Quantitative analyses of precession diffraction data for a large cell oxide.

Abstract

Kinematical and two-beam calculations have been conducted and are compared to experimental precession data for the large unit cell crystal La4Cu3MoO12. Precession electron diffraction intensities are found to exhibit approximate two-beam behavior and demonstrate clear advantages over conventional SADP intensities for use in structure solution.

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Quantitative Analyses of Precession Diffraction Data
for a Large Cell Oxide
Christopher S. Own,* Arun K. Subramanian, and Laurence D. Marks
Department of Materials Science, Northwestern University, 2220 N. Campus Dr., Cook 2036, Evanston, IL 60201, USA
Abstract: Kinematical and two-beam calculations have been conducted and are compared to experimental
precession data for the large unit cell crystal La4Cu3MoO12. Precession electron diffraction intensities are found
to exhibit approximate two-beam behavior and demonstrate clear advantages over conventional SADP intensi-
ties for use in structure solution.
Key words: TEM, electron, precession, diffraction, direct methods
INTRODUCTION
In 1986, Vincent and Bird used focused large-angle conver-
gent beam electron diffraction zone axis patterns to obtain
pseudo-kinematical intensities for use in structure refine-
ments. These experiments stem from earlier work by Vin-
cent et al. ~1984!, who used hollow-cone illumination on
the zone axis to acquire the same type of data from high-
order Laue zone ~HOLZ! reflections. The success ~and diffi-
culties! of these experiments prompted Vincent, with P.A.
Midgley, to develop a technique that uses coil deflections to
sequentially sweep the beam in a conical manner through
successive Bragg reflections to probe first-order reflections,
which have reduced coupling due to their off-zone diffrac-
tion geometry. This is called precession electron diffraction
by virtue of the conical deflection arrangement ~Vincent &
Midgley, 1994!. Analyses to date suggest that precession
pattern intensities are far less sensitive to thickness than
conventional electron diffraction patterns and, if an accu-
rate model can be derived, will simplify structure solution.
In addition, experiments have uncovered a number of hid-
den bonuses provided by the technique, some of which are
outlined in this article.
The geometry of precession electron diffraction is shown
in Figure 1. The beam is rocked in a conical fashion with a
wide symmetrical tilt above the specimen. After interaction
with the specimen, the transmitted and diffracted beams are
descanned by the projector lens deflectors to re-form a
point pattern. This geometry yields several very interesting
features:
• The pattern may be indexed as a conventional diffraction
pattern although the intensities have actually been gath-
ered from off-zone reflection conditions.
• Nonsystematic dynamical effects such as Kikuchi lines
and intensity variations in CBED spots are reduced by
averaging over incident beam directions.
• Because the beam is entering the sample from an off-axis
direction, much of the dynamical scattering that is partic-
ularly strong at the exact Bragg condition ~or zone axis
channeling condition! is avoided.
• Many more FOLZ reflections are illuminated, under more
kinematical conditions, by the Ewald sphere, allowing the
acquisition of an increased number of intensities for use
in structure solution techniques.
• HOLZ reflections are illuminated,yielding expanded three-
dimensional data sets provided that spots from separate
Laue zones do not overlap.
Figure 2a,b demonstrates these characteristics in the
diffraction pattern from a thick Mg3V2O8crystal. A very
moderate precession angle ~;5 mrad! was used to form this
pattern. The extension by precession of the HOLZ ring into
an annulus of width 10 mrad is clearly seen, as well as
blending of nonsystematic dynamical effects into a radially
diffuse background. This dynamical background averaging
can improve intensity measurements by considerably simpli-
fying the problem of background subtraction.
Gemmi ~2001! maintains that two-beam dynamical
corrections can be made to the kinematically expected in-
tensities if necessary and the results of a two-beam correc-
tion using the Blackman formula ~Blackman, 1939! are
valid in the case of electron precession. This assertion was
originally postulated by Vincent and Midgley in their first
Received January 30, 2003; accepted May 28, 2003.
