Ultramicroscopy 111 (2011) 847–853
Contents lists available at ScienceDirect
Ultramicroscopy
journal homepage: www.elsevier.com/locate/ultramic
Experimental and theoretical study of electron density and structure
factors in CoSb3
R. Sæterli a, E. Flage-Larsen b, J. Friis c, O.M. Løvvik b, J. Pacaud d, K. Marthinsen e, R. Holmestad a,n
a
Department of Physics, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway
Department of Physics, University of Oslo, 0316 Oslo, Norway
c
Department of Synthesis and Properties, SINTEF Materials and Chemistry, 7491 Trondheim, Norway
d
Insitut Pprime, UPR 3346 CNRS, Université de Poitiers, SP2MI, Bd P et M Curie, F-86962 Chasseneuil cedex, France
e
Department of Materials Science and Engineering, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway
b
a r t i c l e in fo
abstract
Available online 27 August 2010
We refine two low-order structure factors of the skutterudite CoSb3 using convergent beam electron
diffraction. The relatively large unit cell of this material causes the disks to overlap and introduces a
series of challenges in the refinement procedure. These challenges and future work-arounds are
discussed. The refined structure factors F200 and F600 are compared to X-ray diffraction and density
functional calculated values, the latter calculated using two different functionals. Both relaxed and
experimental lattice parameters are tested to explicitly highlight the impact of the lattice geometry and
atomic position on the structure factors.
& 2010 Elsevier B.V. All rights reserved.
On the occasion of John Spence’s 65 years
anniversary.
Keywords:
CBED
DFT
Skutterudite
Electronic structure
1. Introduction
The importance of accurate experimental structure factor
measurements for determination of the electron density of
materials is well known and a lot of effort has been put into
this task. The advent of powerful calculation schemes such as
density functional theory (DFT) to calculate the electron density
and thus X-ray structure factors from first principles have
contributed significantly to the field [1–3]. It is well known that
the accuracy of DFT calculations relies on the choice of
functionals, potentials, basis sets and technical implementation.
Ultimately, it is also limited by computational resources.
However, if the electron density is accurately determined, a
wealth of electronic, optical and vibrational properties can be
calculated, which is difficult, time consuming and in some cases
impossible to obtain from experiments. In addition, electron
transfer analysis can be done to explicitly investigate the electron
rearrangement due to the setup of bonds [4]. Such electron
rearrangement determines most of the properties of materials
and it is thus very important to not only understand the difference
between ionic and covalent bonded structures, but also the
strength, local geometry and mixing of such bonds. Due to the
array of approximations introduced in these schemes, experimental verification of different functionals, basis sets and
implementation issues is, however, still necessary (see e.g. [5,6]).
n
Corresponding author.
E-mail address: randi.holmestad@ntnu.no (R. Holmestad).
0304-3991/$ - see front matter & 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ultramic.2010.08.002
Employed experimental techniques for accurate structure
factor determination comprise both X-ray methods [7,8] and
different methods based on transmission electron diffraction
(see e.g. [9]). The strength of X-ray synchrotron diffraction lies in
its capability to accurately measure a large number of structure
factors [10]. Generally, this method is very accurate for high-order
structure factors, while for the strong low order reflections it is
severely affected by extinction due to multiple scattering.
To overcome this problem, it has been proven very successful to
combine X-ray diffraction with convergent beam electron diffraction (CBED) studies of the strong low order reflections (see e.g.
[11–15]). The introduction and application of CBED for the
measurement of crystal structure amplitudes and phases are very
much due to the enthusiasm and pioneering work of Spence and
Zuo [16] and Spence [17] in this field in the late eighties and early
nineties. As a result of this and the accelerating development in
hardware (energy filters and detection systems), a lot of work was
done in the field in the nineties, with increasing complexity and
precision (e.g. [18–21]). One of the most versatile methods is the
use of a systematic row of CBED reflections, with high sensitivity
both to the structure factor in question and the critical sample
thickness parameter, and also well developed refinement tools
[18,22,23]. The systematic row approach has been demonstrated
for a range of materials such as GaAs [24], TiAl [25], MgO [26],
Cu2O [11], Cu [13] and SrTiO3 [14]. During the last years, also two
specialised methodologies based on the zone axis approach
[27–30] have been further developed, and represent today’s
‘state-of-the-art’ in structure factor determination by electron
diffraction. The method developed by Tsuda includes refinement
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R. Sæterli et al. / Ultramicroscopy 111 (2011) 847–853
of atom positions as well as structure factors, and has recently
been used to study orbital ordering in the spinel oxide FeCrO2
[30]. In all methodologies, complex structures with large unit cells
are avoided due to the increased overlap of reflections. To the
authors’ knowledge, the systematic row approach has not been
demonstrated on unit cells larger than about 4–5 Å.
