Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 5.2, pp. 649-650   | 1 | 2 |

Section 5.2.8. Eigenvalue approach

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

5.2.8. Eigenvalue approach

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In terms of the eigenvalues and eigenvectors, defined by [{\bf H}_{\rm p}|\,j\rangle = \gamma_{j}|\,j\rangle,]the evolution operator can be written as [{\bf U}(z, z_{0}) = \textstyle\int |\,j\rangle \exp \{\gamma_{j} (z - z_{0})\} \langle \,j|\, \hbox{d}j.]

This integration becomes a summation over discrete eigen states when an infinitely periodic potential is considered.

Despite the early developments by Bethe (1928[link]), an N-beam expression for a transmitted wavefunction in terms of the eigenvalues and eigenvectors of the problem was not obtained until Fujimoto (1959[link]) derived the expression [U_{\bf h} = {\textstyle\sum\limits_{j}} \psi_{0}^{\,j*} \psi_{\bf h}^{\,j} \exp \{-i2\pi \gamma_{j} T\}, \eqno(]where [\psi_{\bf h}^{\,j}] is the h component of the j eigenvector with eigenvalue [\gamma_{j}].

This expression can now be related to those obtained in the other formulations. For example, Sylvester's theorem (Frazer et al., 1963[link]) in the form [f({\bf M}) = {\textstyle\sum\limits_{j}} {\bf A}_{j}\,f(\gamma_{j})]when applied to Sturkey's solution yields [\boldPhi_{\bf h} = \exp (i{\bf M}_{\rm p}z) = \textstyle\sum {\bf P}_{j} \exp \{i2\pi \gamma_{j}z\}](Kainuma, 1968[link]; Hurley et al., 1978[link]). Here, the [{\bf P}_{j}] are projection operators, typically of the form [{\bf P}_{j} = \prod_{n \neq j} {({\bf M}_{\rm p} - {\bf E}\gamma_{n})\over \gamma_{j} - \gamma_{n}}.]On changing to a lattice basis, these transform to [\psi_{0}^{\,j*} \psi_{\bf h}^{\,j}].

Alternatively, the semi-reciprocal differential equation can be uncoupled by diagonalizing [{\bf M}_{\rm p}] (Goodman & Moodie, 1974[link]), a process which involves the solution of the characteristic equation [|{\bf M}_{\rm p} - \gamma_{j}{\bf E}| = 0. \eqno(]


Bethe, H. A. (1928). Theorie der Beugung von Elektronen an Kristallen. Ann. Phys. (Leipzig), 87, 55–129.
Frazer, R. A., Duncan, W. J. & Collar, A. R. (1963). Elementary Matrices, pp. 78–79. Cambridge University Press.
Fujimoto, F. (1959). Dynamical theory of electron diffraction in the Laue case. J. Phys. Soc. Jpn, 14, 1558–1568.
Goodman, P. & Moodie, A. F. (1974). Numerical evaluation of N-beam wave functions in electron scattering by the multislice method. Acta Cryst. A30, 280–290.
Hurley, A. C., Johnson, A. W. S., Moodie, A. F., Rez, P. & Sellar, J. R. (1978). Algebraic approaches to N-beam theory. In Electron Diffraction 1927–1977, edited by P. J. Dobson, J. B. Pendry & C. J. Humphreys, pp. 34–40. Inst. Phys. Conf. Ser. No. 41. Bristol/London: Institute of Physics.
Kainuma, Y. (1968). Averaged intensities in the many beam dynamical theory of electron diffraction. Part I. J. Phys. Soc. Jpn, 25, 498–510.

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