Figs. 4(c) and 5(c), with some of the quality
factors listed in Table 4 column (c). As expected, the results
are qualitatively similar to those without the random noise
[Fig. 5(a) and Table 4 column (a)].
From these seven examples we know which quality criteria
are most useful. S
int
and G
int
do not work on the test data due
to ®nite instrument resolution, and even if they did they are
still not useful for measurements of crystalline materials due
to the large measurement range required to determine them
accurately. As expected, S
disp
and
®t
are scale dependent.
They are then not useful as data quality criteria since, in
general, the absolute scale of the data is not known a priori.
However, they are potentially useful independent measures
for the scale of the data. This leaves S
avg
and G
low
as the
preferred criteria. For perfect data, S
avg
and G
low
are
equivalent. However, for real data including systematic errors
they are not. We have found from the test data that smaller
low-rripples can sometimes be obtained when S
avg
is actually
not equal to unity. Given the problems with determining S
avg
accurately, especially if as sometimes happens S(Q) is curved
in the high-Qregion, we believe that G
low
is a more robust
quality criterion.
3. Data analysis protocols
In this section we use G
low
to compare the quality of PDFs
determined from real data but where different data analysis
parameters were varied to obtain the optimal S(Q). As we
discussed earlier, the parameters that are normally varied for
this purpose, such as sample density, are somewhat arbitrary.
We would like to determine if better results are achieved by
varying a particular parameter or whether arbitrary additive
and/or multiplicative factors could be used.
The data are time-of-¯ight neutron powder diffraction data
from germanium collected on the GLAD at IPNS. Finely
powdered germanium was sealed inside an extruded cylind-
rical vanadium container with helium exchange gas. The
sample weighed 4.472 g and ®lled a container (0.9272 cm
diameter and 5.4 cm high) to a height of 4.0 cm. We therefore
estimate the mass density of the powder sample to be
1.7 g cm
ÿ3
. This was mounted on a closed-cycle helium
refrigerator. Neutron powder diffraction data were measured
at 10 K for 4 h with a collimator of width 0.4636 cm. Parasitic
scattering from the heat shields was estimated by taking data
with the sample environment in place but no sample at the
sample position. Scattering from the sample container was
measured from an empty container. The scattering from a
vanadium rod was also measured to allow the data to be
normalized with respect to the incident spectrum and detector
ef®ciencies. Standard data corrections were carried out as
described elsewhere (Wagner, 1978; Billinge & Egami, 1993)
using the program PDFgetN (Peterson et al., 2000). A repre-
sentative reduced total-scattering structure factor {Q[S(Q)ÿ
1]} is shown in Fig. 6 and was Fourier transformed using Q
max
J. Appl. Cryst. (2003). 36, 53±64 Peter F. Peterson et al. Pair distribution function 59
research papers
Table 5
Results of varying different parameters.
The results of the traditional methods were obtained by using physical parameters (column headed `none'), varying
eff
to minimize G
low
(
eff
column), and
reducing
eff
then varying
eff
to minimize G
low
(
eff
and
eff
column). The second half of the table lists data obtained by varying ,
eff
and
eff
with to obtain
N
m
= 1 and minimize G
low
.
Traditional methods Multiplicative and
Theory None
eff
eff
and
eff
eff
and
eff
and and
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.6672
0.0000 0.0000 0.0000 0.0000 ÿ0.6881 ÿ0.6841 ÿ0.6560
eff
1.7 1.7 1.2820 1.5781 0.7470 1.2820 1.2820
eff
0.4636 0.4636 0.4636 0.4173 0.4636 0.3560 0.4636
S
avg
1.0000 0.7222 0.9555 0.9606 0.9368 0.9481 0.9373
G
low
0.0000 45.7610 0.9430 0.9159 0.8706 0.9043 0.9260
N
m
0.461 (4) 0.6014 (15) 0.6060 (15) 1.000 (3) 1.006 (3) 1.000 (3)
R
wp
0.2857 0.1219 0.1276 0.1274 0.1262 0.1274
a5.6578 (3) 5.65786 (11) 5.65788 (11) 5.65788 (11) 5.65790 (11) 5.65788 (11)
hUi0.002152 (22) 0.002134 (6) 0.002134 (6) 0.002134 (6) 0.002130 (6) 0.002134 (6)
Figure 6
Representative reduced total-scattering structure factor for germanium
at 10 K measured using GLAD at IPNS.
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60 Peter F. Peterson et al. Pair distribution function J. Appl. Cryst. (2003). 36, 53±64
=25A
Ê
ÿ1
to produce the PDF shown in Fig. 7 as the open
circles.
