233102-3
Calvin et al.
Appl. Phys. Lett. 87, 233102 ͑2005͒
not appear in any of the TEM images. Since XRD is particu-
larly sensitive to these large crystallites, it is reasonable that
the XRD results would be at or above the top end of the
range found via TEM, as here. Both of these complications
underscore the difficulty of determining crystallite size dis-
tributions via TEM for samples of this type.
Based on the results of this study, we suggest that cau-
tion is warranted in applying Scherrer analysis to estimate
the relative mean sizes of distributions of nanocrystals when
the distribution may be broad. TEM is, of course, the most
accurate method when sufficient data are collected, and ag-
gregation, polycrystallinity, and contamination are not sus-
pected, but in cases such as the one examined here can be
problematic and very time consuming. EXAFS, although as
yet little used in this capacity, shows promise for being a
robust method for determining relative sizes of samples.23
FIG. 4. Crystallite size found by each technique. TEM shows full range of
crystallites found; error bars on XRD and EXAFS measurements show un-
certainty rather than dispersion.
dispersion. For XRD, the error bars are the standard devia-
tion of the sizes found by applying Scherrer analysis to dif-
ferent diffraction peaks, while for EXAFS they are indicative
of the sensitivity of the fit to particle size. In both cases, the
error bars are a measure of uncertainty in the absolute mean
as determined by each method. Changing the details of the
fitting method in EXAFS or the assumed peak profile in
XRD, for example, leads to all of the determined sizes in-
creasing or decreasing in a correlated way. Thus, the relative
sizes of the samples are determined to a greater accuracy
than is implied by the error bars. In order to estimate the
uncertainty in the relative sizes determined from EXAFS, fits
to an empirical standard using crystallite size as the only
fitting variable were also performed ͑this method is de-
scribed further in Ref. 19͒. These one-parameter fits do not
allow for relaxation of the nanoparticles, and thus may suffer
from systematic bias. Any relaxation effects, however, are
likely to be given by slowly varying monotonic functions of
size; thus, the uncertainties found by the one-parameter fits
are a reasonable estimate of the uncertainty in relative sizes
for the multiparameter fits.
As with other EXAFS studies of particles with broad
size distributions,19–21 it is striking how much larger the
XRD-determined size is than the EXAFS size. This is partly
attributable to the different weighting implied by the two
methods; the difference between the two measurements is
thus a marker for polydispersion. We have previously shown
that for a moderately polydisperse sample XRD becomes
dominated by the size distribution of the largest particles,
while EXAFS is still sensitive to the lower end of the
distribution.9 It can be seen from Fig. 4 that EXAFS cor-
rectly yields relative sizes for the mixed samples between
those of the corresponding as-synthesized samples, while
XRD yields a result more similar to that of the unmixed
sample with the larger mean. This is consistent with behavior
that would be expected for samples with a large degree of
polydispersion.9
1The term nanocrystal is used here for either a monocrystalline nanopar-
ticle or a nanocrystalline inclusion within a matrix.
2C. G. Granqvist and R. A. Burhman, J. Appl. Phys. 47, 2200 ͑1976͒.
3L. K. Kurihara, G. M. Chow, and P. E. Schoen, Nanostruct. Mater. 5, 607
͑1995͒.
4See EPAPS Document No. E-APPLAB-87-031548 for supplemental im-
ages, ͑k͒ data, and ͑R͒ fits. This document can be reached through a
direct link in the online article’s HTML reference section or via the
5It is probable that data with lower noise levels could have been collected
using synchrotron radiation. Our intention here is to compare crystallite
size determination techniques commonly found in the literature; for this
purpose, Scherrer analysis is generally performed using laboratory x-ray
sources.
6Some samples ͑primarily the mixed samples͒ were also analyzed using a
Scintag XDS 2000 diffractometer to test for instrument variability.
Samples analyzed on both instruments produced particle size measure-
ments consistent to within the reported uncertainties.
7A. West, Solid State Chemistry and Its Applications ͑Wiley, West Sussex,
1984͒.
8
JADE 6.1 ͑Materials Data, Livermore, CA, 2002͒.
9S. Calvin, C. J. Riedel, E. E. Carpenter, S. A. Morrison, R. M. Stroud, and
V. G. Harris, Phys. Scr., T 115, 744 ͑2005͒.
10R. B. Greegor and F. W. Lytle, J. Catal. 63, 476 ͑1980͒.
11A. I. Frenkel, C. W. Hills, and R. G. Nuzzo, J. Phys. Chem. B 105, 12689
͑2001͒.
12I. Arcon, A. Tuel, A. Kodre, G. Martin, and A. Barbier, J. Synchrotron
Radiat. 8, 575 ͑2001͒.
13S. Calvin, E. E. Carpenter, B. Ravel, V. G. Harris, and S. A. Morrison,
Phys. Rev. B 66, 224405 ͑2002͒.
14699 independent points, 41 free parameters, and 658 degrees of freedom
for the 12 samples according to the Nyquist criterion. The EXAFS
R-factor, a measure of mismatch between data and fit, was 0.03.
15J. J. Rehr, S. I. Zabinsky, and R. C. Albers, Phys. Rev. Lett. 69, 3397
͑1992͒.
16B. Ravel and M. Newville, J. Synchrotron Radiat. 12, 537 ͑2005͒.
17Relative to the first peak of the derivative of the energy spectrum.
18The mean square radial displacements were initially allowed to vary for
each sample, but were not found to differ significantly between samples;
thus in the final fits all samples were fit to one set of same values:
͑5.0 0.1,6.4+0.2, and 7.9 0.3͒ϫ10−5 nm2 were found for the nearest,
next-nearest, and more distant neighbors, respectively.
Because the particles were polycrystalline, the TEM im-
ages were quite difficult to interpret. An attempt was made to
measure crystallite sizes within each sample, with the result-
ing ranges shown as grey bars in Fig. 4.22 As can be seen,
these ranges generally fall between the XRD and the EXAFS
results. It is not surprising that the ranges do not generally
extend to sizes as small as indicated by EXAFS, as no at-
tempt was made to use TEM to count crystallites smaller
than 1.5 nm radius due to the difficulty in identifying small
crystallites in much larger, irregular particles. There may also
19S. Calvin, M. M. Miller, R. Goswami, S.-F. Cheng, S. P. Mulvaney, L. J.
Whitman, and V. G. Harris, J. Appl. Phys. 94, 778 ͑2003͒.
20C. Calais, M. Matsubayashi, C. Geantet, Y. Yoshimura, H. Shimada, A.
Nishijima, M. Lacroix, and M. Breysse, J. Catal. 174, 130 ͑1998͒.
21A. I. Frenkel, S. Nemzer, I. Pister, L. Soussan, T. Harris, Y. Sun, and M.
H. Rafailovich, J. Chem. Phys. 123, 184701 ͑2005͒.
22Note that, due to the difficulties posed by aggregation and polycrystallin-
ity, observer bias was undoubtedly present for the TEM results.
23The method used here is somewhat time-consuming and requires consid-
erable specialized expertise. In cases in which EXAFS is already being
used to determine other characteristics of the sample, however, an analysis
of this type requires little additional effort. If an easier, quicker, and less
have been occasional large crystallites in the samples that did
accurate method is desired, a one parameter fit is appropriate.
On: Sun, 04 May 2014 08:01:36