In this work, we examine the possibility of refining low-order
structure factors from the skutterudite CoSb3, having unit cell
parameters of 9 Å, using the systematic row approach. Some
skutterudites are promising thermoelectric materials, as their
body-centered cubic unit cells can be filled by so-called ‘‘rattler’’
atoms introduced into the structure to reduce thermal conductivity while at the same time not altering the favorable electronic
structure of the unfilled structures [31]. As some of the highest
performing thermoelectric skutterudites are found among the
filled CoSb3 variants [32], the electronic structure of CoSb3 has
been thoroughly studied both experimentally and theoretically in
the literature [10,33–38] and constitutes a basis on which to test
whether the CBED technique is applicable also to such large unit
cell materials. We refine two bond sensitive low-order structure
factors from CoSb3, and show that the systematic row approach is
close to its useful limit in this material. Comparison to structure
factors retrieved from X-ray diffraction and DFT is made. To shed
light on the sensitivity of the structure factors, the changes in the
lattice geometry and atom positions were studied by relaxing
these parameters from the experimental values. Structure factors
from both the local density and generalized gradient approximation were also calculated.
higher order Laue zone (HOLZ) lines. Simulated intensities were
found through the EXTAL [23] program, which is based on the
Bloch wave method. A goodness of fit parameter
GoF ¼
X ðIexp cItheory Ibackgr Þ2
1
i
i
i
nm1 i
s2i
ð1Þ
was minimized by refining chosen parameters. The sum runs over
all pixels i ¼1, ... ,n, m is the number of refined parameters, Iiexp ,
Iitheory and Iibackgr are experimental, theoretical and background
intensity values, respectively, with c being a scaling factor and
si the standard deviation of Iiexp . The refined parameters include
the amplitude of the desired structure factor(s) Uhkl and
corresponding absorption component(s) U0 hkl, c and Iibackgr
(assumed constant for each disk), beam direction (or conversely,
sample orientation) and sample thickness. Lattice parameters,
atomic positions and anisotropic atomic displacement parameters
(ADPs) were not refined, but taken from X-ray diffraction data,
recorded at a temperature of 100 K [41]. Lattice parameters and
atom positions used in the refinement are given in Table 1. Due to
the large unit cell, a large number of beams had to be included in
the calculations, chosen from the following three criteria (for an
explanation of the terms, see Zuo and Weickenmeier [42]):
(1) proximity to the Ewald sphere 92KSg9max ¼3.0 Å, (2) length of
reciprocal scattering vector 9gmax9¼ 3.0 Å and (3) perturbation
strength 9Ug/2KSg9min ¼25. Strong beams were treated exactly,
while weak beams were treated by Bethe perturbation [43,44].
2. Experimental
2.4. Density functional theory calculations
2.1. Material synthesis
The sample was prepared by melting the elements (Co: SigmaAldrich, 99.995%; Sb: J.T. Baker, 99.8%) in evacuated, sealed silica
glass ampoules, thereafter annealed at 800 1C for four days,
crushed, remelted and further heat treated at 800 1C for 17 days.
The sample was single phase according to powder X-ray
diffraction.
2.2. CBED experiments
TEM specimens were prepared by dimpling and subsequent
Ar ion milling of the CoSb3 polycrystal at liquid nitrogen
temperature. Defect-free areas, as judged from bright field images
and HOLZ lines, were chosen for analysis. CBED experiments were
performed on a JEOL 2200FS equipped with a 2K 2K CCD camera
and the patterns were filtered using an Omega energy filter tuned
on to the zero-loss peak with an energy width of 10 eV to ensure
contributions only from elastically and, unavoidably, thermally
scattered electrons. Further, the sample was cooled to approximately 120 K and allowed to stabilize in order to minimize
thermal diffuse scattering and also to avoid contamination. The
patterns were further deconvoluted using the LUCY algorithm
[39] to correct for the point-spreading of the CCD. This is normally
sufficient for structure factor refinement, although it is acknowledged that the use of direct methods [40] might be better. The
voltage of the microscope was calibrated in advance to 203.5 kV.