The data were modeled using PDFFIT. Pure germanium
has a diamond structure (F
43mwith atoms at 0 0 0 and 1
4
1
4
1
4).
The structure re®nement was carried out over the range 2 < r<
15 A
Êusing the following method. The starting structure was a
crystal structure with lattice parameter of 5.66 A
Êand germa-
nium atoms only at the symmetric sites. The anisotropic
displacement factors were set to be equal (U
11
=U
22
=U
33
=
0.0021 A
Ê
2
). Then the re®nement proceeded by varying para-
meters in ®ve steps.
(i) The scale factor (N
m
), Q-resolution (qsig[1]), and r-
dependent sharpening (delt[1]) are varied.
(ii) N
m
, delt[1] and the lattice parameter (latt[i]) are varied.
(iii) N
m
, qsig[1], delt[1] and latt[i] are varied.
(iv) The isotropic displacement factor (u[i,j]) is varied.
(v) N
m
, latt[i] and u[i,j] are varied.
This method was repeated for all data, however processed,
to obtain values of Nand R
wp
that were reproducible. A
representative ®t is shown in Fig. 7 as the solid line. A
difference curve is shown beneath the data. Notice the
nonphysical peak in the experimental PDF at very low r
coming from the imperfect data corrections. Also, note that
the structural information is not affected in a signi®cant way
by this feature.
3.1. Methods of optimizing PDFs
When processing a given run there are multiple ways of
varying the processing parameters to minimize G
low
. This
discussion is better understood by looking at Table 5. The top
section of Table 5 gives the values used in PDFgetN to analyse
the data and obtain the data PDF. The middle section has the
values of the quality criteria, S
avg
and G
low
, for the resulting
data PDFs. The bottom section of the table gives re®ned
values for N
m
,R
wp
,aand hUifrom the best model ®t to the
data PDFs. The ®rst column gives the theoretical values of
parameters that should be used in an ideal data analysis, and
the resulting optimal values for the quality criteria. In subse-
quent columns, different analysis parameters from the possi-
bilities ,,
eff
and
eff
, have been varied in such a way as to
minimize G
low
. The second column is for the PDF obtained
using the `ideal' parameters. While this does represent the
known experimental setup, due to problems and approxima-
tions in the data analysis, it clearly is not the best PDF: S
avg
is
far from one, G
low
is very large, indicating large low-r
ripples, R
wp
is very large, again indicating signi®cant ¯uctua-
tions from the best ®t model in the region of the PDF
containing structural information, and ®nally the scale factor,
N
m
, is far from the optimal value of unity. Interestingly, the
re®ned structural parameters, aand hUi, are the same, within
the estimated uncertainties, as those determined from the best
PDFs. This illustrates the robustness of the Fourier transform
and the highly constrained modeling in preserving and
extracting the structural information, even in the presence of
signi®cant systematic errors. Nonetheless, it is our objective
here to minimize these errors.
Traditionally, one iteratively adjusts the effective sample
density,
eff
, the effective beam width,
eff
, or both, in such a
way as to make S(Q) asymptote to one at high Q. Generally
this is done `by eye'. The third and fourth columns of Table 5
show the results of varying these traditional parameters, but
here it was done in such a way as to minimize G
low
. In the
third column,
eff
was varied; in the fourth column
eff
was
varied to re¯ect a realistic beam pro®le and then
eff
was
varied. These approaches result in adequate PDFs as
evidenced by the low G
low
and R
wp
values. Use of a more
realistic beam-pro®le results in smaller low-rripples but a
comparable, and slightly worse, R
wp
. However, these correc-
tions did not result in PDFs that had the right scale. This
affected neither R
wp
nor the re®ned structural parameters
because the model contains a re®nable scale factor supporting
the notion that the scale of the data is not closely related to its
`quality', except when model-independent analyses are being
carried out.
The reason for the improper scale is not known. Clearly, it is
a signi®cant effect in the sense that when reasonable values
are used for the known analysis parameters it results in a PDF
that has only 60% of its proper weight. The best quality PDF is
obtained when the asymptotic behavior of S(Q) is best satis-
®ed, but clearly this criterion has been satis®ed with predo-
minantly multiplicative corrections when a mixture of
multiplicative and additive corrections was called for. This
effect can be corrected in an arbitrary way by introducing a
constant additive correction, . Then the data can be corrected
using either
eff
or
eff
and to achieve the proper scale as
well as the proper asymptotic behavior. We ®nally introduce a
correction by multiplicative constant and . This has the
advantage that during the iterative process of optimizing S(Q),
application of and does not require a lengthy re-analysis of
the data. The results are shown in columns 5±7 of Table 5. In
this case the relevant parameters were varied both to minimize
G
low
and to make N
m
= 1. All of the data sets have
Figure 7
Representative PDF (circles), PDFFIT model (line) and difference curve
with 2error bars as dotted lines (offset for clarity) for germanium at
10 K measured using GLAD at IPNS.
comparable or lower G
low
and R
wp
values than those
corrected using traditional methods in columns 3 and 4.
