SSC 4529
PERGAMON
Solid State Communications 109 (1999) 501–505
A simple theory for dynamical electron diffraction in crystals
D. Van Dyck a,*, J.H. Chen b
a
Department of Physics, University of Antwerp (RUCA), Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
b
EMAT, University of Antwerp (RUCA), Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
Received 13 November 1998; accepted 4 December 1998 by M. Cardona
Abstract
A simple but sufficiently accurate expression is obtained for the exit wave of high energy diffracting electrons at a crystal in
zone axis orientation. The exit wave at each atomic column can be parametrised with only one parameter, which is a function of
the projected ‘‘weight’’ of the column. This result allows us to speed up the calculations enormously and make accurate
quantitative structure refinements using electron diffraction and/or high resolution images feasible. 䉷 1999 Published by
Elsevier Science Ltd.
Keywords: C. Scanning and transmission election microscopy; C. Crystal structure and symmetry
1. Introduction
With modern electron microscopes, it is possible
nowadays to achieve a resolution of about oneAngstrom, which is sufficient to discriminate individual atoms. This would make it possible to determine
the atomic structure of matter quantitatively without
much prior knowledge and with a precision that is
comparable to that of X-ray crystallography.
However, contrary to X-ray crystallography, the technique could also be applied to microcrystals and
defective crystals [1]. However, this ambitious goal
is still hampered by difficulties in the quantitative
interpretation of the data.
A quantitative refinement consists of searching for
the best fit between simulated and experimental datasets (images and/or diffraction patterns) in which all
model parameters (atom coordinates, specimen orientation and thickness, imaging parameters, etc.) are
* Corresponding author. Tel.: 0032 3 2180 211; Fax: 0032 3 2180
318.
varied. In fact one searches for a global optimum in
a high dimensional space. This search is done in an
iterative way, in which each step requires full calculation of the dynamical electron diffraction in the
crystal. At present, this is done with standard multislice programs [1], which, if repeated for thousands
of times, presents a real bottleneck for flexible
applications.
A simpler expression for the exit wave, and hence
for the image and diffraction pattern, not only allows
us to speed up the calculation drastically, but also
allows us to calculate the change (gradient) of the
fitness function with respect to change in the parameters in an analytical and hence a more robust way
so as to improve the convergence of the refinement
procedure.
So far attempts to obtain an expression for the exit
wave in closed analytical (the ‘‘holy grail of electron
diffraction’’) form have not been successful, except
for very thin objects (phase object approximation) and
for crystals in two or three beam orientation, but
neither of these approaches is of practical value for
realistic high resolution situations.
0038-1098/99/$ - see front matter 䉷 1999 Published by Elsevier Science Ltd.
PII: S0038-109 8(98)00599-7
502
D. Van Dyck, J.H. Chen / Solid State Communications 109 (1999) 501–505
p
Fig. 1. Scaled 1S wavefunction (normalized) for different types of atoms and different Debye–Waller factors (B), plotted against X ˆ Es r
with Es the 1S eigenvalue.
The reason for this is that most of the theories and/
or simulation programs used till date, are based on
plane wave expansions for the electron wavefield.
This approach stems from X-ray work and is efficient
only if the scattering is weak. However, in a zone axis
orientation, where the projected crystal structure is
simplest, the atom cores exactly superimpose along
the beam direction and hence the scattering is very
dynamical. Therefore, it would be much better to
look for a more appropriate quantum mechanical
base to describe the dynamical wavefield.
For a zone axis orientation of the specimen in a
transmission electron microscope with the accelerating voltage larger than 100 kV, the electrons are
trapped in the electrostatic potential of the atom
columns parallel to the electron beam. Hence, it is
possible to describe the wavefunction of the diffracted
electrons, in terms of the eigenstates of the twodimensional (2D) projected electrostatic potential of
the atom columns. It was shown [2] that this potential
has only one bound eigenstate, which by analogy is
called the 1S state.
In this work, we have found a simple way to parametrise this 1S state and hence the exit wave at the
corresponding column in terms of a minimal number
of parameters. Ideally, the exit wave of a column
should be characterised by three parameters, two
coordinates of projection of the column and one parameter related to the ‘‘weight’’ of the projected
column. However, it is known that high energy electrons almost only encounter an averaged potential
along their path and are not sensitive to potential
variations along this path. Hence, one can expect to
obtain only projected information and ambiguity
about the types and distance of atoms along a column,
which can be removed by combining information
from different zone axis orientations.
