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Accurate Structure Refinement and Measurement
of Crystal Charge Distribution Using Convergent
Beam Electron Diffraction
Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287
energy filtering; imaging filter; 2-D detectors; structure factor
The method for accurate structure refinement from energy-filtered convergentbeam electron diffraction (CBED) patterns is described with emphasis on recent progress in using
imaging filters and 2-D detectors. Details are given about the underlying theoretical model and the
statistical analysis of experimental data. The relationship between crystal potential and charge
density is also derived for crystals at thermal equilibrium. The method is applied to the refinement
of Si (111) and (222) structure factors using various goodness-of-fit (GOF) criteria. Results show that
the refinement method is robust and highly accurate. The importance of the experimentally
measured structure factors is illustrated through the study of the charge density in MgO. With the
measured structure factors, it is possible to obtain details about the charge redistribution due to
crystal bonding. Microsc. Res. Tech. 46:220–233, 1999. r 1999 Wiley-Liss, Inc.
Refinement is an integral part of the structure determination process. Accurate information about atomic
positions, Debye-Waller (D-W) factors, and crystal potentials or charge distributions is obtained through the
refinement process of comparing experimental and
theoretical intensities, after the initial estimation of
atomic positions using direct methods. While considerable difficulties exist for using direct methods with
electron diffraction (ED) due to the multiple scattering,
refinement of ED data is not only possible, but also can
be done with high accuracy. This is because the theory
for dynamical ED is well developed, and, once the
structure model is known, theoretical intensities can be
calculated relatively quickly with today’s fast computers.
In addition to X-ray or neutron diffraction with their
established methods and sophisticated software packages, the ED method is needed for the following reasons: (1) Electron beam can be focused down to a probe
of sub-nanometer size. Such a small probe can be used
to characterize materials of similar dimensions in
combination with other electron microscopic methods.
(2) Electron beam also interacts with crystals differently from X rays or neutrons; with its Coulomb interaction the electron probes both electrons and nuclei in
crystals. (3) The interaction strength with matter is
⬃104 times stronger for 100 kV electrons than for X
Recent developments show that it is possible to
obtain accurate measurements of unit cell parameters
(Tomokiyo et al., 1993; Zuo, 1992; Zuo et al., 1997a),
structure factors (Holmestad et al., 1995; Saunders and
Bird 1995; Zuo and Spence 1991), atomic positions
(Tsuda and Tanaka 1995), and D-W factors (Nüchter et
al., 1995) using quantitative convergent beam electron
diffraction (CBED). This is due to the fact that most
applications in materials science involve the characterr 1999 WILEY-LISS, INC.
ization of structural modifications, for which refinement methods can be readily applied. It is also significant because there is a genuine need of methods for
accurate structural parameter measurement; e.g., to
determine the charge distribution due to crystal bonding. Accurate structure parameters, such as the distribution of charge, provide the indispensable test of the
theoretical approximations (Zuo et al., 1997b). Traditional X-ray methods have been less successful in
measuring charge distributions of crystals, especially
the charge redistribution from bonding. A very small
fraction of charge redistributes when atoms form crystals; this can only be measured with highly accurate
structure factors. The accuracy of X-ray structure factors obtained using the kinematical approximation is
limited by extinction and absorption effects.
Although quantitative analysis of ED is a more
recent development, the history of electron crystallography dates to the early stages of the development of
electron microscopy. The Bloch wave theory for electron
multiple scattering was originally formulated by H.
Bethe (1928) in his thesis. The first CBED pattern was
obtained by Möllenstedt as early as 1939 (Möllenstedt,
1939) from mica. MacGillavry (1940) first attempted
structure-factor measurement from an experimental
CBED pattern by using the two-beam theory of Blackman (1939). During the 1970s, there were a number of
attempts to measure structure factors from CBED
patterns using more sophisticated methods, most noticeably by Cowley, Goodman, Moodie, Lemphful, and
Ichimiya (for a review, see Spence, 1993). Recently, the
dramatic increase in computer speed and the development of new detectors have brought significant progress
Contract grant sponsor: NSF; Contract grant number DMR 9412146.
*Correspondence to: J.M. Zuo, Dept. of Physics and Astronomy, Arizona State
University, Tempe, AZ 85287. E-mail address:
Received 6 October 1997; accepted in revised form 5 April 1999
in this field. The fast computer makes it possible to do a
full dynamical calculation of CBED intensities. The
combination of the energy-filter (Krivanek et al., 1995;
Lanio, 1986; Rose and Krahl, 1995; Tsuno, 1997) and
two-dimensional (2-D) digital detectors—slow-scan CCD
cameras (SSC) (Spence and Zuo, 1988) and imaging
plates (IP) (Ogura et al., 1994)—provides accurate
experimental intensities with a speed that was not
possible before.