Interestingly, this occurred with uniformly lower values of S
avg
.
The data corrected with the arbitrary constant gave R
wp
values that are indistinguishable from those obtained using the
other methods. All of the different analysis methods resulted
in PDFs yielding identical structural parameters within the
errors.
4. Summary
Errors in scattering data, both random and systematic, have an
effect on the PDF. Here we introduced and tested various
quantitative measures of the quality of a PDF in the presence
of systematic errors. The quality criteria (S
avg
,S
disp
,G
low
and
®t
) were evaluated by comparison with synthetic data with
different known systematic errors introduced. The parameters
S
disp
and
®t
vary proportionally with the scale of the data. This
makes them useful for determining an absolute scale factor for
the data. The parameters S
avg
and G
low
(which is a quanti-
tative measure of the ripples in the PDF in the low-rregion)
are scale invariant and are useful for determining the quality
of the S(Q) and resulting PDF. Either one can be optimized to
yield the best possible PDF from a given data set through
direct Fourier transform. Our tests suggest that G
low
is the
more robust of the two criteria.
Data can be optimized using the above criteria by varying
various process parameters. We have investigated a number of
traditional methods (for example, varying the sample density)
as well as the use of arbitrary multiplicative and additive
parameters. All yield essentially comparable results and
structural parameters that are identical within the estimated
uncertainties. We show that it is the high-Qasymptotic
behavior of S(Q) that is the main determinant of the quality of
the PDF (freedom from arti®cial ripples) regardless of the
scale of the data. If properly scaled data are desired it is
necessary to use both multiplicative and additive corrections
to the data. The straightforward use of additive and multi-
plicative constants gave comparably good results to the
traditional methods, making this a desirable alternative
because of the computational speed.
This work was motivated principally by the advent of next-
generation neutron powder diffractometers that can produce
hundreds of data sets in a single experiment. By automating
the data optimization process it is possible to obtain properly
scaled high-quality PDFs from multiple data sets in an ef®cient
manner. Use of the quality-criteria analysis protocols
described here should help this process enormously. These
procedures have been incorporated into the data analysis
program PDFgetN (Peterson et al., 2000).
APPENDIX A
Finite measurement range
As discussed in the main text, data are always measured over a
®nite range. This appendix determines the effect of ®nite
measurement range on the PDF. In addition, the form of (r)
in equation (6) will be determined. To reiterate equation (6)
G0rGcrÿ1r;17
where G
c
(r) is the PDF convoluted with a termination func-
tion:
Gcr 2
Z
Qmax
Qmin
QSQÿ1sinQrdQ
FQSQÿ1
FQ;Qmin;Qmax
:18
The termination function
Q;Qmin;Qmax 1ifQmin QQmax
0 otherwise
n19
has the Fourier transform
r;Qmin;Qmax FQ;Qmin;Qmax
1
Z
1
0
Q;Qmin;Qmax cosQrdQ
sinQmaxrÿsinQmin r
r
Qmax
j0QmaxrÿQmin
j0Qminr;20
where j
0
(Qr) denotes the zeroth-order spherical Bessel func-
tion, shown for completeness. Then the convoluted PDF can
be written as
GcrGrr;Qmin;Qmax :21
As seen in Fig. 8, the high-frequency oscillation from a ®nite
Q
max
is the dominant effect. For this reason, the modeling
program PDFFIT convolutes with the high-Qtermination
function but does not take into account Q
min
.
The second term in equation (6) vanishes if limQ!1 S0Q=
1, i.e. +=1.If+6 1, the form of (r) becomes
important:
J. Appl. Cryst. (2003). 36, 53±64 Peter F. Peterson et al. Pair distribution function 61
research papers
Figure 8
The termination function r;Qmin;Qmax for Q
min
= 0.9 A
Ê
ÿ1
and Q
max
=
40 A
Ê
ÿ1
. Inset are the Q
min
and Q
max
components r;Qmin;1 in (a) and
r;0;Qmaxin (b).
research papers
62 Peter F. Peterson et al. Pair distribution function J. Appl. Cryst. (2003). 36, 53±64
r 2
Z
Qmax
Qmin
QsinQrdQ
2
Qmin cosQminrÿQmax cosQmax r
r
ÿsinQminrÿsinQmax r
r2
2
Q2
max j1QmaxrÿQ2
min j1Qminr
;22
The function j
1
(Qr) in the de®nition of (r) is the ®rst-order
spherical Bessel function. The behavior of (r) can be seen in
Fig. 9.
