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Robust structure and morphology parameters for CdS
nanoparticles by combining small-angle X-ray scattering and
atomic pair distribution function data in a complex modeling
framework
Christopher L. Farrow, Chenyang Shi, Pavol Juhás, Xiaogang Peng and
Simon J. L. Billinge
J. Appl. Cryst. (2014). 47, 561–565
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J. Appl. Cryst. (2014). 47, 561–565
Christopher L. Farrow et al. · Robust structure and morphology parameters for CdS
research papers
Journal of
Applied
Crystallography
ISSN 1600-5767
Robust structure and morphology parameters for
CdS nanoparticles by combining small-angle X-ray
scattering and atomic pair distribution function
data in a complex modeling framework
Received 21 September 2013
Accepted 17 December 2013
Christopher L. Farrow,a Chenyang Shi,b Pavol Juhás,b Xiaogang Pengc and
Simon J. L. Billingeb,c*
a
Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY
10027, USA, bCondensed Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, NY 11973, USA, and cDepartment of Chemistry, Zhejiang University,
Hangzhou, Zhejiang 310027, People’s Republic of China. Correspondence e-mail:
sb2896@columbia.edu
# 2014 International Union of Crystallography
In this work, the concept of complex modeling (CM) is tested by carrying out a
co-refinement of the atomic pair distribution function and small-angle X-ray
scattering data from CdS nanoparticles. It is shown that, compared with either
single technique alone, the CM approach yields a more accurate and robust
structural insight into the atomic structure and morphology of nanoparticles.
This work opens the door for the application of CM to a wider class of
nanomaterials and for the incorporation of additional experimental and
theoretical techniques into these studies.
1. Introduction
The atomic pair distribution function (PDF) of X-ray and
neutron powder diffraction is a powerful tool for the analysis
of the structure of nanoparticles (Egami & Billinge, 2013; Du
et al., 2012; Juhás et al., 2006; Gilbert et al., 2004; Masadeh et
al., 2007). With the advent of synchrotron X-ray techniques
and advanced computing power, the PDF yields structural
information at the ångström level, which is particularly
important for nanocrystalline and poorly crystallized materials
and is gaining popularity in the scientific community (Shyam et
al., 2012; Božin et al., 2010; Dykhne et al., 2011; Michel et al.,
2007; Page et al., 2008; Polking et al., 2012). However, in many
practical cases, the PDF may not contain enough information
to constrain the inverse problem resulting in a unique nanoparticle structure. In this case, a ‘complex modeling’ (CM)
approach (Billinge & Levin, 2007) is needed that incorporates
extra information in the form of both theory and multiple
complementary experimental data sets, such as from transmission electron microscopy (TEM), extended X-ray absorption fine structure (EXAFS), small-angle X-ray scattering
(SAXS) and Raman spectroscopy, in a combined (or
complexed) optimization. There are some steps being taken
towards this goal with attempts to add diffraction (Tucker et
al., 2007) and EXAFS (Krayzman et al., 2008) to PDF data in a
reverse Monte Carlo style ‘big-box’ modeling scheme. To do
this systematically and flexibly is a theoretical and computational challenge but is necessary in problems at the frontier of
materials complexity (Bennett et al., 2010; Cliffe et al., 2010;
Hwang et al., 2012; Corr et al., 2010; Goodwin et al., 2010).
Here we present a first step towards a combined refinement of
J. Appl. Cryst. (2014). 47, 561–565
PDF and SAXS data from CdS nanoparticles using a recently
written CM framework, SrFit (C. L. Farrow, P. Juhas & S. J. L.
Billinge, 2010, unpublished). We show that indeed more
robust structural parameters are obtained this way, demonstrating the viability of this approach.