The principle of structure-factor measurement from
CBED patterns was outlined in the study of GaAs for
structure-factor amplitudes (Zuo et al., 1988) and CdS
for structure-factor phases (Zuo et al., 1989). Spence
(1993) gave a detailed review and a survey of other ED
techniques. Since then, the method has been improved
significantly. Especially, by the combined use of an
imaging energy filter and 2-D detector, 2-D diffraction
patterns can be routinely recorded and used in fitting. A
2-D diffraction pattern recorded in regular arrays gives
significantly more experimental information than a few
line scans. To take advantage of the 2-D information
available from the new detectors, we have developed a
new algorithm (Zuo, 1997a). However, these new technologies have also introduced a new set of problems.
Especially, it is necessary to correct for the nonuniformity and the limited resolution in the filter
and/or detector. It is also necessary to estimate the
amount of noise introduced by the detector. We will
address these problems in this paper.
This paper concentrates on the refinement method
and the measurement of individual structure factors. It
is also possible to refine structure parameters directly
from CBED patterns. Many works in this area have
been published recently. Specifically, for the measurement of atomic positions see Tsuda and Tanaka (1995)
and Tomokiyo et al. (1996); for the measurement of the
D-W factor see Nüchter et al. (1995); and for the
measurement of lattice parameters using the program
described here see Zuo et al. (1997a). An attempt has
also been made to measure atomic positions using the
method described here; for an initial report see Holmestad et al. (1997). For the application of quantitative
CBED to other areas of crystallography and materials
problems, see Holmestad et al. (1999). The method
discussed here is general and can be applied to both
systematic and zone-axis cases. However, in a zone-axis
orientation, additional benefits can be obtained using
symmetry. For refinement using zone-axis patterns, see
Saunders and Bird (1995).
independent electron in a potential field:
[K 2 ⫺ (k ⫹ g)2] Cg ⫹
⫽ 0.
Here we have
K 2 ⫽ k 2o ⫹ Uo,
Ug ⫽ U gC ⫹ U 9g ⫹ iU8g,
Ug ⫽ 2m 0 e 0 Vg /h2
is the optical potential for beam electrons, which consists of the crystal potential V, absorption V8, and a
correction to the crystal potential due to virtual inelastic scattering V9. The theory of optical potentials for an
electron beam was developed by Yoshioka (1957). The
most important contribution to U8 and U9 for g ⫽ 0
comes from inelastic phonon scattering. Both U’ and U9
can be calculated, given the phonon distribution in the
crystal. Details on the evaluation of the absorption
potential are given by Bird and King (1990), by Weickenmeier and Kohl (1991) for atoms with isotropic D-W
factors, and by Weickenmeier and Kohl (1998) and Peng
(1997) for anisotropic D-W factors. The virtual scattering term U9 has been estimated by Rez (1978b) for
limited cases, and more recently by Anstis (1996). The
U9 term is significantly smaller than U’ for silicon
according to Rez. However, for heavy elements the U9
may become important (Anstis, 1996).
For an electron beam with a sufficiently small diameter, the diffraction geometry can be approximated by a
crystal slab with surface normal n. Letting
k ⫽ K ⫹ ␥n,
where ␥ is the dispersion of the wavevector inside the
crystal, and inserting equation 5 into 1 leads to K 2 ⫺
(K ⫹ g)2 ⫺ 2Kn␥ ⫺ ␥2 ⫽ 2KSg ⫺ 2Kn␥ ⫺ ␥2, inside the
square bracket. Neglecting the backscattering term of
␥2, we obtain from equation 1
2KSgCg ⫹
⫽ 2Kn 1 ⫹
The Bloch wave method is used to calculate theoretical intensities. For ED from perfect crystals with
relatively small unit cells, the Bloch wave method is
preferable because of its flexibility and accuracy. The
multi-slice method or other similar methods are best in
the case of diffraction from strained crystals or crystals
containing defects. A more detailed description of the
Bloch wave method can be found in Spence and Zuo
(1992). For high-energy electrons, the exchange and
correlation between the beam and crystals electrons
can be neglected (Rez, 1978a), and the problem of ED
reduces to solving the Schrödinger equation for an
2 ␥C .
