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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 87   | 1 | 2 |

Section 1.3.4.3.5.2. Hermitian-antisymmetric or pure imaginary transforms

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

1.3.4.3.5.2. Hermitian-antisymmetric or pure imaginary transforms

| top | pdf |

A vector [{\bf X} = \{X ({\bf k}) | {\bf k} \in {\bb Z}^{n} / {\bf N}{\bb Z}^{n}\}] is said to be Hermitian-antisymmetric if[X ({\bf k}) = -\overline{X (-{\bf k})} \hbox{ for all } {\bf k.}]Its transform [{\bf X}^{*}] then satisfies[X^{*} ({\bf k}^{*}) = -\overline{X^{*} ({\bf k}^{*})} \hbox{ for all } {\bf k}^{*},]i.e. is purely imaginary.

If X is Hermitian-antisymmetric, then [{\bf F} = \pm i{\bf X}] is Hermitian-symmetric, with [\rho\llap{$-\!$} = \pm i{\bf X}^{*}] real-valued. The treatment of Section 1.3.4.3.5.1[link] may therefore be adapted, with trivial factors of i or [-1], or used as such in conjunction with changes of variable by multiplication by [\pm i].








































to end of page
to top of page