Download Obtaining structural information from the atomic pair

130
Z. Kristallogr. 219 (2004) 130–135
# by Oldenbourg Wissenschaftsverlag, München
Obtaining structural information
from the atomic pair distribution function
Thomas Proffen* and Katharine L. Page
Los Alamos National Laboratory, Lujan Center, MS H805, Los Alamos, NM 87545, U.S.A.
Received October 30, 2003, accepted November 12, 2003
Dedicated to Takeshi Egami, recipient of the 2003 Warren Award
Powder diffraction / Medium range order /
Pair distribution function / Disorder / Diffuse scattering
Abstract. The knowledge of the detailed atomic structure of modern materials is the key to understanding the
their macroscopic properties. The atomic pair distribution
function (PDF) reveals short-range and medium-range
structural information. In this paper we present an overview of refinement and modelling techniques. In short, we
will be trying to answer the question: What can I learn
from my PDF?
Introduction
The papers in this section of Zeitschrift für Kristallographie are dedicated to Takeshi Egami, winner of this years
Warren Award presented by the American Crystallographic
Association. The award honors Takeshi’s contribution to
structural analysis of disordered materials using the atomic
pair distribution function (PDF). It is timely that this collection of papers coincides with the publishing of a text
book on the subject by Egami and Billinge (2003).
The power of the PDF technique is the fact that it uses
the complete diffraction pattern, i.e. diffuse and Bragg
scattering, and reveals information about the local and
medium range structure of the material. The method has is
origins in the study of glasses and liquids. The application
of this method to disordered crystalline materials was largely pioneered by Takeshi Egami and is now developing
into a more widely used characterization tool for materials, crystalline or otherwise. Most recently, Takeshi Egami
was the principle investigator (Fig. 1) of the upgrade the
Neutron Powder DiFfractometer (NPDF) at the Lujan
Neutron Scattering Center (Proffen et al., 2002a). The instrument NPDF is dedicated to the study of disordered
and nanocrystalline materials using the PDF method and
can be called the best instrument of its kind in the world.
With advances in instrumentation and data collection,
we want to address the question in this paper how to ex-
* Correspondence author (e-mail: [email protected])
tract structural information from a PDF using some very
simple and straight forward examples. Keeping this the
focus of our paper, we will concentrate on our own work
and modelling software. The experimental PDF is the
Fourier transform of the normalized X-ray or neutron scattering intensities (for details see Egami and Billinge
(2003), Proffen et al. (2003)). For the purpose of this paper, we start assuming an experimental PDF was successfully obtained. Some local structural information can simply be obtained by inspection. The PDF is a measure of
the probability of finding an atom at a distance r from
another atom. In a study by Billinge et al. (1996), the
authors used the temperature dependence of the oxygenoxygen PDF peak height to demonstrate the existence and
Fig. 1. Sponsors of the upgrade of the Neutron Powder DiFfractometer at the Lujan Center at Los Alamos National Laboratory. Top
row from the left: Takeshi Egami (University of Tennessee) and Despina Louca (University of Virginia). Bottom row from the left: Thomas Proffen and Simon Billinge (Michigan State University). Investigators not shown are Tony Cheetham (University of California, Santa
Barbara) and John Parise (State University of New York at Stony
Brook).
Unauthenticated
Download Date | 8/4/17 2:02 AM
131
Structural information from the pair distribution function
behavior of polarons in calcium doped manganites. Simple
peak fitting will reveal nearest neighbor bond distances
and the integral of a suitably normalized PDF corresponds
to the coordination number. A simple analysis of the peak
width dependence on the distance r reveals information
about correlated motion (Jeong et al., 1999). However, in
many cases the comparison or even refinement of a structural model will be desired. As mentioned before, the PDF
is simply the bond length distribution of the material
weighted by the respective scattering powers of the contributing atoms. The PDF can be calculated from a structural
model using the relation
"
#
1 P P bi bj
Gc ðrÞ ¼
dðr rij Þ 4prr0 ;
ð1Þ
r i j hbi2
where the sum goes over all pairs of atoms i and j within
the model crystal separated by rij. The scattering power of
atom i is bi and hbi is the average scattering power of the
sample.