In a bulk material one does not consider the sample size or
shape because of its macroscopic size in all directions. Crystallography reveals information about the intrinsic atomic
geometry. However, in a nanomaterial knowledge of both
atomic structure and also particle shape and size are important. The PDF contains information about particle size and
shape as well as its internal structure (Egami & Billinge, 2013;
Masadeh et al., 2007). Practically, however, the PDF is insensitive to information about morphology. The PDF yields the
range of coherence of the structure which, in general, may be
smaller than the actual particle size (Egami & Billinge, 2013;
Zhang et al., 2003). The refinement of particle size and shape
from the PDF may also not be very robust owing, for example,
to parameter correlations and a lack of information about
particle size distributions (Shi et al., 2013). On the other hand,
SAXS gives the shape and size of nanoparticles with good
precision but contains no information at the atomic scale
(Glatter & Kratky, 1982). In order to obtain precise information for both, it is preferable to use a CM approach such as
a combination of SAXS and PDF. Recently the theoretical
relationship between the two techniques has been established
(Farrow & Billinge, 2009). Estimates of nanoparticle shape
and size from SAXS data have been compared with PDF data
to provide insight into nanoparticle disorder (Gilbert et al.,
2004), but SAXS is not commonly used to corroborate PDF
doi:10.1107/S1600576713034055
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561
research papers
data and has never been co-refined with PDF data in a CM
scenario, before this study. However, with the advent of
equipment that facilitates the collection of SAXS with wideangle X-ray scattering data suitable for PDF work (Daniels et
al., 2010) this approach can be expected to become more
common in the future. This software development will
increase the power of such measurements.
2. Experimental methods
Here we study the structure of CdS nanoparticles that were
synthesized and purified using the method reported by Yu &
Peng (2002). A TEM image of the nanoparticles is shown in
Fig. 1.
The CdS nanoparticles are evident as darker regions in
some of which it is possible to resolve lattice fringes. From the
image, they have a rather uniform size distribution and appear
to have a slightly spheroidal shape with the largest dimension
around 3 nm and an aspect ratio between 1:1 (spherical) and
2:1 (slightly elongated).
X-ray PDF data were collected at beamline 11-ID-C of the
Advanced Photon Source at Argonne National Laboratory.
The PDF data were measured in a flowing nitrogen cryocooler
at a temperature of 100 K using the rapid acquisition pair
distribution function (RaPDF) method (Chupas et al., 2003)
with an X-ray wavelength of 0.10798 Å and a sample-todetector distance of 391.1 mm. A large area two-dimensional
fast readout Perkin Elmer image plate detector was used. The
raw two-dimensional data were azimuthally integrated and
converted to one-dimensional intensity versus 2 using FIT2D
(Hammersley et al., 1996). An empty capillary was measured
for background subtraction. PDFgetX3 (Juhas et al., 2013) was
used to correct and normalize the diffraction data and to
Fourier transform
R Qmax them to obtain the PDF, G(r), according to
Q½SðQÞ 1 sin Qr dQ. Here Q is the
GðrÞ ¼ ð2=Þ Qmin
magnitude of the momentum transfer on scattering and S(Q)
is the properly corrected and normalized powder diffraction
intensity measured from Qmin to Qmax (Egami & Billinge,
2013). G(r) gives the probability of finding a pair of atoms
separated by a distance r,
GðrÞ ¼ 4r½ðrÞ 0 0 ðrÞ;
where ðrÞ is the atomic pair density, 0 is the average atomic
number density and 0 ðrÞ is the characteristic function for the
sample shape (Farrow & Billinge, 2009). The characteristic
function is the autocorrelation of the shape function of the
sample, which is a geometric function that has value unity
inside the (average) nanoparticle and value zero outside. For
isotropically scattering samples such as powders of nanoparticles, we use the orientationally averaged characteristic
function (Gilbert, 2008; Farrow & Billinge, 2009). This quantity is obtained directly from SAXS (Guinier, 1963) and in the
PDF it describes the attenuation of a nanoparticle PDF with
increasing r (Masadeh et al., 2007; Gilbert, 2008; Farrow &
Billinge, 2009). Before we can extract accurate values for the
particle size it is necessary to correct for the finite resolution of
the measurement. Both standard nickel and cerium oxide
samples were measured to extract Qdamp, which is the parameter that corrects the PDF envelope function for instrument
resolution effects (Proffen & Billinge, 1999; Farrow et al.,
2007). The standard approach was used based on the knowledge that the calibration samples are crystalline and so any
fall-off in PDF peak intensity with increasing r from these
samples is due to the experimental resolution. The particle size
in the fit is set to infinity and the Qdamp resolution parameter is
varied. The refined value of Qdamp = 0.0396 Å1 is then fixed in
model refinements to the nanoparticle data.