In general, and are not Q-independent. By reference to
equation (18) it can readily be seen that, providing (Q) and
(Q) have ®nite Fourier transforms,
G0rFfQg GcrFfQQÿ1gr:23
APPENDIX B
Derivation of quality criteria
In the main text several quality criteria were mentioned. This
appendix will present derivations for those criteria (in the
same order as in the text).
Presented ®rst were the reciprocal-space criteria in equa-
tions (8), (9) and (10). To ®nd the value of S
int
one starts from
the de®nition of the PDF in equation (1), rewritten as
4rrÿ02r
Z
1
0
Q2SQÿ1sinQr
Qr dQ;
22rÿ0Z
1
0
Q2SQÿ1sinQr
Qr dQ:
24
If we then consider the case when r!0, then
220ÿ0Z
1
0
Q2SQÿ1dQ:25
In the derivation of the PDF equation, equation (1), the self-
scattering was neglected (Wagner, 1978) and (0) = 0. Again,
this is because no atoms can be found separated by distances
less than the nn separation. Thus, we ®nd
Z
1
0
Q2SQÿ1dQÿ220:26
This is the Q-space relationship equivalent to equation (34)
and was used to ®nd normalization constants in earlier studies
(Kartha, 1953; Norman, 1957; Krogh-Moe, 1956; Petkov &
Danev, 1998). This integral can be most accurately evaluated if
the coherent intensity in S(Q)atQ
max
is small. Bragg peaks
disappear at this point in crystalline materials.
Then S
avg
and S
disp
can both be found starting from the
de®nition of S(Q),
SQIQ
hbi2ÿhb2iÿhbi2
hbi2; 27
where comes from the compressibility of the sample and is
effectively negligible (Wagner, 1978). The second term in the
sum is called the normalized Laue monotonic diffuse scat-
tering, L. For X-rays, replace the neutron scattering lengths in
equation (27) with X-ray form factors. This de®nition of S(Q)
is a sample-dependent quantity, independent of whether it was
determined from X-ray or neutron diffraction. Since we here
discuss how S(Q) can be optimized by multiplicative and
additive corrections before the Fourier transform, many of the
principles we describe can equally well be applied to S(Q)
functions derived from X-rays or neutrons, though the tests
were applied using neutron data analysis techniques. None-
theless, the quality criteria and process for optimizing S(Q) are
equally valid for X-ray and neutron data. In the high-Qlimit
the scattering is incoherent and the measured intensity, I(Q),
becomes the total sample scattering cross section (per atom),
hb
2
i. Equation (27) then becomes
lim
Q!1 SQhb2i
hbi2ÿL1:28
Taking an average of S(Q) once it has achieved its high-Q
asymptote yields
Savg lim
Q!1hSi1:29
The range of Qover which the average is calculated should
extend down from the maximum Qvalue that will be Fourier
transformed, Q
max
. Ideally it should extend over a range of
data where most of the scattering is incoherent due to Debye±
Waller effects. The Debye±Waller effect comes from the
thermal motion of atoms in the material and damps out the
Bragg peaks in the high-Qregion. In general, there will be
some coherent scattering (since in general Q
max
is chosen
where the coherent signal to noise ratio becomes unfavor-
Figure 9
The function (r) (solid line) for Q
min
= 0.9 A
Ê
ÿ1
and Q
max
=40A
Ê
ÿ1
.The
amplitude of the oscillations at r= 0.9 A
Êis 3 A
Ê
ÿ2
. The broken lines are
shown to demonstrate the decay of the components of (r).
able); however, hSishould still have a value of one if deter-
mined over a suf®ciently wide range of Q.
In the limit of low Q, one can see that, since limQ!0IQ=0
(neglecting the self scattering),
lim
Q!0SQ 0
hbi2ÿLÿL:30
Then it is seen that the dispersion of S(Q) between its low-Q
and high-Qasymptotes must be
Sdisp lim
Q!1 SQÿ lim
Q!0SQ1L:31
Next derived are the real-space quality criteria. Starting
again with equation (1) we can see that
Z
1
0
rGrdr4Z
1
0
r2rÿ0dr:32
Note that the second term in the integral is the volume of an
in®nite sphere times the average number density,
0
, which
will be the total number of atoms, n. Then the integral
becomes
Z
1
0
rGrdr4Z
1
0
r2rdrÿn:33
When integrating r
2
(r), one must remember that (r)isa
correlation function so the integral will be nÿ1 and the atom
in the pair located at the origin is not included in the integral.