Here Kn ⫽ K · n and gn ⫽ g · n. Equation 6 reduces to an
eigen equation by renormalizing the eigenvector
(Metherell, 1975):
Bg ⫽ 1 ⫹
To simulate a CBED pattern, a coordinate system is
needed to specify each beam direction, K. We use the
zone-axis coordinate system, in which the incident
beam direction is specified by its tangential component
(Kx, Ky) in the closest zone axis. For each point inside
the CBED disk of g, the intensity is given by:
Ig(Kx, Ky) ⫽ 0 ␾g(Kx, Ky) 02
Vg ⫽
ci(Kx, Ky) C gi(Kx, Ky) exp [2␲i␥i(Kx, Ky) t] ,
U geff ⫽ Ug ⫺
with eigenvalues ␥i and eigenvector C gi from diagonalizing equation 6. Here ci is the first column of the inverse
of the eigenvector matrix or the excitation coefficient as
determined by the incident-beam boundary condition
(Spence and Zuo, 1992). The calculations converge with
increasing number of beams included. Straightforward
use of equation 6 with a large number of beams is often
impractical since the matrix diagonalization is very
time-consuming. The computer time needed to diagonalize an N ⫻ N matrix is proportional to N3. A solution to
this is to use the Bethe potential:
兰 d r exp (2␲ig · r)7V(r)8
r8 exp (2␲ig · r8) 57Z(r8)8 ⫺ 7␳(r8)86 (12)
exp [2␲ig · (r ⫺ r8)]
兰d r
0 r ⫺ r8 0
4␲ ⑀o g2
(Zg ⫺ Fg).
Here we have
Fq ⫽
兰 d r exp (2␲iq · r) 7␳(r)8
exp (2␲iq · R ) 兺 f (q) exp (2␲iq · r )
N 兺
⫻ 7exp (2␲iq · ui)8
⫽ ␦(q ⫺ g)
兺 f (q) exp (2␲iq · r )
2KS geff ⫽ 2KSg ⫺
⫻ 7exp (2␲iq · ui)8,
and similarly,
for weak beams. Here the summation is overall weak
beams h. The Bethe potential accounts for the perturbation effects of weak beams. A weak beam is defined
0U 0ⱖ␻
The ␻max is selected for the best theoretical convergence
(Birkeland et al., 1996; Zuo and Weickenmeier, 1995).
Beams are selected subject to cutoff criteria of maximum g length and their perturbation strength. These
should be tested to check their effects on the calculated
For a 100-kV electron traveling at a speed 1⁄3 that of
light, the time it spends inside a TEM specimen is
about 10⫺15 seconds. The typical phonon frequency is
about 1012 Hz. Thus, the fast electron sees individual
frozen configurations of a vibrating lattice. The experimentally observed diffraction pattern at a 1-second
exposure time is an average of individual diffraction
patterns from ⬃1012 instant configurations of the crystal. It can be shown that the average diffracted beam at
a reciprocal lattice point is equal to that scattered by an
average potential 7V(r)8 under both the kinematical and
dynamical diffraction conditions. In general we have,
7V(r)8 ⫽
兰 d r8
7Z(r8) ⫺ ␳(r8)8
0 r ⫺ r8 0
Zq ⫽ ␦(q ⫺ g)
兺 Z exp (2␲iq · r )
⫻ 7exp (2␲iq ·ui)8.
Here u is the displacement from the equilibrium atomic
position and g is a reciprocal lattice point. The quantity,
7exp (2␲iq · ui)8, is the temperature factor. For harmonic vibrations, it can be shown that (Willis and
Pryor, 1975)
T(g) ⫽ 7exp (2␲ig · u)8 ⫽ exp (⫺2␲27(g · u)28).
For isotropic vibrations, this reduces to the familiar
D-W factor
T(g) ⫽ exp (⫺2␲2g27u28) ⫽ exp (⫺Bg2/4),
with B ⫽ 8␲2 7u28. For the anisotropic case, see Willis
and Pryor (1975). Equation 12 is the basic relationship
between potential and charge density as measured by
electrons or x-rays for the study of crystal bonding. It is
clear from equation 13 that the thermal effects on both
potential and charge density can be described by the
same temperature factors.
The experimental setup using an in-column ⍀ energyfilter is illustrated in Figure 1. A probe (P) is formed on
top of a thin crystal plate by de-magnifying the source
Fig. 1.
Schematic diagram of experimental setup using a in-column imaging energy filter.
using the condenser lens. A circular aperture is placed
after the condenser lens, which forms a cone of illumination. A diffraction pattern consisting of disks is formed
at the back focal plane of the objective lens. A point in
the CBED disk and all conjugate points in other disks
(related by reciprocal lattice vectors) constitute a diffraction pattern for one incident beam direction in the
illuminating cone. In Figure 1, the filter accepts the
image of the probe formed by the objective lens. In the
LEO 912 electron microscope, there is an additional
intermediate lens between the filter and objective lens.
