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Electron CrystallographyЧNow a Handy Method.

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Electron Crystallography-Now
a Handy Method
Wilhelm Mertin*
The term electron crystallography denotes the structure analysis of crystalline samples from electron diffraction patterns and
electron microscopy images. The possibility of solving crystal
structures ab initio by electron crystallography has been discussed several times." -41 In physics and materials science the
problems associated with electron diffraction that arise because
of the strong interaction between electrons and matter, particularly multiple scattering etc. (dynamical effects), have been
highlighted and dealt with,". 5 . 6 ] while in molecular biology
where samples consist of only light elements, the possibilities of
kinematical electron diffraction have been exploited['. '. *I based
on principles applied in X-ray crystallography.
X-rays interact with shell electrons of atoms, while electrons
are deflected mainly by the potential of a nucleus of an atom.
Therefore, while ideally X-rays are diffracted only once (kinematical diffraction) in a suitable sample and the diffracted rays
are not further modified by additional diffraction in the passage
through the crystal, electrons undergo multiple scattering in
that crystal. The diffracted beams interfere not only with the
central beam but also with each other. This is called dynamical
diffraction. The phenomenon requires a much more complicated theoretical treatment". 61 than the kinematical case, which
is usually the basis for calculations in X-ray crystallography.
Nevertheless, many studies have been published in which structures were solved by using quantitative electron diffraction intensities. However, the commonly held view was that dynamical
effects were too
and that the initial phase models had
been gained by employing results from corresponding X-ray
For several years a new generation of electron microscopes
has been generally available with a 1-2 A resolution and an
acceleration voltage of 200-400 kV (higher than that of previous instruments). Now rows of atoms viewed end-on can be
resolved with high-resolution electron microscopy (HREM).
Such electron micrographs can be interpreted directly in terms
of the projected crystal structure if they are recorded at an
optimal defocus condition (Scherzer focus) and if the object is
sufficiently thin.[5.6.91 In the course of this development the
accumulation of experimental data has led to growing confidence that "quasi kinematical" conditions can be assumed even
for inorganic substances, minerals, and ceramics, if the area
under investigation is thin enough (weak phase approximation).
Based on principles applied in molecular biology, a system of
programs has been developed["] that offers the possibility of
evaluating micrographs and electron diffraction patterns on a
personal computer and of determining the averaged structure of
a single crystal. Other image processing systems also exist.["l
A recent publication "A crystal structure determined with
0.02 A accuracy by electron microscopy" by T. E. Weirich
Dr. W. Mertin
Institut fur Anorgdmsche und Analytlsche Chemie der Universitst
Heinrich-Buff-Ring 58, D-35392 Giessen (Germany)
Fax: Int. code +(641)99-34156
VCH Verlagsgesell.xhafI mbH. 0-69451 Wernherm. 1997
et aI.["] demonstrates the possibilities of electron crystallography. The structure of Ti, [Se, was solved and refined by utilizing
Fourier transforms of high-resolution micrographs and data
from single-crystal electron diffraction patterns.["]
In a first step an electron microscope acts as a diffractometer.[l3] As with X-rays one can register the intensities of the
deflected beams; however, one loses the phases cp of these
waves. Under conditions that are suitable for imaging a structure in projection, the diffracted intensities are[31 Z(h,k ) =
F ( h , k ) F * ( h , k )= IF(h,k)12 and F ( h , k ) = /F(h,k)lexp(icp).
IF(h,k)l is the magnitude of the structure factor amplitude
F(h,k). cp is the resulting phase of the deflected beam in the
direction h, k . The origin of a unit cell of a crystal is the reference
point from which the phases of all waves originating from atoms
are counted, and therefore the positional parameters of the
atoms in a unit cell are contained in cp. The different phases
relate the different reflections to each other. The advantage of
electrons over X-rays is that they can be diverted by electromagnetic lenses. The diffracted beams can be reunited with the primary beam to form an image of the projected structure. Phases
and amplitudes of the waves are thus utilized. When the weak
phase object approximation is valid, the image intensity is directly proportional to the projected potential of the crystal
structure.[31This means in the reverse case that not only amplitudes but also the phases of structure amplitudes can be gained
from the computed Fourier transform of a digitized electron
microscope image if it is recorded under the right conditions, as
mentioned above. But even nonideal images can be computationally improved by correcting for wrong defocus, astigmatism, etc.['*] An important step is the refinement of the
origin.["] The origin of the unit cell of the structure projection
is placed so that the experimental phases are most reasonably
related by a symmetry as high as possible; this symmetry is then
regarded as the symmetry of the structure projection. The symmetry is now imposed on both amplitudes and phases, and a
density map is calculated as an inverse Fourier-transformation.