The PDF of the CdS nanoparticle was calculated theoretically from atomic models by multiplying the bulk Gbulk(r) by a
spheroidal characteristic function (r) (Gilbert, 2008; Farrow
& Billinge, 2009; Masadeh et al., 2007; Yang et al., 2013):
Gnano ðrÞ ¼ ðrÞGbulk ðrÞ:
Figure 1
(a) A TEM image of a representative region of the CdS nanoparticle
sample. (b), (c) Spheroidal morphology and atomic structure of CdS
nanoparticles obtained (b) from a PDF fit and (c) from the PDF–SAXS
co-refinement. Cd is shown as gray and S as blue in the image.
562
Christopher L. Farrow et al.
ð1Þ
ð2Þ
Here Gbulk(r) is a PDF calculated using periodic boundary
conditions and the model used for the bulk structure was the
one appropriate for II–VI nanoparticles that contain significant numbers of stacking faults. As discussed by Yang et al.
(2013), the local structure in these compounds is well
explained by a model that mixes wurtzite and zinc blende
PDFs with a mixing fraction that may be refined, but which
depends on the stacking fault density. In this approach
isotropic atomic displacement parameters are used and the
lattice parameters of the hexagonal wurtzite and cubic zinc
blende structures are constrained to be consistent. The
modeling was carried out using the SrFit program.
The SAXS data of the same samples were collected at
beamline X10A of the National Synchrotron Light Source
(NSLS) at Brookhaven National Laboratory. A Bruker
Robust structure and morphology parameters for CdS
electronic reprint
J. Appl. Cryst. (2014). 47, 561–565
research papers
SMART 1500 CCD detector (10 10 cm, 1024 1024 pixels)
was placed 695.8 mm from the sample. The incident beam had
a wavelength of 1.095 Å. The samples were exposed for 10 min
at room temperature to obtain a two-dimensional diffraction
image. Similar to the PDF, the raw two-dimensional data were
azimuthally integrated and converted to one-dimensional
intensity versus 2 curves using FIT2D. The empty capillary
was measured to determine the background signal.
The SAXS data are shown in Fig. 2. In order to ensure that
we measured the same sample under the same conditions, the
sample was measured as a solid aggregate. As a result there is
a pronounced interparticle interference function, Sp(Q), that
results in a well defined correlation peak in the data at around
0.18 Å1. The actual measured signal, I(Q), is then the
product of Sp(Q) and P(Q), where Sp(Q) is an analytic function appropriate for a disordered hard-sphere structure model
and P(Q) is the Fourier transform of the orientationally
averaged characteristic function (Glatter & Kratky, 1982). In
detail, IðQÞ ¼ PðQÞSp ðQÞ þ BEðQÞ, where EðQÞ is the
experimentally measured empty capillary SAXS profile that
was scaled by a prefactor B during the fit to the SAXS data.
One should note that the strong interparticle correlation,
Sp(Q), makes the precise determination of the SAXS form
factor fit difficult. This is a drawback of measuring the SAXS
data on the same sample as the PDF was measured on. One
motivation was to test the principle that the PDF and SAXS
data could be collected from the same sample and combined in
a complex. The approach establishes a proof of principle for
this method, but SAXS data from dilute samples are preferred
for the most accurate SAXS information.
3. Complex modeling of PDF and SAXS data
Co-refinement of the PDF and SAXS data is carried out by
constructing a ‘complex’ in the SrFit CM framework, as shown
schematically in Fig. 3. The structural model is specified by
parameters. Additional parameters that incorporate morphological, thermal and experimental effects are also included in
the complex. The parameters are linked to variables which are
passed to the optimizer. Constraints may be applied to individual variables and to connect different parameters and
variables. Constraints may be applied within a single calculated function, or ‘leaf’, of the complex. Also, parameters in
different leaves may be constrained together. For example, in
this case the nanoparticle major and minor axes were
constrained to be the same in the SAXS and PDF calculations.
The SrFit implementation is highly modular and makes heavy
use of object-oriented software design, which makes it highly
flexible and easily extensible and thus ideal for an application
such as the heterogeneous refinement described here. This
example demonstrates the utility of the complex modeling
approach for increasing the robustness of structure models.
For this study it was necessary to implement a new SAXS
function calculator in SrFit, which calculates the SAXS
intensity given a morphological model that is shared between
the SAXS and PDF leaves of the complex. The parameters
that the SAXS leaf uses are the major and minor axes of the
spheroidal nanoparticle shape, as well as mean particle
separation and effective radius that goes into the Sp(Q)
calculation. Additionally, there were independent scale factors
for the sample and for the subtraction of the measured
background. The spheroid parameters are shared with the
PDF calculator but the particle packing information is not.