The filter forms a series of probe images at its exit
plane, dispersed according to the beam energy. An
energy-selection slit is placed around the zero-loss
image to obtain the energy-filtered CBED pattern,
which is recorded with a 2-D detector. The condensor
aperture determines the CBED disk size. The disk
should be comparable to the reciprocal-lattice spacing
to avoid excessive overlap, which makes the interpretation of experimental data difficult. For large unit-cell
crystals, the large angle CBED (LACBED) technique
may be useful. LACBED selects a single diffracted
beam by using a small selected-area aperture. LACBED
also removes part of the large-angle inelastic scattering
due to phonons, which is not possible with the current
energy-filter (Morniroli et al., 1997).
The diffracted beams pass through an energy filter,
which is placed after the final intermediate lens and
before the projector lens of the microscope. Alternatively, a post-column energy filter can be placed below
the viewing screen (Krivanek et al., 1995). With the
in-column filter, it is possible to have a combination of
different detectors, and it also takes the full advantage
of the projector lens of the microscope. Uhlemann and
Rose (1994) made a theoretical comparison between
different filters. Details about the ⍀-filter can be found
in Lanio (1986), Rose and Krahl (1995), and Tsuno et al.
(1997). The LEO 912 ⍀ filter uses the design of Lanio
(1986). A description of the post-column Gatan Imaging
Filter (GIF) is given by Krivanek et al. (1995). For the ⍀
filter, the filter takes an image (in diffraction mode) of
the diffraction pattern at the entrance pupil plane and
forms a diffraction pattern at the exit pupil plane and
an image at the energy-slit aperture. Figure 2 shows a
zero-loss energy-filtered Si[111] CBED pattern and the
corresponding spectrum (see inset in Fig. 2) using the
be 40 mm. At such a small CL, it is often difficult to
adjust for lens distortions. Additionally, there are some
residual distortions in the GIF, which are difficult to
characterize. The geometric distortion is almost absent
in the ⍀ filter owing to its mid-plane symmetry (Rose
and Krahl, 1995), in principle.
For quantitative analysis, the energy-filtered CBED
pattern is recorded digitally with an SSC or IP in an
array of pixels. Both detectors have excellent linearity
and dynamic range in each pixel. For a comparison
between these two detectors, see Zuo (1996) and Zuo et
al. (1996). Two important characteristics of a linear
detector for quantitative analysis are the point-spread
function (PSF) and detector quantum efficiency (DQE).
The PSF determines the resolution of as-recorded images, while DQE measures the amount of noise introduced in the detection process. Methods for measuring
these are given by Zuo (1996) and Weickenmeier et al.
(1995). Figure 4 shows the measured modulation transfer function (MTF) and DQE for Gatan model 679 SSC.
In general, the recorded image can be expressed as
␨ ⫽ Hf ⫹ n.
Fig. 2. Energy-filtered Si [111] zone axis pattern using the ⍀-filter
of Leo 912 electron microscope at 120 kV with 15 eV energy window.
The corresponding spectrum is shown at lower right corner. The white
lines indicate the position of the energy window. Arrows indicate the
energy-filtering envelope. (The size of the envelope can be enlarged
with free lens control).
LEO 912 electron microscope with an energy slit 16 eV
wide. The envelope seen in Figure 2 is due to the
non-isochromaticity caused by the axial second-order
aberrations at the energy-selection plane (Rose and
Krahl, 1995). The effects of this aberration make a
circle in the object plane into an ellipse; the center of
which is displaced by a distance proportional to the
square of the radius. This is illustrated in Figure 3. A
disk forms a flattened cone-shaped aberration figure.
Figure 3c shows the experimentally observed aberration figure for silicon. For a large object, the aberration
figure overlaps on the energy-dispersion plane with
those for different energy losses. With a finite energy
slit, the primary beam energy of the filtered pattern
differs at different radii from the optical axis (called
non-isochromaticity). By using a narrower energy slit,
it is possible to reduce the contributions from electrons
of different energies. However, this also results in a
smaller field of view. For energy filtering with the
zero-loss peak, the problem is most severe for metals
with large plasmon losses. In the CBED mode, the
image is made of diffracted beams. They are sufficiently
separated in the Leo 912 ⍀ filter for the filtering of a few
reflections (see Fig. 2). We observed no significant
difference between the zero-loss disk intensities with 8and 16-eV windows. The plasmon energy of silicon is
about 18 eV. Krivanek et al. (1995) reported a better
performance for isochromatic imaging using the GIF.
The field of view of the GIF is limited by the smallest
camera length (CL) available in the microscope and the
size of the CCD used for detection. The GIF has a
magnification of 15⫻, so to record a diffraction pattern
of 600 mm CL with the GIF, the microscope CL needs to
Here ␨ and f are the recorded and original image,
respectively, n is the noise, and H is the PSF operator.
The effects of the PSF can be removed by deconvolution.
