International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 4.3, pp. 425427

The general formulations for the dynamical theory of electron diffraction in crystals have been described in Chapter 5.2 of IT B (2001). In Section 4.3.6, the computing methods used for calculating diffractionbeam amplitudes have been outlined.
Given the diffractedbeam amplitudes, , the image is calculated by use of equations (4.3.8.2), including, when appropriate, the modifications of (4.3.8.13b).
The numerical methods that can be employed in relation to crystalstructure imaging make use of algorithms based on (i) matrix diagonalization, (ii) fast Fourier transforms, (iii) realspace convolution (Van Dyck, 1980), (iv) RungeKutta (or similar) methods, or (v) powerseries evaluation. Two other solutions, the Cowley–Moodie polynomial solution and the Feynman pathintegral solution, have not been used extensively for numerical work. Methods (i) and (ii) have proven the most popular, with (ii) (the multislice method) being used most extensively for HREM image simulations. The availability of inexpensive array processors has made this technique highly efficient. A comparison of these two Nbeam methods is given by Self, O'Keefe, Buseck & Spargo (1983), who find the multislice method to be faster (time proportional to ) than the diagonalization method (time proportional to ) for N 16. Computing space increases roughly as for the diagonalization method, and as N for the multislice. The problem of steeply inclined boundary conditions for multislice computations has been discussed by Ishizuka (1982).
In the Blochwave formulation, the lattice image is given by where and are the eigenvector elements and eigenvalues of the structure matrix [see Hirsch, Howie, Nicholson, Pashley & Whelan (1977a) and Section 4.3.4].
Using modern personal computers or workstations, it is now possible to build efficient singleuser systems that allow interactive dynamical structureimage calculations. Either an image intensifier or a cooled scientific grade chargecoupled device and singlecrystal scintillator screen may be used to record the images, which are then transferred into a computer (Daberkow, Herrmann, Liu & Rau, 1991). This then allows for the possibility of automated alignment, stigmation and focusing to the level of accuracy needed at 0.1 nm point resolution (Krivanek & Mooney, 1993). An imagematching search through trial structures, thickness and focus parameters can then be completed rapidly. Where large numbers of pixels, large dynamic range and high sensitivity are required, the Image Plate has definite advantages and so should find application in electron holography and biology (Shindo, Hiraga, Oikawa & Mori, 1990).
For the calculation of images of defects, the method of periodic continuation has been used extensively (Grinton & Cowley, 1971). Since, for kilovolt electrons traversing thin crystals, the transverse spreading of the dynamical wavefunction is limited (Cowley, 1981), the complex image amplitude at a particular point on the specimen exit face depends only on the crystal potential within a cylinder a few ångströms in diameter, erected about that point (Spence, O'Keefe & Iijima, 1978). The width of this cylinder depends on accelerating voltage, specimen thickness, and focus setting (see above references). Thus, small overlapping `patches' of exitface wavefunction may be calculated in successive computations, and the results combined to form a larger area of image. The size of the `artificial superlattice' used should be increased until no change is found in the wavefunction over the central region of interest. For most defects, the positions of only a few atoms are important and, since the electron wavefunction is locally determined (for thin specimens at Scherzer focus), it appears that very large calculations are rarely needed for HREM work. The simulation of profile images of crystal surfaces at large defocus settings will, however, frequently be found to require large amounts of storage.
A new program should be tested to ensure that (a) under approximate twobeam conditions the calculated extinction distances for smallunitcell crystals agree roughly with tabulated values (Hirsch et al., 1977b), (b) the simulated dynamical images have the correct symmetry, (c) for small thickness, the Scherzerfocus images agree with the projected potential, and (d) images and beam intensities agree with those of a program known to be correct. The damping envelope (product representation) [equation (4.3.8.17)] should only be used in a thin crystal with ; in general, the effects of partial spatial and temporal coherence must be incorporated using equation (4.3.8.13a) or (4.3.8.13b), depending on whether variations in diffraction conditions over are important. Thus, a separate multislice dynamicalimage calculation for each component plane wave in the incident cone of illumination may be required, followed by an incoherent sum of all resulting images.