Again, this results from the self scattering being ignored. This
leads to
Gint Z
1
0
rGrdrÿ1:34
This relationship can be taken over a limited range of r, but
only in the limit of r!1when (r)!
0
[and G(r)!0] will
it give useful results. For most crystalline materials, while there
is a ®nite crystal size, the distance for the correlations to
approach the average value occurs at very high r(100 A
Ê).
We know that G(r) should have the proper scale. If the
number of nearest neighbors is known in a particular situation,
this can be used to determine the proper scale for G(r) using
Z
rb
ra
4r2rdrX
ij
bibj
hbi2Z
rb
ra
rÿrijdr:35
This is the coordination number, n(r), between r
a
and r
b
weighted by the scattering lengths of the atoms. For mona-
tomic materials, equation (35) reduces to
Z
rb
ra
4r2rdrnrbÿnra:36
This quantity is often known in crystals and ideal network
glasses (Wright et al., 1995). However, for many studies the
coordination number is exactly what is being determined.
In the region where ris less than the nearest-neighbor
distance, r
low
, there are no structural correlations and it is
readily seen from equation (1) that
Grÿ4rN0:37
If the scale factor, N, is unity, this becomes G(r)=ÿ4r
0
.As
we discussed previously, it is exactly in this very low rregion of
G(r) where features appear that can be directly attributed to
systematic errors from experimental uncertainties. Thus,
measuring the mean-square ¯uctuations of the measured G(r)
from equation (37) and summing them in this region below the
nearest-neighbor peak will be a sensitive measure of the
quality of the experimental PDF.
One problem is that, in general, Nis not known precisely a
priori. In the case of crystalline materials (such as in the
current case) a good estimate of Nis possible by ®tting the
crystallographic model to G(r). Then it is possible to estimate
the ¯uctuations from equation (37) using this value of N.In
this case, evaluating R
wp
from the best-®t model in the region
below the ®rst peak could serve this purpose.
A more useful quality criterion does not require prior
knowledge of the structure and can be automated in a
computer, allowing multiple data sets to be evaluated. Here
we propose a model-independent way to determine the extent
of the low-rripples. The philosophy is laid out below and the
practical implementation discussed in the next paragraph.
First, to estimate Nin equation (37) we ®t a straight line, ®xed
at G(0) = 0, through the low-rdata below the low-redge of the
nearest-neighbor peak, r
low
. From this we extract a value
®t
that will take the place of N
0
in equation (37). Then we would
like to estimate the amplitude of the ¯uctuations of the
experimental G(r) from equation (37), for example, by inte-
grating the square of the difference [G
exp
(r)ÿ(ÿ4r
®t
)]
2
over the low-rregion. This is then normalized to make it
dimensionless and scale invariant (i.e. it does not depend on
N). By minimizing this residuals function with respect to
allowed changes in S(Q)(e.g. the parameters and ) the
PDF can be optimized.
In implementing this scheme a number of practical
considerations became apparent. First, we noticed that large
¯uctuations at very low r(e.g. see Fig. 7) biased the linear
®tting and an incorrect
®t
was obtained. This effect was
removed by weighting by r
2
in the ®tting routine to obtain
®t
.
We also note that larger ¯uctuations at very low rthat die out
very quickly are less harmful to the data region of the PDF
than lesser ¯uctuations that, nonetheless, propagate further.
We therefore tried including higher-power rweightings in the
de®nition of the residuals function itself, G
low
. These were
tested by using them to optimize a data set that was then ®t by
a model to obtain R
wp
. The results improved signi®cantly when
an r
1
weighting was introduced, but did not improve signi®-
cantly for higher rweightings. We therefore chose to de®ne
G
low
as follows in the practical implementation:
J. Appl. Cryst. (2003). 36, 53±64 Peter F. Peterson et al. Pair distribution function 63
research papers
research papers
64 Peter F. Peterson et al. Pair distribution function J. Appl. Cryst. (2003). 36, 53±64
Glow R
rlow
0
rGr4r2fit
2dr
R
rlow
0
4r2fit
2dr
:38
The authors would like to thank V. Petkov for informative
discussions. ESB thanks V. Levashov for discussions. This
work has bene®tted from the use of the Intense Pulsed
Neutron Source at Argonne National Laboratory. This facility
is funded by the US Department of Energy, BES-Materials
Science, under Contract W-31-109-ENG-38. Los Alamos
National Laboratory is funded by the US Department of
Energy under contract W-7405-ENG-36.
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