However, the direct deconvolution of a recorded image
using the inverse of the PSF leads to an excessive
amplification of noise. To overcome this, we use the
constrained image-restoration technique by employing
a filter of type (Ren et al., 1997):
Q(u, v) ⫽
H *(u, v)
H *(u, v) H(u, v) ⫹ ␥P(u, v)
with P(u, v) as a smoothing function and ␥ adjusted to
satisfy the constraint
00 g ⫺ Hf̂ 00 ⫽ 00 ⌬n 00 .
Here f̂ is the deconvoluted image and ⌬n is the estimated additional noise introduced in the detection
process. There are a number of choices for P(u, v). The
optimum in the least-square measure is the Wien filter,
in which P(u, v) is the ratio of the power spectra of
signal and noise. This requires a prior knowledge of the
signal, which is often unknown. Alternatively, we may
take P(u, v) ⫽ [⫺4␲(u2 ⫹ v2)]2, which makes the restored image as smooth as possible. The value of ␥,
which measures the degree of removal of the PSF, with
0 as complete, depends on the amount of noise in the
recorded image—the smaller the noise, the better the
deconvolution. Other procedures for deconvolution are
also available. The Richardson-Lucy method described
by Snyder et al. (1993) is specifically targeted for
Poisson processes, which can be applied to CCD images.
Figure 5 shows the relative importance of energyfiltering and image deconvolution. The example used is
the TiA1 (200) systematics. The top is the unfiltered
CBED pattern, which shows a large amount of background in the line profile. The filtering removes most of
the background. The remainder is mostly due to the
Fig. 3. The aperture aberration in uncorrected ⍀ energy filter. a: Aberration figure in energy selection
plane. b: Corresponding circle in the object plane for ellipse in a. c: Experimentally observed spectrum for
Silicon in image mode.
limited resolution of the detector, which is removed by
deconvolution. The deconvolution also makes edges
sharper. For comparison, we show the fitted theoretical
pattern at the bottom. The effects of filtering and
deconvolution are large compared to the amount of
noise expected from the detector.
The quality of a CBED pattern depends on the
sample and the microscope. Critical alignment of the
microscope is required. Special attention is needed for
the optimum objective lens setting, specimen z-position, correction for projector lens distortion, and the
choice of probe size. For the use of an SSC, a highquality image is needed to correct for the gain variations. For a relatively flat sample with no defects, such
as silicon, acquiring a good-quality CBED pattern in a
non-zone-axis orientation for quantitative analysis is
fairly easy. Acquiring a good zone-axis pattern is considerably more difficult; the strong dynamical channeling
condition is very sensitive to defects and surface imperfections or contamination. Distortions of CBED patterns due to crystal faults such as stacking faults and
dislocations are easy to spot. The deficiency line tends
to split owing to the strain field associated with these
faults. Thickness variation under the probe also distorts the CBED pattern; this can be reduced with a
smaller probe. The ultimate check of the quality of
CBED patterns is the refinement itself, which measures the degree of agreement between theory and
Fig. 4. Experimentally measured (a) MTF and (b) DQE of Gatan model 679 SSC. The dashed lines in
b are projected DQE with zero noise in gain image.
experiment. Additional assessment of experimental
diffraction pattern quality can be made in a zone-axis
orientation, where the symmetry-related points can be
used to detect distortions.
Automated refinement from CBED intensities is
achieved by finding the best fit between experiment and
theory. The best fit is defined by using a goodness-of-fit
(GOF) parameter:
GOF ⫽ ⑀
兺 h(I
i ,
␴i, I it).
tion of (I t ⫺ I exp)/␴. Box and Tiao (1992) defined a class
of probability distributions
␳(I exp 0 p, ␴) ⬀
exp ⫺
⌫[(1 ⫹ ␤)/2]
Here the function h measures the agreement between
the experimental intensity, Iexp, with estimated uncertainty ␴ and the theoretical intensity, It. The theoretical
intensity is calculated with parameters, p. ⑀ is the
normalization coefficient. In the maximum-likelihood
approach, h is the logarithm of the probability distribu-
0 2
with ⫺1 ⬍ ␤ ⱕ 1. For ␤ ⫽ 0, equation 22 reduces to the
Gaussian distribution. On the basis of equation 22, we
can define a series of GOF’s using different values of ␤:
⌫[3(1 ⫹ ␤)/2] 1/(1⫹␤) I exp ⫺ I t( p)
GOF ⫽ ⑀
I iexp ⫺ I it( p)
For constant standard error, ␴, the lowest GOF gives
the maximum likelihood of the probability distribution
of equation 22. The definition of equation 23 encompasses the two most commonly used measures in
Fig. 5. A comparison between the effects of energy-filtering and deconvolution. For reference, the
bottom shows the fitted theoretical pattern.