The outlook for obtaining higher resolution at the time of writing (1997) is broadly as follows. (1) The highest point resolution currently obtainable is close to 0.1 nm, and this has been obtained by taking advantage of the reduction in electron wavelength that occurs at high voltage [equation (4.3.8.16)]. A summary of results from these machines can be found in Ultramicroscopy (1994), Vol. 56, Nos. 1–3, where applications to fullerenes, glasses, quasicrystals, interfaces, ceramics, semiconductors, metals and oxides and other systems may be found. Fig. 4.2.8.6 shows a typical result. High cost, and the effects of radiation damage (particularly at larger thickness where defects with higher free energies are likely to be found), may limit these machines to a few specialized laboratories in the future. The attainment of higher resolution through this approach depends on advances in highvoltage engineering. (2) Aberration coefficients may be reduced if higher magnetic fields can be produced in the pole piece, beyond the saturation flux of the specialized iron alloys currently used. Research into superconducting lenses has therefore continued for many years in a few laboratories. Fluctuations in lens current are also eliminated by this method. (3) Electron holography was originally developed for the purpose of improving electronmicroscope resolution, and this approach is reviewed in the following section. (4) Electron–optical correction of aberrations has been under study for many years in work by Scherzer, Crewe, Beck, Krivanek, Lanio, Rose and others – results of recent experimental tests are described in Haider & Zach (1995) and Krivanek, Dellby, Spence, Camps & Brown (1997). The attainment of 0.1 nm point resolution is considered feasible. Aberration correctors will also provide benefits other than increased resolution, including greater space in the pole piece for increased sample tilt and access to Xray detectors, etc.
The need for resolution improvement beyond 0.1 nm has been questioned – the structural information retrievable by a single HREM image is always limited by the fact that a projection is obtained. (This problem is particularly acute for glasses.) Methods for combining different projected images (particularly of defects) from the same region (Downing, Meisheng, Wenk & O'Keefe, 1990) may now be as important as the search for higher resolution.
References
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Downing, K. H., Meisheng, H., Wenk, H. R. & O'Keefe, M. A. (1990). Resolution of oxygen atoms in staurolite by three dimensional transmission electron microscopy. Nature (London), 348, 525.
Grinton, G. R. & Cowley, J. M. (1971). Phase and amplitude contrast in electron micrographs of biological materials. Optik (Stuttgart), 34, 221–233.
Haider, M. & Zach, J. (1995). Multipole correctors. Proceedings of Microscopy and Microanalysis, edited by G. Bailey, pp. 596–567. New York: Jones and Bigell.
Hirsch, P. B., Howie, A., Nicholson, R. B., Pashley, D. W. & Whelan, M. J. (1977a). Electron microscopy of thin crystals. New York: Krieger.
Hirsch, P. B., Howie, A., Nicholson, R. B., Pashley, D. W. & Whelan, M. J. (1977b). Electron microscopy of thin crystals, p. 190. London: Butterworth.
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Ishizuka, K. (1982). Multislice formula for inclined illumination. Acta Cryst. A38, 773–779.
Krivanek, O. L., Dellby, N., Spence, A. J., Camps, R. A. & Brown, L. M. (1997). Aberration correction in the STEM. Proc EMAG 1997, edited by S. McVitie. London: Institute of Physics.
Krivanek, O. L. & Mooney, P. E. (1993). Applications of slowscan CCD cameras in HREM. Ultramicroscopy, 49, 95–108.
Self, P. G., O'Keefe, M. A., Buseck, P. R. & Spargo, A. E. C. (1983). Practical computation of amplitudes and phases in electron diffraction. Ultramicroscopy, 11, 35–52.
Shindo, D., Hiraga, K., Oikawa, T. & Mori, N. (1990). Quantification of electron diffraction with imaging plate. J. Electron Microsc. 39, 449–453.
Spence, J. C. H., O'Keefe, M. A. & Iijima, S. (1978). On the thickness periodicity of atomicresolution images of dislocation cores. Philos. Mag. A38, 463–482.
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