The electronic structure of CoSb3 was calculated using the Vienna
Ab-initio Simulation Package (VASP) [45,46]. Two functionals, the
local-density approximation LDA-PZ [47] and generalized gradient
approximation GGA-PBE [48] were employed together with the
projector augmented wave (PAW) method [49]. Experimental lattice
parameters were used as input to a quasi-Newton residual
minimization scheme with direct inversion in the iterative subspace
(RMM-DIIS) relaxation. An energy cutoff of 550 eV and Monckhorst–
Pack k-point sampling of 8 8 8 were sufficient to obtain
converged results to within a few meV of the total energy. The allelectron charge density was regenerated on a dense 480 480 480
real-space grid to improve the resolution of localized charges and
was passed to a fast Fourier transform routine in the FFTW package
[50] to obtain the structure factors. To improve the resolution of the
all-electron density and thus the structure factors, a dense real-space
grid is necessary. However, such grids have an upper bound limited
by computational resources, in particular memory usage. In this
work, we opted for a dense real-space grid such that relative errors
in the 000 structure factor from the chemical electron count of the
valence electrons were within 10 4. Relaxed lattice parameter and
atom positions from VASP are given in Table 1 together with the
experimental parameters of Krebs Larsen [41].
Table 1
Experimental (at 100 K) and relaxed DFT lattice parameter a from [41] and atomic
positions.
Experimental values
2.3. Refinement of CBED data
Refinements of three low-order non-extinct structure factors,
U110, U200 and U600, were attempted from systematic row
orientations chosen on the basis of relatively few intersecting
a (Å)
Sb y
Sb z
a
Ref. [41].
a
Relaxed DFT values
9.02
9.11
0.3354
0.1579
0.3334
0.1595
R. Sæterli et al. / Ultramicroscopy 111 (2011) 847–853
849
2.5. Comparison of CBED and DFT data
The electron structure factors Uhkl are not directly comparable
to the structure factors Fhkl obtained from DFT and X-ray
diffraction. The two quantities are however related by the Mott
formula [16], taking into account the ADPs for the given
temperature and the core contribution to the electron structure
factors:
Fhkl ¼
X
16p2 eh2 Os2
Zj Tj ðsÞe2pigrj Uhkl
m0 gjej2
j
ð2Þ
here, Tj(s) is the temperature factor, s ¼ sin y=l is the scattering
angle and other notations follow that of Spence and Zuo [16].
0
X-ray structure factors at 0 K are referred to as Fhkl
and are
0
120 K 0
100 K
calculated as Fhkl ¼ Fhkl ðFSC =FSC Þ, assuming that the ADPs are
sufficiently similar for 120 K and 100 K.
One can define the bonding charge density, rB, as the charge
density of the real crystal, rC, minus the charge density of a
reference crystal, rR. This reference should be free of bonding
features and is often referred to as an independent atom model
(IAM). The IAM model can be found from Hartree–Fock calculations
as in the parameterized version of Su and Coppens [51] (SC), or
calculated by for example DFT by considering the procrystal (PC),
a superposition of free atomic electron densities embedded in the
crystal unit cell. The quantities FB, Fhkl and FR can then be defined as
the Fourier transforms
Z
ri ðrÞe2pigr dr
ð3Þ
Fi ¼
Fig. 1. (a) Experimental CBED image of systematic 110 row of reflections. (b)–(g)
Bloch wave simulated images from the same conditions as in (a) using
approximately 25, 50, 125, 300, 500 and 850 beams, respectively.
of rB, rC and rR, respectively.
The direct comparison of experimental and calculated Fhkl values
can be challenging due to the limitation of the plane-wave
methodology to represent core states and/or all-electron densities
accurately. However, as core contributions to the electron density do
not usually contribute in the self-consistent procedure, we compare
the experimental and theoretical values of FB rather than Fhkl.
3. Results
Fig. 1(a) shows one of the experimental CBED patterns of the
systematic 110 row of disks. The disk overlap is substantial due to
the relatively large lattice parameter, although a small beam
convergence is used. Any further reduction in beam convergence
causes the disk area to be unsatisfactorily small, and would in
addition extend the coherence fringes now seen at the edges of
the disks further towards the center. Simulated images from the
same thickness and orientation are shown in Fig. 1(b)–(g), using
an increasing number of beams from about 25 to 850 in the
calculations. The refined sample thickness was about 95 nm. As
can be seen from the simulations, a high number of beams is
clearly needed in order to reproduce the experimental intensity.