crystallography, the ␹2 (␤ ⫽ 0) and normalized R-factor
(␤ ⫽ 1). For large ␤, the probability distribution falls off
slower than the Gaussian, which gives a more robust
measure against systematic errors in theory and experiment due to its long-tailed distribution. For cases
where differences between theory and experiment are
mostly normally distributed, except for a few outlier
points, the ␹2 refinement can be distorted owing to the
presence of these outlier points (for discussion on this,
see Huber, 1981). Huber proposed an alternative measure considering a small percentage of outlier points:
I iexp ⫺ I it( p)
(for example 95%) is covered by the normal distribution. In next section we will show how the choice of GOF
affects the refinement results using a test case of
The experimentally recorded intensity in each detector pixel (i, j) is the sum of all intensities within the
pixel. Intensity mixing between different beam directions is also expected from the finite spatial resolution
of the detector and energy window of the filter; in
particular, both small- and large-angle thermal diffuse
scattering are not removed by the energy filter. A
general expression for the theoretical intensity considering all these factors is:
I i,t j ⫽
h(t) ⫽
2at ⫺ a2
兰兰 dx8dy8t(x8, y8) C(x ⫺ x8, y ⫺ y8)
⫹ B(xi, yj).
This definition adopts the normal distribution for the
center and exponential distribution for the tail. The
parameter a is selected so that the majority of points
Here, the diffracted intensity, t, convoluted with the
response function, C, and the background intensity, B,
are integrated over the pixel. Simple models of C are
the delta and Gaussian functions, which can be implemented easily. The background function is often as-
Fig. 6. A simplified flow chart of EXTAL program. Two optimization cycles are employed for structural
and geometric parameters respectively. The optimization for geometric parameters is repeated for each
individual diffraction pattern.
sumed constant within the CBED disk (Saunders and
Bird, 1995), which seems sufficient for the refinement
Depending on the experimental intensities collected,
the parameters can be structure-factor amplitudes
and/or phases, absorption coefficients, lattice constants, atomic position parameters, and D-W factors.
Other parameters include crystal orientation, thickness, electron accelerating voltage, detector function,
and residual diffuse background intensities. For structure-factor measurements, we can choose between systematic, three-beam, or zone-axis diffraction conditions. We prefer the systematic condition for its
simplicity, short computational time, and good accuracy.
Estimation of errors in refined parameters is quite
involved in the case that the distribution is not normal.
For discussions on this, see Box and Tiao (1992). In the
normal case, the precision of the measured parameters
is given by
␴a2k ⫽ ␹2D kk
Dkl ⫽
1 21 2
1 ⭸I it ⭸I it
⭸pk ⭸pl
The partial derivative of the intensity with respect to a
parameter can be calculated numerically or by using
perturbation theory. General formulas using the firstorder perturbation method for these derivatives are
given by Zuo (1997b).
There are a number of refinement algorithms (Zuo
and Spence, 1991; Nuchter et al., 1995; Saunders and
Bird, 1995; Birkeland 1997). The REFINE program by
Zuo and Spence (1991) uses line scans of intensities.
Modifications of this program have also been made to
include non-parallel line scans (Nuchter et al., 1995;
Birkeland 1997). The algorithm of Saunders and Bird
(1995) uses a Fourier-transform method for the evaluation of derivatives and has been mostly applied to
zone-axis patterns.
We have developed an new algorithm EXTAL (Electron crys(X)TALlography) for quantitative analysis of
2-D CBED patterns (Zuo, 1997a; Zuo, 1998b). The
simplified flowchart of the program is shown in Figure
6. This program is capable of dealing with multiple
CBED patterns by separating the structural parameters of structure factors, atomic positions, and D-W
factors from the geometrical parameters of crystal
orientation, image position, camera length, and sample
thickness. Since ED patterns are generally twodimensional owing to the short wavelength of the
Fig. 7. a: Experimentally recorded energy-filtered Si (1 1 1) systematic CBED pattern. b: Intensities along lines indicated in a and the
best theoretical fit. c: Difference between theory and experiment
plotted in b normalized by the estimated error in experimental
Fig. 7.
TABLE 1. Dependence of Si (1 1 1) and (2 2 2) Refinement Results on the Probability Distributions of Equation 22
U(1 1 1) 10⫺1
U8(1 1 1) 10⫺3
U(2 2 2) 10⫺3
␤ ⫽ 0.25
␤ ⫽ 0.5
␤ ⫽ 0.75
␤ ⫽ 1.0
incident electrons, using multiple diffraction patterns
in different projections enables the measurement of
three-dimensional parameters such as atomic positions
and the parameters of the three-dimensional unit cell.
The separation between structural and geometrical
parameters is achieved using two optimization cycles.