Using the relation above, the PDF can be calculated
and tested against an average structural model or the result
of molecular dynamics simulations. This allows one to test
those models, however, often a refinement of that model is
desired. In this paper we will focus on two different cases:
(1) The structural model contains a large number of atoms
and we use Reverse Monte Carlo (RMC) to create a configuration agreeing with the observed PDF. (2) The model
can be described with just a few atoms and the structural
parameters are refined using a least-square algorithm. The
following sections describe these two approaches and give
examples. A good resource for more information as well
as to obtain the programs mentioned in this paper is Billinge (2001).
Reverse Monte Carlo simulations
In this section we deal with modelling that requires a
large model structure, typically containing tens of thousands of atoms or even more. The Reverse Monte Carlo
(RMC) technique used to create a structure consistent with
the observations can be understood as simply “shaking”
the model structure and keeping good configuration. The
RMC method was originally developed by McGreevy and
Pusztai (1988) and a comprehensive summary of the method applied to crystalline materials is given in Tucker et al.
(2001a). The RMC method minimizes the difference between observed and calculated data (in our case PDFs).
The algorithm in more detail is schematically shown in
Fig. 2 and works as follows: First, the PDF is calculated
from the chosen crystal starting configuration and a goodness-of-fit parameter c2 is computed.
c2 ¼
N ðG ðr Þ G ðr ÞÞ2
P
e i
c i
:
s2
i¼1
ð2Þ
The sum is over all measured data points ri , Ge stands for
the experimental and Gc for the calculated PDF. The RMC
simulation proceeds with the selection of a random site
within the crystal. The system variables associated with
this site, such as occupancy, sðiÞ or displacement, rðiÞ, are
Fig. 2. Schematic flow diagram of RMC simulation. For details see
text.
changed by a random amount, and then the model PDF
and the goodness-of-fit parameter c2 are recalculated. The
change Dc2 of the goodness-of-fit c2 before and after the
generated move is computed. Every move which improves
the fit (Dc2 < 0) is accepted. ‘Bad’ moves worsening the
agreement between the observed and calculated PDF are
accepted with a probability of P ¼ exp ðDc2 =2Þ. As the
value of Dc2 is proportional to 1=s 2 , the value of s has
an influence on the amount of ‘bad’ moves which will be
accepted. Obviously there are two extremes: For very
large values of s, the experimental data are ignored
(c2 0) and with very small values of s the fit ends up
in the local minimum closest to the starting point, because
there is a negligible probability for ‘bad’ moves. The
RMC process is repeated until c2 converges to its minimum. The result of a successful RMC refinement is one
real space structure which is consistent with the observed
PDF. In order to exclude chemically implausible resulting
structures additional constrains, e.g. minimal allowed distances between atoms, may be introduced.
As example to illustrate the RMC method, we will consider a sample with chemical short range order (SRO).
Further details of this example can be found in Proffen
(2000). We will extract chemical ordering information
from a simulated PDF, which we will refer to as ‘data’.
This allows one to test the capability of the PDF method
to extract such information for crystalline samples. The
RMC simulations were carried out using the program DISCUS (Proffen and Neder, 1997). Figure 3a shows a section of the disordered structure used to calculate the “experimental” data. The SRO chosen is characterized by a
negative correlation in the 11-direction which results in a
local superstructure as indicated by the circle in Fig. 3a.
The calculated PDFs for this configuration and a random
distribution are shown in the bottom panel of Fig. 3a as
solid and dashed line, respectively. The difference curve is
plotted below the PDFs and one can see the “signal” that
carries the SRO information. Starting from a random configuration, a RMC simulation was carried out, allowing
the two different atom types to order. The resulting calculated PDF and the input PDF are shown in the top panel
Unauthenticated
Download Date | 8/4/17 2:02 AM
132
T. Proffen and K. L. Page
revealing the structure on different length scales. It is important to consider the range in r used for the refinement
when evaluating a structural result. In this section we will
first discuss an example of short range order and then extend the refinements to medium range order using a simulated domain structure as example. All refinements in this
section were carried out using the program PDFFIT
(Proffen and Billinge, 1999).