*Corresponding author. E-mail: csown@northwestern.edu
Microsc. Microanal. 10, 96–104, 2004
DOI: 10.1017/S1431927604040322
MicroscopyAND
Microanalysis
© MICROSCOPY SOCIETY OF AMERICA 2004

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paper on precession. Gjønnes in 1997 made a comprehen-
sive analysis of the precession diffraction geometry and
proposed geometrical correction factors for the two-beam
model to account for the excitation error. She also calcu-
lated the geometrical factor due to beam convergence and
warned that the primary errors in precession patterns are
still dynamical in nature. In this article, we investigate the
validity of the two-beam model for precession by studying
patterns acquired from a novel cuprate in a precession-
capable Hitachi UHV H-9000 TEM and comparing them to
nonprecessed diffraction data gathered from the same re-
gion of the specimen.
EXPERIMENTAL
The original paper of Vincent and Midgley described the
use of a convergent beam to obtain nonoverlapping pre-
cessed CBED patterns. The intensities were acquired by
using a line scan procedure similar to that used for conven-
tional CBED analysis, where integrated line intensity was
converted into integrated area using a least squares method
~Berg et al., 1998!. Use of CBED allows a very small probe
size provided that the instrument can very accurately
precess the beam about a specific region of the sample.
However, convergence is not strictly necessary in electron
precession, and can be considered a hindrance because
some specimens are susceptible to radiation damage from a
focused probe and some instruments may have difficulty
maintaining a pivot point at a precise area on the specimen
due to condenser and objective prefield aberrations.
Figure 1. Precession electron diffraction geometry.
Figure 2. a: Selected area DP of the @5 t 32# zone axis of Mg3V2O8.
b: Precessed SADP of Mg3V2O8using a moderate precession angle
of 5.2 mrad. Several HOLZ annuli are apparent and nonsystematic
effects in the ZOLZ are averaged into a radially diminishing
background.. ~Note: Images a and b have identical exposure times
and digitizing conditions and have received the same digital image
processing, so they can be directly compared.!
Quantitative Analyses of Precession Diffraction Data
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Due to the limitations inherent to CBED precession, as
well as practical considerations during implementation, we
use a selected area precession mode with near-parallel illu-
mination to gather sharply defined precessed spots. This
allows collection of data from specific areas of the speci-
men, ease in aligning precession geometry ~by virtue of
sharp diffraction points!, and loose constraints on the im-
age plane alignment capabilities of the instrument such that
precession mode can be retrofitted onto almost any existing
TEM. The real-space sampling resolution in this mode of
operation is limited by SA aperture size and by SA errors.
~To avoid confusion, conventional SAD patterns will be
referred to in this article as SADPs, whereas precession data
will be henceforth referred to as “precessed,” with the SA
mode used to form sharp spots implicit.!
The precession hardware is a data acquisition and con-
trol ~DAQ! system built around a National Instruments data
control board. New electronics were designed to insert a
clean external control signal generated by the PCI card.
Great attention was given to preservation of full microscope
functionality such that routine instrument alignment and
everyday operation were not affected in any way.The H-9000
beam deflection amplifiers were not designed to handle
rapid AC signals and, as such, were redesigned and re-
placed with high-bandwidth amplifiers ~Fig. 3!. The sys-
tem can output 1 MS/s per channel, resulting in a scan/
descan system that operates distortion-free beyond 2 kHz
before noticeable attenuation, and routinely demonstrates a
scanned/descanned spot resolution of less than 0.5 mrad in
the diffraction plane. It is typically operated with an angular
resolution of 18 ~360 subdivisions per cycle! at 60 Hz with
input deflections of a few volts, translating to no more than
about 800 mA of current per deflection coil. Software was
written in Labview visual language and offers an infinitely
configurable, user-friendly interface that includes compen-
sations for aberration symmetries up to threefold ~Fig. 4!.