Fig. 2(a) shows an experimental CBED image of a systematic
200 row of disks, used to refine U200 and U600. The best U200 and
U600 structure factor refinement yielded values of 0.01549 and
0.03985 Å 2, respectively, with a GoF value of 1.59 for 368
intensities/pixels and 897 beams included in the refinement. The
result of the refinement is shown in Fig. 2(b), portraying the
intensity profiles of the linescans marked and numbered in
Fig. 2(a).
Table 2 summarizes the results of the U200 and U600
refinements. In addition to the structure factors Uhkl and
absorption U0 hkl, the structure factors are also given as X-ray
0
structure factors Fhkl
that have been corrected for the finite
experimental temperature. These are compared to experimental
Fig. 2. (a) experimental CBED image of systematic 200 row of reflections with the
linescans used to refine U200 and U600. (b) Linescan intensity profiles of the
linescans marked in (a). Red crosses mark experimental intensities while green/
light curves are best fit refined intensity values. Blue/dark curves mark the
difference between theoretical and experimental intensities. (For interpretation of
the references to colour in this figure legend, the reader is referred to the web
version of this article.)
X-ray diffraction values [41] and the SC IAM values. No reliable
refinement of the 110 structure factor was obtained.
To check the consistency of the refined structure factor values,
the GoF was evaluated as a function of the two refined structure
factors for the measurement yielding the lowest GoF. Fig. 3 plots
the GoF versus ten values of U200 and U600 in an interval around
850
R. Sæterli et al. / Ultramicroscopy 111 (2011) 847–853
Table 2
Experimentally refined best fit U200 and U600 structure factors (in Å 2) with scattering angles s ¼ sin y=l and absorption U0 hkl(in Å 2), and the conversion to X-ray structure
0
factors Fhkl
(in electrons per unit cell) at 0 K. Experimental X-ray structure factors [41] also with the temperature factors set to zero and the IAM values of Su and Coppens
(SC) [51] are included for comparison.
h
k
2
6
0
0
a
l
0
0
s
0.11
0.33
XRD
a
IAM (SC)
CBED
Fhkl
Uhkl
U0 hkl
0
Fhkl
0
Fhkl
149.50
663.22
0.01549 (4)
0.0398 (3)
6.71 10 4
3.31 10 3
153.2 (4)
670 (6)
153.8
666.2
Ref. [41].
Fig. 3. Contour plot of GoF versus the two refined structure factors U200 and U600
for an interval around the final refined values, calculated using about 900 beams.
the obtained minimum values. All refinable parameters other
than the structure factors are fixed.
Table 3 shows FB from DFT calculations using the LDA and GGA
functionals, and for both functionals using both SC and PC as
reference IAM valence values. Additionally, for GGA, two sets of
lattice parameters and atomic positions were used as input to the
DFT calculations, one with the experimental parameters used in
the refinements and one set of relaxed parameters. The results of
the CBED refinements and X-ray diffraction values [41] are shown
for comparison, also these with both SC and PC used as reference
values.
In Fig. 4, the electron density difference rB in the 130 and 200
planes from the relaxed GGA calculations is shown. The electron
accumulations between Co and Sb, and between the Sb positions
in the Sb4 ring caused by bonding are clearly visible. To
compensate, electron depletion occurs in the areas not covered
by these bonds.
4. Discussion
The best U200 and U600 structure factor refinement yielded
values 0.01549 and 0.03985 Å 2, respectively, with a GoF of 1.59.
Several linescan positions and experimental images were tested,
which all came out with higher GoF values. Using refinements
which resulted in GoF values less than 2 gives intervals of
0
0.01549–0.01632 Å 2 and 152.5–153.2 for U200 and F200
, respectively. Corresponding numbers for the 600 structure factor are
0.03985–0.04123 Å 2 and 659.2–670.1. Using these intervals as
0
estimates of the standard deviation gives uncertainties in the Fhkl
values are 0.23% and 0.81% for the 200 and 600 structure factors,
respectively. In Table 2, we have chosen to give these intervals
around the value with lowest GoF rather than the mean value
(corresponding to the refinement run with the largest number of
beams). One benefit of the large unit cell is that even with a
relatively large error in Uhkl, the s2 factor in the second term of the
Mott formula (Eq. (2)) ensures that for the low-order factors
the error in Fhkl remains small. The plot in Fig. 3 shows that the
minimum in GoF is broad, however with no other distinct minima
than the one found. The variations in the refined values do,
however, suggest that other local minima can be found in the
parameter space comprising the other refinable parameters.