SrFit was configured to use differential evolution (DE) as a
regression algorithm. In DE, populations of candidate structure solutions are retained and combined with each other (the
analog of reproduction) as well as mutated. In this case, we
retained a population of 160 individuals (ten times the number
of parameters) during the regression. All parameters,
including the diameters, were varied when generating the
population. After 105 function evaluations (roughly 625
generations) the minimization was stopped and a downhill
simplex algorithm used to find the local minimum.
4. Results and discussion
We first show the results that are obtained when the SAXS
and PDF data are fitted separately. The structural results are
Figure 2
Spheroidal model fit to SAXS data. Intensity is calculated as
Sp ðQÞPðQÞ þ BEðQÞ, where Sp(Q) (black dashed line) is the particle–
particle interference function, P(Q) is the spheroidal shape factor (gray
line) and E(Q) is the SAXS profile of the empty capillary (green dashed
line) that was scaled by a constant B during the fit. BE(Q) is shown as a
green curve. The SAXS data are shown as a thick blue line and the model
fit is in red.
J. Appl. Cryst. (2014). 47, 561–565
Figure 3
Schematic of the complex modeling setup in SrFit, described in the text,
which combines PDF and SAXS data in a single optimization problem.
Christopher L. Farrow et al.
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Robust structure and morphology parameters for CdS
563
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Table 1
Structural and shape parameters from spheroidal CdS nanoparticle
model fits to the PDF and SAXS data.
Results for the wurtzite phase are shown below [Cd at (1/3, 2/3, 0), S at (1/3, 2/
3, z)].
Rw(PDF)
Rw(SAXS)
a (Å)
c (Å)
Sz
Cd Uiso (Å2)
S Uiso (Å2)
Wurtzite fraction
Equatorial radius (Å)
Polar radius (Å)
SAXS
PDF
Complex
–
0.0148
–
–
–
–
–
–
11.36
20.77
0.146
–
4.134
6.753
0.441
0.0094
0.0158
0.366
10.17
32.43
0.149
0.0158
4.135
6.752
0.434
0.0095
0.0173
0.376
11.38
23.45
shown in Table 1. The fits to the experimental PDFs are shown
in Fig. 4 with the difference curve offset below. The fits are
quite good. The high-r region is shown on an expanded y scale
to highlight the small differences. This quality of fit would be
considered to be quite satisfactory in a regular PDF refinement of nanoparticle samples (Masadeh et al., 2007; Yang et
al., 2013; Du et al., 2012; Gilbert et al., 2004; Polking et al.,
2012). The fits result in a nanoparticle aspect ratio that is 3.2:1
as is shown in Fig. 1(b).
The fits to the SAXS data alone are shown in Fig. 2 as the
solid red line, with Sp(Q) and P(Q) (Glatter & Kratky, 1982)
shown separately, offset below. The best fit ellipsoid has an
equatorial radius and polar radius of 11.36 and 20.77 Å,
respectively, giving an aspect ratio of 1.8:1. It is striking that
the best fit PDF model results in major and minor axes for the
spheroid that are quite different from the SAXS-derived ones.
Although the ‘size’ obtained from the PDF may be smaller
than that obtained from SAXS because of disorder, it cannot
be bigger, as we see here for the major axis which is 32.43 Å
from the PDF but only 20.77 Å from SAXS. This indicates an
aberration in either the PDF or the SAXS result, which may
Figure 4
Experimental (blue) and corresponding simulated PDFs (red) with
difference curve (green) offset below for CdS nanoparticles (a) fitted
using the PDF alone and (b) co-refined to PDF and SAXS data. Data in
the high-r range (>20 Å) are magnified ten times for clarity.
564
Christopher L. Farrow et al.
occur if the information content of the data that constrains
that parameter is weak, if there are systematic errors unaccounted for in the model or if refinement parameters are
significantly correlated (or some combination of these). For
example, in the current case the information in the PDF is
somewhat weak because an approximate Gaussian is used for
the effects of the measurement resolution, as we discuss below,
and the information content on the nanoparticle size and
shape from the SAXS is weakened by the fact that it derives
from a fit to data from a concentrated sample rather than a
dilute nanoparticle dispersion optimized for nanoparticle form
factor determination. The idea behind complex modeling is
that it both pinpoints such systematic errors, which are
sometimes unavoidable, and mitigates their effects.