The software is written for real-time signal update,
permitting a dynamic alignment procedure for quick distor-
Figure 3. a, b: Buffered inverting mixer for tilt and descan. c: Descan amplifier.
98
Christopher S. Own et al.

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tion correction. In the current software revision, modifica-
tions of the control signal can be applied during alignment
through a visual minimization procedure to bring precessed
reflections down to sharply focused spots. Consequently,
precession alignment including distortion correction can be
conducted rather painlessly within 10 min. A more detailed
description of the system and its implementation will be
given elsewhere ~C.S. Own, submitted!.
Diffraction patterns along the @001# zone axis of a
La4Cu3MoO12crystal were taken in both selected area and
precession modes ~;20 mrad precession angle!. Nine nega-
tives comprised each exposure series and were digitized
using an Optronics microdensitometer at a scale of 25 mm
per pixel. SADP intensities were then acquired by pattern
matching a unitary spot motif to the digitized diffraction
spots and then integrating the intensity of the matched
pattern. Precession intensities were quantified in a slightly
different way by first applying a background subtraction
algorithm within a masked area around each spot and then
integrating the remaining intensity.
Reduction of nine data sets into a single set was con-
ducted by scaling the data from each negative while mini-
mizing the overall error between data sets. Details of this
procedure can be found in a previous paper from our group
~Xu et al., 1994!. Intensities from the precession exposure
series matched very well when scaled, resulting in extremely
low error between negatives. Compared to the conventional
SAD data set, the precession data set had errors at least an
order of magnitude better. This is an important practical
point, and can be attributed to the fact that the precession
technique is much less sensitive to minor variations in the
experimental conditions and is consequently less suscepti-
ble to systematic errors than the SAD methods. Precession
provides nearly the same intensities for symmetry-equivalent
reflections and is fairly tolerant of a slightly off-zone diffrac-
tion condition.
RESULTS AND DISCUSSION
La4Cu3MoO12is a complex oxide exhibiting “frustrated”
magnetic behavior. The structure is a homeotype of YAlO3,
a rare-earth hexagonal phase. The crystal is very slightly
monoclinic but was assumed to have a rectangular cell for
the calculations. Unit cell parameters are a ? 6.86 Å, b ?
10.98 Å, and c ? 7.9147 Å, and b ? 90.028 ~Vander Griend
et al., 1999!. The structure is shown in Figure 5. Intratri-
angle antiferromagnetic interactions align two of three spins,
yielding trimers with Stotal?1
2
ordering between adjacent cells that doubles the unit cell
along the a direction.
Figure 6a–c shows the diffraction patterns for calcu-
lated, SADP, and precessed reduced diffraction data sets
~non-symmetry-averaged!. The intensity of each reflection
is represented using gray scale as well as spot size to visually
demonstrate relative spot intensities. ~040!-type reflections
in the kinematical pattern are considerably more intense
than other reflections in the pattern. This high intensity is
not apparent in the SADP pattern nor in the precessed
pattern. Clearly, dynamical interaction cannot be avoided
near the transmitted beam, even with precession. However,
the fine details in the diffraction map are rather well pre-
served in the precessed DP.With precession, the variation in
intensity between adjacent reflections is clearly displayed,
and the alternating spots show, qualitatively, appropriate
ratios. The SAD pattern, in contrast, loses some fine infor-
_. This spin coupling causes an
Figure 4. Version 0.51 of the precession control software.
Quantitative Analyses of Precession Diffraction Data
99

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mation such as the subtle intensity ordering in the reflec-
tions that occur between the ~230! and ~2 t 30! spots.
To better quantify the relationship between precessed
and nonprecessed patterns, a Blackman curve was calcu-
lated for a crystal thickness of 50 nm using the two-beam
equation
I2-beam
Ikin
@
1
Ahkl?