The relative difference in the calculated FB values between the
two lattice parameters and atom position sets is around 3–4%
(see Table 3) for both the 200 and the 600 structure factor, most
likely caused by the temperature difference for which the lattice
parameters have been obtained (e.g. 100 K versus 0 K) and/or
density functional related errors, stemming from the choice of
functional, basis set or implementation issues associated with
representing the all-electron density on a finite grid. The relative
difference between the LDA and PBE functionals for both 200 and
600 structure factors is smaller, but similar in magnitude.
Even though the fundamental approach of superpositioning
the electron density at atomic positions is similar in the PC and SC
models, the way the single atomic electronic density is addressed
is different. The SC reference employs relativistic Hartree–Fock
approximations, while LDA and PBE density functional calculations were used for the PC reference. Hartree–Fock calculations
include exact exchange, while correlation effects are disregarded.
The PBE functional mixes exchange and correlation in an
approximate manner. Consequently, it is reasonable to expect
that the SC and the calculated PC reference in this work yield
different results. Thus, from a calculation point of view, since the
same calculation methods are used, it can be argued that the PC
reference is better founded, such that the difference between the
electron density of the crystal and PC represent a minimum of
approximation errors. Using the DFT structure factors of the
crystal with the SC reference would likely introduce additional
errors related to the different calculation schemes. From an
experimental point of view it is difficult to argue which reference
to use. This is also confirmed in Table 3, where both the SC and PC
references for the experimental data yield results that are similar
in magnitude to the calculations.
The fact that the calculated FB is consistently lower than
measured values (given the same reference) might be caused by
effects not included in the calculations, for example dynamic core
contributions, lack of proper description of the exchange and
correlation effects, or particularly lattice dynamics related to the
finite temperature during the experiments. Future studies with
R. Sæterli et al. / Ultramicroscopy 111 (2011) 847–853
851
Table 3
Comparison of FB ¼ FCvalence FRvalence from LDA and GGA-PBE functionals, where FRvalence found using both the procrystal (PC) and Su-Coppens (SC) [51] approach, with the
core
core
experimentally found FB ¼ FC FRvalence FSC
values for the 200 and 600 structure factors. The core values FSC
were obtained from SC as these are not obtainable through the
PC approach and comprise the same core electrons, while the reference valence values FRvalence were obtained from both SC and PC approaches. Values are in electrons per
unit cell.
a
IAM reference
h
k
l
LDA
GGA-PBE
GGA-PBE (relaxed)
CBED
XRD
SC
2
6
2
6
0
0
0
0
0
0
0
0
3.313
2.395
1.596
2.283
3.288
2.379
1.639
2.308
3.228
2.305
1.582
2.240
3.70
6.88
2.05
6.81
4.30
2.98
2.65
2.53
PC
a
Ref. [41].
Fig. 4. (a) The unit cell of CoSb3 with the 200 (horizontal) and 130 (inclined) lattice planes. Co atoms are black, while Sb atoms are gray. (b), (c) CoSb3 electron density
difference rB plots in the 130 and 200 planes, respectively, calculated using relaxed structural parameters in the GGA-PBE approach. The 200 plane is shifted horizontally
by half a unit cell to encompass a whole Sb4 ring. Contour lines are drawn from 0.01 to 0.01 Bohr 3 in 0.002 increments, with black and gray lines denoting positive and
negative contour lines, respectively. The electron buildup due to the Co–Sb bond in the octahedra is clearly visible between the Co and Sb position. Similarly visible is the
electron buildup between the Sb positions due to the Sb–Sb bond in the Sb4 ring.
different calculation schemes or a larger number of structure
factors might reveal the source of the discrepancies. The
comparison of experimental and theoretical structure factors is
today burdened by the need of a common bond-free reference. By
for example combining plane-waves and atomic orbitals, or using
a continuous electron density model for the core region, better
bond-free electron density references might be obtained, so that
the absolute structure factors can be used as the common
reference.