a
b
Short range structure
c
Fig. 3. Simulation of chemical SRO: (a) shows the structure corresponding to the simulated “experimental” data. (b) is the result of the
RMC refinement. (c) The top panel shows the experimental PDF and
the RMC result. The difference is show below the curves. The bottom panel shows the PDF of the structure showing SRO versus a
random distribution of atoms. The difference is shown below the
curves.
of Fig. 3c. The flat difference curve shows the excellent
agreement. A section of the resulting structure after the
RMC refinement is shown in Fig. 3b. A similar arrangement of the two atom types is observed. This demonstrates the ability of the RMC method to extract disorder
information from the PDF. In more complicated cases,
however, it is important to apply sensible physical and
chemical constraints to the system to avoid unlikely resulting configurations. For a much more detailed discussion
of these simulation as well as discussion of systems with
additional distortions, please refer our earlier paper Proffen (2000).
As an example for a refinement of the near neighbor region of a PDF, we choose the semiconductor alloy
ZnSe1x Tex . These materials are of great interest since the
electronic band gap can be tuned via the doping level
x. It is evidently very important to have accurate local
structural information of these systems. Details of this
study can be found in Peterson et al. (2001). For the
purpose of this paper, we concentrate on the nearest
neighbor (NN) peak. Both endmembers ZnTe and ZnSe
have the zinc-blende structure (F 43m) where the Zn
atoms and Te, Se atoms occupy the two interpenetrating face-centered-cubic (fcc) lattices. The NN bond
lengths found in the end-member compounds are
L0ZnTe ¼ 2.643(2) A and L0ZnSe ¼ 2.452(2) A. Obtaining
scattering data out to a large momentum transfer, Q, allows one to resolve this bond length difference of only
Dr ¼ 0:11 A in the PDF. The near neighbor region of a
PDF of ZnSe0:5 Te0:5 measured on NPDF at T ¼ 15 K is
shown in Fig. 4 as filled circles. One can clearly see the
two NN bond distances corresponding to Zn––Se and
Zn––Te distances. Two models were refined. First a model
based on the average structure. Lattice parameters and
atomic displacement parameters were refined. The result is
shown in Fig. 4 as dashed line. Although the refinement
shows a good agreement for the second and third PDF
peak, the refinement does not show the split of the first
peak. This, of course, is not surprising, since the average
structure is constraint to have only one NN bond length.
Real space Rietveld
As discussed in the introduction, an alternative way to obtain structural information from the PDF is the refinement
of parameters from a relatively small structural structural
model. This has been referred to as Real-Space-Rietveld
(Billinge, 1998). The name points out the similarity with
conventional Rietveld refinement (Rietveld, 1969) in terms
of the structural parameters refined such as lattice parameters, atomic position and atomic displacement parameters. However, one needs to take care not to take the
similarity any further. Rietveld refinements use Bragg
peaks and an arbitrary background function and the result
will be the average structure of the material. PDF analysis
on the other hand is based on the total scattering pattern,
Fig. 4. Refinement of the PDF of ZnSe0:5 Te0:5 . The data are shown
as filled circles. The refinement of the average and the distorted structural model are shown as dashed and solid lines, respectively. For
details see text.
Unauthenticated
Download Date | 8/4/17 2:02 AM
133
Structural information from the pair distribution function
The second model adds h111i displacements to the Zn as
well as Se/Te atoms. On order to allow different NN bond
length, we need to specify the sites the Se and Zn atoms
occupy rather than using an occupancy of 0.5. We retain
the original unit cell and place the atoms on their specific
sites and allow a displacement from the average atomic
sites in h111i directions. The refinement result is shown as
solid line in Fig. 4. The resulting distortions are
xZn ¼ 0:06 A and xSe; Te ¼ 0:03 A. For this model, the
split of the NN peak shows up in the refined PDF (Fig. 4).
