International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 5.2, p. 649
Section 5.2.7. Two-beam approximation^{a}Department of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia,^{b}Arizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287–1504, USA, and ^{c}School of Physics, University of Melbourne, Parkville, Australia |
In the two-beam approximation, as an elementary example, equation (5.2.6.2) takes the form 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, Here, the are the Pauli matrices and , are the real and imaginary parts of the complex scattering coefficients appropriate to a noncentrosymmetric crystal, i.e. . Expanding, using the anti-commuting properties of : and putting , , so that and , 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
This result was first obtained by Blackman (1939), 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 ).
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 , deriving from the shape function of the crystal emerges, as it does, albeit less obviously, in the N-beam theory.
This derivation of equation (5.2.7.2) 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).
References
Blackman, M. (1939). Intensities of electron diffraction rings. Proc. Phys. Soc. London Sect. A, 173, 68–72.Gilmore, R. (1974). Lie Groups, Lie Algebras, and Some of Their Applications. New York: Wiley–Interscience.