For each trial in the structural parameter optimization,
an independent optimization is carried out for the
geometrical parameters. This is repeated for each diffraction pattern included. Considerable saving of computer time is achieved by fixing the beam direction in
the theoretical calculation and locating the corresponding directions in the 2-D pattern as suggested by
Gribelyuk and Ruhle (1993). When additional image
processing is applied to both experimental and theoretical CBED patterns, the program has shown to be very
effective in measuring lattice parameters in the presence of dynamical effects. The advantage of this program, compared with the REFINE program, is that
multiple line or area scans can be taken, which allows
the user flexibility in designating the experiment for
the maximum sensitivity to the measured structural
parameters. An example of using this program for
structure-factor measurement is shown next. For an
example of measuring lattice parameters, see Zuo et al.
As an example, we describe the structure factor
measurements of Si (111) and (222) using the methods
described here. Silicon is a good test case because of
existing X-ray pendellosung measurements. The experimental CBED pattern of silicon (1 11) systematics was
recorded by tilting about 12° off the [110] zone at room
temperature and with a beam energy of 196.7 kV. This
is shown in Figure 7a. The pattern was taken on a
Philips CM200 with a field-emission gun and equipped
with a GIF (BEML, China). The characteristics of the
CCD were measured following the procedures described
by Zuo (1996). Figure 7a was deconvoluted using the
method described in section 3 (Ren et al., 1997). The
orientation was estimated by indexing the diffraction pattern using the IdealMicroscope program (Zuo et al., 1993).
The amount of experimental data in figure 7a far
exceeds the number of parameters to be determined, so
we select only some of the pixels in the experimental
pattern for fitting. The typical number of data points is
about a few thousand. We chose these data points by
line scans. These lines are drawn with consideration
given to their sensitivity to certain parameters. For
example, intensities along a line across a diffracted disk
are sensitive to structure factors and specimen thickness, intensities along a line across the zero disk are
sensitive to absorption parameters, and intensities
along a line across ZOLZ and HOLZ lines are sensitive
to orientation and magnification. The line scans used
TABLE 2. Dependence of Si (1 1 1) and (2 2 2) Refinement Results
on the M-Estimator of Equation 24 and Comparison
With the Experimental X-Ray Results*
A ⫽ 3.0
A ⫽ 2.0
A ⫽ 1.5
X ray
U(1 1 1) 10⫺1
U8(1 1 1) 10⫺3
U(2 2 2) 10⫺3
*For a list of X-ray structure factors of Si, see Zuo et al. (1997b).
for silicon (1 11) systematics are shown in Figure 7a.
The refinement program EXTAL was used with different GOF criteria. Theoretical intensity was calculated
with 279 beams, selected using the beam selection
criteria described in Zuo and Weickenmeier (1995). The
refinement program was run repeatedly with different
starting points to ensure that the real minimum was
The refinement depends on the GOF criterion selected. To check the effects of this and to see which
criterion may suit our purpose best, we refined Figure
7a with various GOF defined in Refinement. Table 1
shows the refinement results with different ␤ for the
GOF of equation 23. The estimated error is quoted in
parentheses for the normal distribution (␤ ⫽ 0). The
refinement results depends on the selected GOF or ␤
value. The column ‘‘Mean’’ shows the average of five
different refinements and its standard deviation. The
difference between the (222) structure factors determined using the ␹2 and R-factor is somewhat large
compared with the estimated error.
Table 2 compares the results using the M estimator of
Huber with different cutoff values of a. As a decreases,
the results approach the ones obtained with the Rfactor. Compared with the X-ray value, the mean from
Table 1 gives the best agreement. Of the individual
GOFs, the R-factor seems the best. The results from the
standard ␹2 are acceptable within the estimated error;
although the (222) is on the low side compared with the
X-ray value.
To check the success of the selected GOF, we look at
the distribution of the residual difference between
experiment and theory. Figure 8 shows the histogram of
the normalized difference obtained with ␹2, the Huber
estimator of a ⫽ 2.0, and the normalized R-factor. For
comparison, we also fitted the histogram with Gaussian
(Fig. 8a and b) and exponential (Fig. 8c) distributions.
The R-factor resulted in the largest number of points
having deviations within the estimated error. In obtaining the mean from different distributions in Table 1, we
assumed an equal probability for the distribution. A
more rigorous averaging using Bayesian statistics would
require the evaluation of the probability for the distribution with the given data. This would require integration
in the parameter space, which is difficult to do in this
case. For a full discussion on the robust statistics in
Bayesian approach, see Box and Tiao (1992).
Fig. 8. Histogram of residual difference, (I t ⫺ I exp)/␴, as obtained from (a) chi-square fit (b) M fit with
a ⫽ 2.0, and (c) R-factor fit. For details, see text.