It should be noted that the refinement range r does not
exceed the size of the unit cell. Refining over a wider
range using this simple model is not sensible, since we
would create a long range ordered crystal using the modified unit cell which is clearly not consistent with the diffraction pattern. A similar type of refinement of X-ray
PDF data of semiconductor alloys Inx Ga1x As can be
found in Petkov and Billinge (2001). In the next section
we use the fact that the refinement models a certain distance range in the PDF and discuss what information can
be extracted in case of domain structures.
Medium range structure
In order to investigate what information the PDF can reveal in case of domain structures, we use the PDF calculated from a simulated domain structure as “experimental”
data. In our case, the matrix consists of a cubic lattice of
atoms M with a lattice parameter of a ¼ 3 A. The model
contains spherical domains containing atoms D on the
same lattice, coherently imbedded in the matrix structure.
The diameter of the domains is d ¼ 15 A. The overall
composition of the model structure is 85% M and 15% D
atoms. The domain structure as well as the resulting PDF
were created using the program DISCUS (Proffen and Neder, 1997). All subsequent refinements of the simulated
data were carried out with the program PDFFIT as in the
last section. The structural model is refined is the same as
the matrix structure, only each site is occupied by atoms
M and D and the occupancy parameter, oM , is refined.
The first refinement was carried out over a range or
2:5 < r < 27:5 A. The refined occupancy is oM ¼ 0:85
just as expected. The refinement is shown in Fig. 5 in the
top panel as solid line. The data are represented as filled
circles and the difference between data and calculated
PDF is displayed below. The overall R-value for this refinement is R ¼ 0:21% 2. Inspection of the top panel of
Fig. 5 shows residuals between calculated PDF and the
data, mainly at distances r < 7:5 A. Note, that the difference curves in the Figure are enlarged by a factor of fifty
for clarity. Below the difference curve, R-values for individual sections with a width of Dr ¼ 5 A are listed. As already obvious from the difference curve, the agreement up
to r ¼ 7:5 A is the worse. This can be understood considering that at distances smaller than the radius of the do-
2
In the context of this paper we use the R-value only as relative
measure to compare refinements. No attempts have been made to
compute sensible standard deviations for the simulated data.
mains (d=2 ¼ 7:5 A) more B––B pairs are sampled than in
the random case. At distances larger than r ¼ 7:5 A only
M––M and M––D pair are sampled. We neglect at this
point correlations M––M from different domains. Since the
refinement range chosen extends to r ¼ 27:5 A, the model
assumes the correct overall composition at the expense of
worse agreement at low r.
However, we are free to choose the refinement range
and rather than using the complete range of the PDF in a
single refinement, we refine sections of the PDF individually. A set of five refinements over a range of Dr ¼ 5 A
each is shown in the bottom panel of Fig. 5. The refinement ranges are indicated by the vertical dotted lines.
Again, the data are represented as filled circles and the
refinement result as solid line. The difference and individual R-values shown below the data indicate a much better overall agreement. For the first range up to r ¼ 7:5 A
we see the biggest improvement. The refined occupancy is
lower, oM ¼ 0:71 and the R-value has significantly improved. Observing a lower occupancy of atom type M or
a higher occupancy of atom type D is consistent with the
view that sampling with distances smaller than the domain
radius yields more D––D pairs. For all other refinement
ranges we find the overall occupancy of oM ¼ 0:85. It
should also be noted that the agreement at higher values
of r in this second refinement is somewhat better compared to the refinement of the complete PDF range.
As we have seen, refining the PDF in sections, we can
extract structural information as a function of length scale.
In this simple example, it was evident from the agreement
of the data with a simple model, that there were structural
differences when sampling with distances below and
Fig. 5. PDF refinement of a simulated structure containing domains.
The top panel shows the result of a refinement of the complete range.
The bottom panel shows refinement results of individual regions, all
with a width of D ¼ 5
A. The R-values are shown for each region.
For comparison R-values for the same regions are calculated and displayed in the top panel. In both cases, the “experimental” data are
shown as filled circles and the calculated result as solid line. The
difference between observed and calculated values multiplied by a
factor of 50 is shown below the curves in both panels. For mode
details see text.