Finally, we can admit that the accuracy of an
Fig. 2. Periodicity D1S of the electron wave along the atom column, plotted against …d2 =Z† ⫹ 0:276B: The linear relationship is clear.
D. Van Dyck, J.H. Chen / Solid State Communications 109 (1999) 501–505
503
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D. Van Dyck, J.H. Chen / Solid State Communications 109 (1999) 501–505
Fig. 3. Amplitudes of the diffracted beams from Cu [0 0 1] plotted against the crystal thickness. The symbols ‘‘MS’’ and ‘‘1S’’ denote the
results calculated by the multislice method and the present method, respectively.
analytical exit wave expression has to be of the order
of say 5% as many other sources of error such as
noise, inaccuracy of atom potentials, inelastic scattering, thermal effects, etc. are at least of the same order
of magnitude.
that for a crystal in zone axis orientation, the exit wave
at one particular column located at the origin can be
approximated as
c…r; z† ˆ 1 ⫹ fs …r†‰e⫺2p iEs z ⫺ 1Š;
1†
where f s(r) is the 1s eigenstate of the Hamiltonian
H(r) of the projected column potential.
2. Theory
It was shown by Van Dyck and Op de Beeck [2]
H…r†fs …r† ˆ Es fs …r†;
2†
where r is the vector in the 2D plane perpendicular to
D. Van Dyck, J.H. Chen / Solid State Communications 109 (1999) 501–505
the electron beam, Es the 1S eigenvalue and z the
crystal thickness.
A 2D potential of this form has one strongly bound
state, which by analogy to atoms, is called the 1S
state. Note that the other states are not neglected,
but for thin crystals these will not interfere and are
incorporated in the term 1 in Eq. (1). If needed we can
improve Eq. (1) so as to account for these states up to
first order [3]. The 1S state is radially symmetric and
can be expressed as a function of the radial coordinate
r.
By comparing the 1s states calculated for various
atom column types, we observed empirically that all
the 1s states obey an empirical scaling behaviour [3].
Fig. 1 shows a superposition of the scaled f s functions
obtained from various types of columns. It is clear that
within an acceptably small error, all functions are
identical.
p 1
p fs …r† ˆ w0 Es r ˆ w0 …X†;
3†
Es
p
where both X ˆ Es r and w 0 are pure numbers without units.
From Eq. (1), the motion of the electron along the
column direction is perfectly periodical and the periodicity is given by
D1S ˆ
1
:
Es
4†
From the previous equation, we expect that the exit
wave at a particular column is radially symmetric and
has a universal shape, with only one extra parameter
Es (or D1S) related to the projected ‘‘weight’’ of the
column, which is a function of the atomic number (Z),
the distance d between successive atoms along the
column, and the temperature coefficient B (Debye–
Waller factor).
To our surprise (Fig. 2), we observed empirically
that D1S is a rather simple function of Z, d, B, as shown
in Fig. 2.
" 2
#
d
⫹ 0:276B :
5†
D1S ˆ a
Z
Fig. 3 shows the amplitudes of various diffracted
beams in Cu [1 0 0] as a function of thickness,
505
calculated both with the present method and with
the classical multislice program. The difference
between the results is only marginal and smaller
than the experimental accuracy.
From fitting with experimentally obtained exit
waves, it should therefore be possible to derive accurate values for the column positions, and reasonable
values for their weights. The extinction distance Eq.
(5) is related in a simple way to Z, d and B, but does
not allow us to unambiguously determine these values
individually without prior knowledge.
The physical reason behind the simple relation(s) is
still to be investigated.
3. Conclusion
We established empirically a simple analytical
expression for the exit wave of a crystal in a zone
axis orientation. The exit wave at each column can
be parametrised with only one parameter which is
related to the ‘‘weight’’ of the column and which is
a simple function of the atom number, atom distance
in the column and temperature factor. Details are
published in a separate article [4].
These preliminary results have to be tested in dynamical calculations for complex crystals, and possibly
extended so as to include small tilts from the zone
axis.
Acknowledgements
One of the authors (JHC) is grateful to Professors J.
Van Landuyt and G. Van Tendeloo for their enduring
support and IUAP 4/10 (Belgium) for financial
support.
References
[1] H.W. Zandbergen, S.J. Andersen, S. Jansen, Science 277
(1997) 1221–1225.
[2] D. Van Dyck, M. Op de Beeck, Ultramicroscopy 64 (1996) 99–
107.
[3] M. Op de Beeck, D. Van Dyck, Phys. Stat. Sol. 150 (1995)
587–602.
[4] D. Van Dyck, J.H. Chen, Acta Cryst., in press.