The results here show that the refinement method is
highly accurate for the structure-factor measurement
of both strong and weak reflections. The accuracy for
the Si (111) reflection is comparable to the X-ray
pendellosung method. Unlike the X-ray method, the ED
method discussed here is general. With the small probe
of CBED, it can be applied to most crystals with small
unit cells. There is no fit-for-all magic solution in
choosing the best GOF criteria. Results here show that
the refinement does depend on the GOF within the
estimated error bar. The R-factor gives better results
than ␹2 in this case.
One of the important applications of quantitative
CBED is the study of crystal bonding by accurate
measurement of low-order structure factors. An ex-
ample of this is the study of the charge density in MgO
(Zuo et al., 1997c; Zuo 1997b). For applications of
charge-density measurements to important materials
problems, see Holmestad et al. (1999). Magnesium
oxide is a prototype of ionic bonding with its NaCl
structure. There is a general lack of experimental
knowledge on the charge distribution in ionic crystals.
In studying MgO, we want to see whether the charge
distribution of MgO can be described in the traditional
picture of ionic bonding with a charge transfer of two
electrons from Mg to O. We also want to see the charge
distribution of oxygen. With oxides playing an increasingly important role in materials with new properties,
such as high-Tc superconductivity and colossal magnetoresistence, understanding oxygen states and their dependence on crystal structure can provide an important
clue to the mechanism for these exotic phenomena.
The MgO specimen was prepared by burning magnesium ribbons in air and collecting MgO smoke on a
copper grid covered with a thin carbon film. A platelet of
micrometer size and about 80 nm thick was selected
from different shapes of MgO particles. The experiment
was performed on a Philips 400T electron microscope.
The intensities were measured by deflecting the CBED
pattern over the entrance aperture of a Gatan 607
serial energy-loss spectrometer. This was done before
the energy filters and parallel detectors were available.
This setup still offers unsurpassed resolution and signalto-noise ratio when conditions in the microscope are
stable during the scan. Details about the measurement
have been published (Zuo, 1997b). Table 3 lists all
experimentally measured structure factors together
with theoretical models for comparison.
The theoretical structure factors are calculated using
different models. The Dirac-Fock (DF) spherical ions
are calculated using the multi-configurational DF program of Rez et al., 1994 and a Watson ⫹2 well for
oxygen, which is an atomic calculation. In this model,
we found that the radius of 1.2 Å gives the best fit to
experimental structure factors. Theoretical structure
TABLE 3. Listing of Present Measured Low-Order Structure Factors
of MgO and Comparison With Theory*
*Superimposed spherical 1atoms and 2ions calculated using Dirac-Fock method;
Crystal structure factor calculated using LAPW and the 3LDA or the 4GGA.
factors are calculated using the full-potential linearized
augmented plane wave (LAPW) method as implemented in the WIEN95 package (Blaha et al., 1990) for
crystals. The calculation uses approximated functionals for the exchange and correlation energies. Two
approximations are used. One is the local density
approximation (LDA) and the other is the generalized
gradient approximation (GGA). The LDA gives a slightly
better fit than the GGA, but the difference is small.
The deformation map, formed by the difference between the crystal charge density and the charge density
of superimposed atoms, is used to show the bonding
charge distributions. Figure 9 shows the difference in
charge density between the experimental data and
superimposed neutral atoms. The continuous lines
show excess electrons and the broken lines show the
depletion of electrons. Figure 9 clearly shows the
charge transfer from Mg to O. The charge distribution
around the oxygen is slightly non-spherical. A multiple
analysis of the MgO charge density shows a small
hexadecapole component. It was found that the charge
distribution in MgO can be in the full ionic form.
However, the distribution of transferred electrons is
significantly broader than predicted by theory. This
suggests some effects due to the polarization of charge
in the ionic lattice due to thermal vibrations. For
details about this, see Zuo et al. (1997c).
Recent developments in quantitative ED have shown
that it is possible to extract quantitative structure
information from CBED patterns. The success relies on
quantitative analysis of experimental and theoretical
intensities. The experimental setup for the measurement of intensities was found sufficient. The theoretical
intensity is calculated taking the full account of dynamical effects. Experimentally measured structure factors
provide an accurate description of charge distributions
Fig. 9. MgO charge density difference map between crystal and superimposed neutral atoms in a-b
plane for (a) experiment and (b) theory using LAPW and LDA.
in real crystals. For MgO, the experimental charge
density can be approximated by a superposition of
spherical Mg2⫹ and O2⫺ ions. The additional difference
between theory and experiment is due to the details of
the charge distribution around oxygen.
Many thanks to Dr. R. Holmestad for providing
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