0
Ahkl
J0~2x! dx
~1!
where the integration limit Ahkl? 2pem0tVg/h2. Electron
scattering factors were computed from values provided with
the NCEMSS multislice software ~Kilaas, 2003!, and experi-
mental isotropic Debye-Waller factors acquired from X-ray
data were included in the calculation of the lattice potential.
This ratio is directly comparable to the nonprecessed diffrac-
tion intensity ~ISADP! divided by the calculated kinematical
intensity. The precessed intensity, however, cannot be di-
rectly compared to the Blackman calculation because it is
integrated over angle a ~see Fig. 1! rather than excitation
error. Intensities measured from precession patterns must
be converted using the relationship
Iprec@ g?1??
g
2R0?
2
{?
0
2p
Ig~a! da.
~2!
The geometrical prefactor accounts for excitation error only;
an extra factor of 1/g{!1?~g/2R0!2, suggested by Gjønnes
for beam convergence,has been omitted,because the preces-
sion patterns were acquired using parallel illumination. R0
is the radius of the zeroth-order Laue zone, calculated to be
1.015 Å?1for this experiment. The results are plotted in
Figure 7, wherein the y-axis represents the ratio of calcu-
lated as well as measured dynamical-to-kinematical inten-
sity ratios and is unitless.
Comparing two-beam theory ~dots! to the precession
data ~circles!, the intensity ratios cluster along the curve but
do not appear to exactly follow two-beam theory. At higher
~.3! Ahklvalues, intensity ratios are too low. However, the
general trend of the precession data is distinctively Blackman-
like in the critical region before the first oscillation of the
Bessel integral: Intensity ratios decrease with increasing
Ahkl. Compared to the nonprecessed SADP intensities ~tri-
angles!, the precessed data exhibits behavior far closer to
two-beam theory than SA diffraction. Figure 7 does not
show all SAD data points collected in the current x-axis
range. Roughly 50 data points reside above the maximum
intensity ratio scale shown on the plot and are distributed
over various Ahkl. The precession data also contains about
half a dozen outliers at very low Ahklthat are several orders
of magnitude too large ~not shown!. However, these occur
only for Ahkl , 0.1—caused by division by very small
kinematical intensities—and do not express the apparently
random scatter of SADP intensity ratios.
The above result is not sufficient to dismiss two-beam
theory as a good model of precession intensity behavior.
Given the clustering behavior of the precession outliers at
small Ahkl, the deviations from the two-beam calculation
are likely due to two related sources of error:
1. The experimental diffraction patterns for the @001# zone
were found to be slightly different from the expected
structure proposed by Vander Griend et al. ~1999!. In the
calculated kinematical pattern, h ? odd reflections man-
ifest as extremely dim spots in between columns of
strong reflections. In the experimental patterns, they
were absent, indicating that the crystal under study was
likely a different phase from that which was expected.
The presence of the strong h ? even reflections and
preservation of the expected symmetry in experimental
patterns indicates that the overall lanthanum framework
structure is identical. However, loss of odd-h reflections
in experimental patterns signifies a relaxation of the
magnetically frustrated structure and, hence, a very slight
change in the arrangement of metal-oxygen tetrahedra
within the La framework. This will cause a modification
of structure factors, with the largest errors correspond-
ing to small structure factors that represent small scatter-
ing contributions.
2. ForLa4Cu3MoO12,themajorityof structurefactorscorre-
spond to large Ahkl. Unfortunately, with increasing Ahkl,
the two-beam approximation begins to break down. Low
intensity generally corresponds to small structure factor,
which corresponds to low Ahkl. This means that for this
crystal, few low Ahklreflections exist that can strongly
demonstrate the Blackman trend, rendering it difficult to
precisely map out the behavior in the region of Ahkl
where two-beam behavior is most dominant ~very high
angle reflections and/or reflections with low structure fac-
Figure 5. @001# zone axis of La4Cu3MoO12.
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Christopher S. Own et al.