Unauthenticated
Download Date | 8/4/17 2:02 AM
134
T. Proffen and K. L. Page
above r ¼ 7:5 A. This is of course exactly the radius of
the domains we have created in our model used to create
the input data. This is a very simple demonstration and
more complex models could be constructed using the multi phase refinement capabilities of PDFFIT. However,
more complicated modelling was outside the scope (and
submission deadline) of this paper. An nice example of
refining two distinct ranges of the PDF of a Xerogel is
Petkov et al. (2002).
Future
Diffraction data contain a great deal of information. Bragg
reflection data reveal the average structure of crystalline
materials with increasing complexity and accuracy. But the
diffraction data contain even more information in form of
diffuse scattering allowing one to study the short and medium range order in materials. A recent review (Proffen
et al., 2003) lists a number of studies employing the PDF
technique. The ability to refine a structural model is key
to this type of studies. In cases of “bad” crystals the
Bragg information might not be usable to extract the
atomic structure from the diffraction data. In cases of nanocrystals, the extend of the long range order is not sufficient to produce sharp Bragg reflections. In those cases a
simple refinement of a structural model using e.g. PDFFIT
is a very simple and straight forward way to obtain the
local structure of the material. With the growing number
of compounds that show drastically different properties
when present as nanoparticles compared to as bulk materials, the PDF method and structural refinements as we have
discussed in this paper will soon become an important and
routine characterization tool. More extended disorder can
be understood using RMC refinements (see e.g. Tucker
et al. (2001b) or Proffen et al. (2002b)). RMC methods are
also the main simulation route to model PDF data from
liquids or glasses (Petri et al., 2000).
Another application of PDF analysis is the study of
domain structures via r dependent structural refinements
of the PDF similar to the simple example discussed in the
last section. One should keep in mind that on an instrument such as NPDF, PDFs extending to distances in excess of r ¼ 150 A can be obtained. The simulations presented here are just a first step of a systematic exploration
a
b
Fig. 6. (a) Section of the simulated structure containing domains.
The matrix (M) atoms are shown as empty circles, the domain (D)
atoms as filled circles. (b) This panel shows the fraction of M-M, MD and D-D pairs occurring in the domain structure shown in (a) as a
function of distance r. For details see text.
by the authors to advance refinement techniques for systems containing such nanodomains. A section of the simulated structure used in the last section is shown in Fig. 6a.
The matrix atoms M are shown as open circles and the
domain atoms D are shown as filled circles. As a representation of the matrix-matrix (M––M), domain-domain
(D––D) and matrix-domain (M––D) contribution to the
PDF, the fraction of each type of atom pairs (M––M,
D––D and M––D) as a function of sampling distance r are
shown in Fig. 6b. If one would construct a model for the
matrix, domain and domain boundary structure, a refinement of these three phases to the observed PDF should be
possible. The phase fraction obtained in the different refined regions of the PDF contain information about domain size and domain-domain correlations. Starting with
short distances, the PDF is dominated by M––M and D––D
contributions. As r increases beyond the domain size, the
contribution of D––D decreases as seen in Fig. 5b. For larger domains this approach seems much more feasible than
RMC simulations of an enormous model system since the
structural parameters are limited to the matrix and domain
structures as well as a model for the domain boundary.
More studies of model systems are refinement techniques
or more complex nanodomain structures are currently under way.
Acknowledgments. This work has benefited from the use of the Lujan
Center at Los Alamos Neutron Science Center, funded by Department
of Energy Office of Basic Energy Sciences and Los Alamos National
Laboratory funded by Department of Energy under contract W-7405ENG-36. Katharine P. acknowledges her support by NASA through
NASA-USRP.
References
Billinge, S. J. L.: Real-Space Rietveld: Full Profile Structural Refinement of the Atomic Pair Distribution Function. In: Local structure
from Diffraction (S. J. L. Billinge; M. F. Thorpe, Eds.) page 137.
Kluwer, New York 1998.
Billinge, S. J. L.: The total scattering homepage 2001.
http://www.totalscattering.org/.
