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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 88-89   | 1 | 2 |

Section 1.3.4.3.5.6. Generalized multiplexing

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.3.5.6. Generalized multiplexing

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So far the multiplexing technique has been applied to pairs of vectors with similar types of parity-related and/or conjugate symmetry properties, in particular the same value of [epsilon].

It can be generalized so as to accommodate mixtures of vectors with different symmetry characteristics. For example if [{\bf X}_{1}] is Hermitian-symmetric and [{\bf X}_{2}] is Hermitian-antisymmetric, so that [{\bf X}_{1}^{*}] is real-valued while [{\bf X}_{2}^{*}] has purely imaginary values, the multiplexing process should obviously form [{\bf X} = {\bf X}_{1} + {\bf X}_{2}] (instead of [{\bf X} = {\bf X}_{1} + i{\bf X}_{2}] if both had the same type of symmetry), and demultiplexing consists in separating[\eqalign{ {\bf X}_{1}^{*} &= {\scr Re}\, {\bf X}^{*}\cr {\bf X}_{2}^{*} &= i{\scr Im}\,{\bf X}^{*}.}]

The general multiplexing formula for pairs of vectors may therefore be written[{\bf X} = {\bf X}_{1} + \omega {\bf X}_{2},]where ω is a phase factor (e.g. 1 or i) chosen in such a way that all nonexceptional components of [{\bf X}_{1}] and [{\bf X}_{2}] (or [{\bf X}_{1}^{*}] and [{\bf X}_{2}^{*}]) be embedded in the complex plane [{\bb C}] along linearly independent directions, thus making multiplexing possible.

It is possible to develop a more general form of multiplexing/demultiplexing for more than two vectors, which can be used to deal with symmetry elements of order 3, 4 or 6. It is based on the theory of group characters (Ledermann, 1987[link]).

References

Ledermann, W. (1987). Introduction to Group Characters, 2nd ed. Cambridge University Press.








































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