Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 5.2, pp. 553-554   | 1 | 2 |

Section 5.2.7. Two-beam approximation

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 3052

5.2.7. Two-beam approximation

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In the two-beam approximation, as an elementary example, equation ([link] takes the form [\pmatrix{u_{0}\cr u_{\bf h}\cr} = \exp \left\{i \pmatrix{0 &V^*({\bf h})\cr V({\bf h}) &K_{\bf h}\cr} T\right\} \pmatrix{0\cr 1\cr}. \eqno(] If this expression is expanded directly as a Taylor series, it proves surprisingly difficult to sum. However, the symmetries of Clifford algebra can be exploited by summing in a Pauli basis thus, [\eqalign{ &\exp \left\{i \pmatrix{0 &V^*({\bf h})\cr V({\bf h}) &K_{\bf h}\cr} T\right\}\cr &\quad = \exp \left\{i {K_{\bf h} T\over 2}\right\} {\bf E} \exp \left\{i \left({K_{\bf h}\over 2} \boldsigma_{3} + V^{R} \boldsigma_{1} - V^{I} \boldsigma_{2}\right) T\right\}.}] Here, the [\boldsigma_{i}] are the Pauli matrices [\displaylines{\boldsigma_{1} = \pmatrix{0 &1\cr 1 &0\cr},\;\boldsigma_{2} = \pmatrix{0 &i\cr -i &0\cr},\;\boldsigma_{3} = \pmatrix{-1 &0\cr 0 &1\cr},\cr {\bf E} = \pmatrix{1 &0\cr 0 &1\cr},}] and [V^{R}], [V^{I}] are the real and imaginary parts of the complex scattering coefficients appropriate to a noncentrosymmetric crystal, i.e. [V_{\bf h} = V^{R} + iV^{I}]. Expanding, [\eqalign{ &\exp \left \{i \left({K_{\bf h}\over 2} \boldsigma_{3} + V^{R} \boldsigma_{1} - V^{I} \boldsigma_{2}\right) T\right\}\cr &\quad = {\bf E} + i \left({K_{\bf h}\over 2} \boldsigma_{3} + V^{R} \boldsigma_{1} - V^{I} \boldsigma_{2}\right) T\cr &\qquad - {1\over 2} \left({K_{\bf h}\over 2} \boldsigma_{3} + V^{R} \boldsigma_{1} - V^{I} \boldsigma_{2}\right)^{2} T^{2} + \ldots,}] using the anti-commuting properties of [\boldsigma_{i}]: [\left.\matrix{\boldsigma_{i} \boldsigma_{j} + \boldsigma_{j} \boldsigma_{i} &= 0\cr \phantom{\sigma_{i} \sigma_{j}+ }\boldsigma_{i} \boldsigma_{i} &= 1\cr}\right\}] and putting [[(K_{\bf h}/2)^{2} +V({\bf h}) V^*({\bf h})] = \Omega], [{\bf M}_2=[(K_{\bf h}/2)\boldsigma_3] [+V^{R}\boldsigma_1-V^I\boldsigma_2]], so that [{\bf M}_{2}^{2} = \Omega {\bf E}] and [{\bf M}_{2}^{3} = \Omega {\bf M}_{2}], the powers of the matrix can easily be evaluated. They fall into odd and even series, corresponding to sine and cosine, and the classical two-beam approximation is obtained in the form [\eqalignno{ &{\bf Q}_{2} = \exp \{i (K_{\bf h}/2) T\} {\bf E} \left[(\cos \Omega^{1/2} T) {\bf E} + i \left({\sin \Omega^{1/2} T\over \Omega^{1/2}}\right) {\bf M}_{2}\right].\cr & &(}]

This result was first obtained by Blackman (1939[link]), using Bethe's dispersion formulation. Ewald and, independently, Darwin, each with different techniques, had, in establishing the theoretical foundations for X-ray diffraction, obtained analogous results (see Section 5.1.3[link] ).

The two-beam approximation, despite its simplicity, exemplifies some of the characteristics of the full dynamical theory, for instance in the coupling between beams. As Ewald pointed out, a formal analogy can be found in classical mechanics with the motion of coupled pendulums. In addition, the functional form [(\sin ax)/x], deriving from the shape function of the crystal emerges, as it does, albeit less obviously, in the N-beam theory.

This derivation of equation ([link] exhibits two-beam diffraction as a typical two-level system having analogies with, for instance, lasers and nuclear magnetic resonance and exhibiting the symmetries of the special unitary group SU(2) (Gilmore, 1974[link]).


Blackman, M. (1939). Intensities of electron diffraction rings. Proc. Phys. Soc. London Sect. A, 173, 68–72.Google Scholar
Gilmore, R. (1974). Lie groups, Lie algebras, and some of their applications. New York: Wiley–Interscience.Google Scholar

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