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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 35   | 1 | 2 |

Section 1.3.2.4.2.3. Conjugate symmetry

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

1.3.2.4.2.3. Conjugate symmetry

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The kernels of the Fourier transformations [{\scr F}] and [\bar{\scr F}] satisfy the following identities:[\exp (\pm 2\pi i {\boldxi} \cdot {\bf x}) = \exp \overline{[\pm 2\pi i {\boldxi} \cdot (-{\bf x})]} = \exp \overline{[\pm 2\pi i (-{\boldxi}) \cdot {\bf x}]}.]As a result the transformations [{\scr F}] and [\bar{\scr F}] themselves have the following `conjugate symmetry' properties [where the notation [\breve{f}({\bf x}) = f(-{\bf x})] of Section 1.3.2.2.2[link] will be used]:[\displaylines{{\scr F}[\,f] ({\boldxi}) = \overline{{\scr F}[\bar{\, f}] (-{\boldxi})} = \breve{\overline{{\scr F}[\bar{\, f}] ({\boldxi})}}\cr {\scr F}[\,f] ({\boldxi}) = \overline{{\scr F}[\breve{\bar{\,f}}] ({\boldxi})}.}]Therefore,

  • (i) f real [\Leftrightarrow f = \bar{f} \Leftrightarrow {\scr F}[\,f] = \breve{\overline{{\scr F}[\,f]}} \Leftrightarrow {\scr F}[\,f] ({\boldxi})  =\overline{{\scr F}[\,f] (-{\boldxi})}:] [ {\scr F}[\,f]] is said to possess Hermitian symmetry;

  • (ii) f centrosymmetric [\Leftrightarrow f = \breve{f} \Leftrightarrow {\scr F}[\,f] = \overline{{\scr F}[\bar{\, f}]}];

  • (iii) f real centrosymmetric [\Leftrightarrow f = \bar{f} = \breve{f} \Leftrightarrow {\scr F}[\,f]  =\overline{{\scr F}[\,f]} =] [\breve{\overline{{\scr F}[\,f]}} \Leftrightarrow {\scr F}[\,f]] real centrosymmetric.

Conjugate symmetry is the basis of Friedel's law (Section 1.3.4.2.1.4[link]) in crystallography.








































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