The potential of electron density analysis is illustrated in Fig. 4,
where the bonds can be seen qualitatively through electron
accumulation and depletion. It is also possible to quantify the
magnitude of these accumulation and depletion areas by studying
the inter-atomic line scans of the electron density difference. Such
analyses have previously been used to study the real-space
geometry and quantitative determination of the bond strength in
skutterudites and Zintl compounds [4,52]. The task of generating
the electron density with reasonable resolution from experiments
is demanding and sometimes impossible. It is thus more foreseeable to calculate the electron density and structure factors and
then compare the calculated and experimental structure
factors, which are most sensitive to the bond rearrangements.
Choosing these structure factors might be difficult, but a
general idea can be obtained from the geometry of the lattice
and atomic positions.
The systematic row of 110 reflections, with lattice spacing of
6.4 Å, in Fig. 1(b) illustrates nicely the disk overlap challenges and
the need for a high number of beams in such large unit cell
systems. We note that the difference between the two images
simulated using 500 beams and 850 beams is as much as 4% in
certain areas. It is possible that an even higher number of beams
has to be included to obtain converged structure factor values.
Also, the lack of results can be attributed to the very small area
available for refinement due to the disk overlap.
For the CBED refinements, the large number of excited beams
leads to large calculations that need to consider several hundred
beams, so that the result depends on the exact knowledge of
several structure factors. Also the refinement of parameters other
than the structure factors is severely affected by the large unit
cell, as the small area encompassed in each disk complicates the
refinement of the exact direction of the incoming beam. In
addition, the line scan intensities are likely to be influenced by
overlapping weak background disks. The issue with the background is addressed by Nakashima and Muddle [30], who propose
that their differential approach partly solves this problem. An
interesting future prospect would be to try this approach on large
unit cell systems to see whether any improvement can be
obtained.
Experimentally, the closeness of the reflections calls for a small
beam convergence in order to avoid overlap of the disks, so that
the reciprocal area spanned by each disk is limited. The number of
HOLZ lines is large due to the large number of excited beams, and
areas without two or more intersecting HOLZ lines are rare. The
small convergence angle of the beam also introduces the
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R. Sæterli et al. / Ultramicroscopy 111 (2011) 847–853
coherence fringes seen around the perimeter of each disk, limiting
the useful area of quantitative analysis even further. The fringes
could also be due to some slight instrumental misalignment of
apertures or lenses, and are not present at the convergence angles
regularly used in CBED measurements. The existence of these
fringes has been verified to be present in patterns from JEOL
2010F and JEOL 2200FS instruments.
The large angle CBED (LACBED) technique [53] has been
applied to materials characterization in order to overcome the
overlap of reflections, however the introduction of a shadow
image of the sample overlaid with the reciprocal space information makes it less useful to the determination of structure factors,
where precise measurement of intensities is vital. With everincreasing computational power, a more likely path to accurate
low-order structure factors is the consideration of interference
effects and the overlapping disk areas, however the theoretical
refinement tools to achieve this still awaits invention. The
invention of such tools could be the next challenge in the area
of quantitative electron diffraction for great minds such as
John Spence or those following in his steps.
5. Summary
We have refined the 200 and 600 structure factors of CoSb3
using convergent beam electron diffraction (CBED). These results
were compared to structure factors obtained from plane-wave
based density functional calculations and also X-ray diffraction.
The low-order structure factors are important both for verification
of various theoretical approaches, and also to complement X-ray
diffraction results that suffer from the effects of extinction for
these reflections. Technical aspects regarding the calculations of
first principle structure factors were discussed. Both experimental
and relaxed lattice parameters were included in the calculations
to shed light on the sensitivity of the structure factors to geometry
changes in the lattice. Two functionals were also compared and
reasonable agreement with experiments was shown for all
calculations. Reasons why other CBED low-order structure factor
refinements are hard to converge in such large unit cell systems
are discussed, together with possible future improvements.
Acknowledgements
The authors would like to thank Ole-Bjørn Karlsen for
synthesizing the material and Bjørn G. Soleim for preparation of
the TEM sample. Further, we are indebted to Finn Krebs Larsen
who has been kind enough to supply the two relevant X-ray
structure factors as well as temperature factors, lattice parameters and atom positions. In addition, we would like to thank
Krebs Larsen, Øystein Prytz and Johan Taftø for valuable
discussions. The NOTUR consortium should be acknowledged for
computational resources and The Research Council of Norway for
financial support.
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