This density map should correspond to the projected potential
of the crystal structure from which the atomic positions can be
determined. Figure 1 depicts how potential maps obtained
by crystallographic image processing of a micrograph of
Nb,W,,0,,['41 can vary with the imposed symmetry or by
shifting the origin of the unit cell. The correction of the original
electron microscopy images and all other computational steps
are by no means mere routine. Good electron micrographs and
electron diffraction patterns are still the best prerequisites.
T. E. Weirich et al.["] demonstrated that once the phase model is established the more accurate intensities and amplitudes
from electron diffraction patterns can be used to refine the structure to a level that is comparable to that obtained by X-ray
methods. In the case of Ti, ,Se, the lattice constant in the direction of projection of the structure was conveniently short
( b = 3.4481 A); thus projected atoms did not overlap within the
unit cell. Therefore the projection in one direction was suffi-
0570-0833/97/360l-O04b$ 15.00+ .25/0
Angew. Chem. Int. Ed. Engl. 1997, 36, No. 112
Figure 1 . Schematic representation and computed projection maps of Nb,W,,O,,
calculated for different conditions of symmetry and origin. a) Projection map on
(001). black dots represent NbJW atoms, symmetry pyg; mean phase error
q R e s = 18.9’. b) Structure principle [lS] of Nb2W,,0,, in projection along [OOl];
lattice parameters: u = 36.7, h = 12.3, c = 3.8 A. c) p2 Symmetry leads t o only a
slightly worse fit of the data ( q R e s =19.1’) [12] than the true symmetrypyy in a).
d) Imposed symmetryp2; origin shifted by 3/10 of lattice parameter a ; mean phase
error enlarged to q R r s = 30.9.
cient to solve the structure. However, the programm system
used[’” also offers the possibility to combine Fourier transforms of crystal projections of different directions; that is to
proceed in a way as usually done in X-ray crystallography to
solve a three-dimensional structure.
The branch of electron crystallography that cannot always
make use of high-resolution electron microscopy images and its
Fourier transforms has also advanced rapidly. It has been applied to structures of mineralsC6I and especially to the determination of structures of small crystals like those often encountered
for organic molecules and polymers which, in addition, are sensitive to radiation damage.13. 16] Direct methods, like those
known from X-ray crystallography, the maximum entropy
method, and other likelihood ranking methods have been applied successfully to the elucidation of structures.
Electron crystallography is valuable addition to the tools
available for solving crystal structures. Although the interaction
of electrons with crystalline material is very complicated and
theoretical concerns are justified, the method should be further
A n g w . Chrm. lnt. Ed. Engl. 1997. 36, No. 112
tested to establish the range of its applicability and to work out
how the problems associated with it can be solved. Crystal structures should be solved by using data from different projections.
A long-term goal would be the three-dimensional reconstruction of an inorganic real crystal volume.[’71 Materials that, for
example, are very sensitive to radiation damage, oxygen, and
moisture, or easily undergo phase changes during the preparation of the sample for electron microscopy are difficult to investigate especially since the very thin transparent areas of a sample
should remain intact. For very small crystals, however, electron
crystallography is the method of choice. Particularly in combination with X-ray, image simulation, and crystallographic image processing (CIP) methods, it can be a valuable tool. Even
micrographs whose resolution is not particularly high can be
used to obtain phase information on sublattices, for example,
when strongly scattering building elements of a structure are
present. A modern computer, simple digitizing equipment (e.g.
a C C D camera and a frame grabber) are requirements which
make the method accessible to nearly every laboratory. Of
course, one also needs to have access to a high-resolution electron microscope, but the number of these instruments available
is increasing all the time.
German version Angen Chrm 1997, 109, 46 -48
Keywords: electron crystallography electron diffraction electron microscopy * image processing structure elucidation
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VCH Verlagsgesellsrhafr mbH, 0-69451 Weinheim, 1997
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