International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 5.2, p. 648   | 1 | 2 |

Section 5.2.5. Projection approximation – real-space solution

A. F. Moodie,a J. M. Cowleyb and P. Goodmanc

aDepartment of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia,bArizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287–1504, USA, and cSchool of Physics, University of Melbourne, Parkville, Australia

5.2.5. Projection approximation – real-space solution

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Many of the features of the more general solutions are retained in the practically important projection approximation in which [\varphi (x, y, z)] is replaced by its projected mean value [\varphi_{\rm p} (x, y)], so that the corresponding Hamiltonian [{\bf H}_{\rm p}] does not depend on z. Equation (5.2.4.1b[link]) can then be solved directly by iteration to give [{\bf U}_{\rm p} (z, z_{0}) = \exp \{- i{\bf H}_{\rm p} (z - z_{0})\}, \eqno(5.2.5.1)]and the solution may be verified by substitution into equation (5.2.4.1a[link]).

Many of the results of dynamical theory can be obtained by expansion of equation (5.2.5.1)[link] as [{\bf U}_{\rm p} \equiv {\bf 1} - i{\bf H}_{\rm p} (z - z_{0}) + {i^{2}\over 2!} {\bf H}_{\rm p}^{2} (z - z_{0}) - \ldots,]followed by the direct evaluation of the differentials. Such expressions can be used, for instance, to explore symmetries in image space.








































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