Billinge, S. J. L.; DiFrancesco, R. G.; Kwei, G. H.; Neumeier, J. J.;
Thompson, J. D.: Direct observation of lattice polaron formation
in the local structure of La1 xCaxMnO3. Phys. Rev. Lett. 77
(1996) 715–718.
Egami, T.; Billinge, S. J. L.: Underneath the Bragg-Peaks: Structural
Analysis of Complex Materials. Elsevier Science B.V. Amsterdam
2003.
Jeong, I.-K.; Proffen, Th.; Mohiuddin-Jacobs, F.; Billinge, S. J. L.:
Measuring Correlated Atomic Motion using X-Ray Diffraction. J.
Phys. Chem. A103 (1999) 921–924.
McGreevy, R. L.; Pusztai, L.: Reverse Monte Carlo Simulation: a
New Technique for the Determination of Disordered Structures.
Mol. Simul. 1 (1988) 359–367.
Peterson, P. F.; Proffen, Th.; Jeong, I.-K.; Billinge, S. J. L.; Choi, K.S.; Kanatzidis, M. G.; Radaelli, P. G.: Local Atomic Strain in
ZnSe1 xTex from High Real Space Resolution Neutron Pair Distribution Function Measurements. Phys. Rev. B63 (2001) 165211.
Petkov, V.; Billinge, S. J. L.: Local structure of random Inx Ga1x As
alloys by full-profile fitting of the atomic pair distribution function. Physica B305 (2001) 83–89.
Petkov, V.; Trikalitis, P. N.; Božin, E. S.; Billinge, S. J. L.; Vogt, T.;
Kanatzidis, M. G.: Structure of V2O5 n H2O xerogel solved by
the atomic pair distribution function technique. J. Am. Chem.
Soc. 124 (2002) 10157–10162.
Unauthenticated
Download Date | 8/4/17 2:02 AM
135
Structural information from the pair distribution function
Petri, I.; Salmon. P. S.; Fisher, H. E.: Defects in a Disordered World:
The Structure of Glassy GeSe2. Phys. Rev. Lett. 84 (2000) 2413–
2416.
Proffen, Th.: Analysis of Occupational and Displacive Disorder using
the Atomic Pair Distribution Function: a Systematic Investigation.
Z. Kristallogr. 215 (2000) 661–668.
Proffen, Th.; Billinge, S. J. L.: PDFFIT, a Program for Full Profile
Structural Refinement of the Atomic Pair Distribution Function. J.
Appl. Cryst. 32 (1999) 572–575.
Proffen, Th.; Billinge, S. J. L.; Egami, T.; Louca, D.: Structural analysis of complex materials using the atomic pair distribution function – a practical guide. Z. Kristallogr. 218 (2003) 132–143.
Proffen, Th.; Egami, T.; Billinge, S. J. L.; Cheetham, A. K.; Louca,
D.; Parise, J. B.: Building a high resolution total scattering powder diffractometer – Upgrade of NPD at MLNSC. Appl. Phys. A
74 (2002) S163–S165.
Proffen, Th.; Neder, R. B.: DISCUS, a Program for Diffuse Scattering and Defect Structure Simulations. J. Appl. Cryst. 30 (1997)
171–175.
Proffen, Th.; Petkov, V.; Billinge, S. J. L.; Vogt, T.: Chemical short
range order obtained from the atomic pair distribution function. Z.
Kristallogr. 217 (2002) 47–50.
Rietveld, H. M.: A Profile Refinement Method for Nuclear and Magnetic Structures. J. Appl. Crystallogr. 2 (1969) 65.
Tucker, M. G.; Dove, M. T.; Keen, D. A.: Application of the Reverse
Monte Carlo Method to Crystalline Materials. J. Appl. Cryst. 34
(2001) 630–638.
Tucker, M. G.; Squires, M. P.; Dove, M. T.; Keen, D. A.: Dynamic
structural disorder in cristobalite: neutron total scattering measurement and reverse Monte Carlo modelling. J. Phys. Cond. Matt. 13
(2001) 403–623.
Unauthenticated
Download Date | 8/4/17 2:02 AM