We now consider the results of the complex modeling
approach where the information from PDF and SAXS is used
together in the same refinement. The fit is shown in Fig. 4(b)
and summarized in Table 1. A comparison of the morphology
obtained by the PDF alone and the PDF–SAXS complex is
shown in Figs. 1(b) and 1(c). The quality of the PDF fit, as
measured by the agreement factor, Rw, is slightly worse in the
complex. However, the addition of the SAXS data constrains
the particle size and shape to values that better reflect the
TEM image evident in Fig. 1(a). Thus, the complexed model
has produced a more physically reliable result than the PDF
refinement alone. There are corresponding small adjustments
to the refined values of the wurtzite fraction and isotropic
displacement parameters.
The PDF-only and the complexed refinements converged to
slightly different minima in the physical-parameter space. To
explore the robustness of the refinements we tried varying
starting values in an attempt to steer each refinement into the
other parameter space minimum, without success. Further, we
tried fixing the wurtzite fraction or atomic displacement
parameters in the PDF-only model to those values refined in
the complexed model, but in each case the PDF-alone
refinement preferred the elongated nanoparticle shape and
did not find the alternative minimum. This indicates that these
are stable minima for each model and the problem is not due
to a lack of model convergence. By fixing the particle shape
parameters to those obtained from the complex, the PDFalone refinement did refine to the same minimum as the
complex, albeit with a higher Rw. This suggests that there is
some systematic error in the model used for the PDF calculation with respect to the particle shape, which may originate
from an imperfect correction for measurement resolution
effects or perhaps comes from inadequacies in the Morningstar–Warren approximation (Warren et al., 1936; Egami &
Billinge, 2013). To test the former we allowed the Qdamp
parameter to vary in the refinement rather than fixing it to the
value obtained in the calibration measurement. In this case,
the value refined for the short axis did not change much, but
Qdamp and the long-axis parameters became highly correlated,
resulting in a large increase in both Qdamp and the value of the
long-axis parameter, which went up to 127 Å. This shows that
the values refined for the particle shape functions are rather
highly correlated with Qdamp, which must be determined
Robust structure and morphology parameters for CdS
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J. Appl. Cryst. (2014). 47, 561–565
research papers
carefully from calibration data for accurate results. Further, it
shows that the stability of the refined particle size parameters
decreases as the particle dimensions get larger. By introducing
the information from small-angle scattering data in a modeling
complex, the uncertainty on these parameters is greatly
reduced, leading to both more physical refined structural
parameters and a better understanding of the uncertainties
and limitations of our models. For the most reliable corefinement, SAXS data from a dilute suspension are preferred
to avoid uncertainties from the separation of the form factor
and the structure factor in the SAXS. However, in some cases
it may not be possible to make a separate dilute sample, and
this proof of principle demonstration indicates that SAXS
data from a concentrated sample may still add value in a
complex modeling framework.
5. Conclusion
Many experiments are being carried out to elucidate the
structure of complex materials and the data being analyzed
quantitatively using model refinements, whether it is Rietveld
refinement of powder diffraction data or fits to PDF or
EXAFS data, for example. The accuracy (as opposed to the
precision) of the results is often limited by systematic errors or
uncertainties in the measurement, model description or data
processing, rather than the statistical uncertainty on the data.
This study shows how complex modeling can help to test the
reliability of physical parameter estimates as well as giving
insight into the limitations of calculated and experimentally
determined spectra by using multiple data/information
sources. This is an important step towards obtaining more
reliable quantitative estimates of structural parameters that
will be used to understand the properties of complex materials. The SrFit complex modeling framework is designed to
make this approach more straightforward to apply.
The SrFit framework was developed and the data were
collected under the National Science Foundation (NSF)
funded DANSE project (DMR-0520547). The samples were
prepared at Zhejiang University funded by the Columbiabased US-DOE-BES-funded Energy Frontier Research
Center under award No. DE-SC0001085. The data analysis
was carried out and the project finalized as part of the
Complex Modeling LDRD project at Brookhaven National
Laboratory, which is funded by the US-DOE-BES under
contract No. DE-AC02-98CH10886. Use of the National
Synchrotron Light Source at Brookhaven National Laboratory is also funded under contract No. DE-AC02-98CH10886.
Use of the Advanced Photon Source is supported by the UUSDOE-BES under contract No. DE-AC02-06CH11357.
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