International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 1.1, pp. 16-17

Section 1.1.4.7.4. Trigonal, tetragonal, hexagonal and cylindrical systems

A. Authiera*

a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.4.7.4. Trigonal, tetragonal, hexagonal and cylindrical systems

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We remarked in Section 1.1.4.6.2.3[link] that, in the case of tensors of rank 2, the reduction is the same for threefold, fourfold or sixfold axes. It suffices therefore to perform the reduction for the tetragonal groups. That for the other systems follows automatically.

1.1.4.7.4.1. Groups [{\bar 3}], [3]; [4/m], [{\bar 4}], [4]; [6/m], [{\bar 6}], [6]; [(A_{\infty}/M)C], [A_{\infty}]

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If we consider a fourfold axis parallel to [Ox_{3}] represented by the matrix given in (1.1.4.3)[link], by applying the direct inspection method one finds [Scheme scheme5] where the symbol ⊖ means that the corresponding component is numerically equal to that to which it is linked, but of opposite sign. There are 3 independent components.

1.1.4.7.4.2. Groups [{\bar 3}m], [32], [3m]; [4/mm], [422], [4mm], [{\bar {4}}{2}m]; [{6}/mm], [622], [6mm], [{\bar{6}}{2}m]; [(A_{\infty}/M) \infty (A_{2}/M)C], [A_{\infty}\infty A_{2}]

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The result is obtained by combining the preceding result and that corresponding to a twofold axis normal to the fourfold axis. One finds [Scheme scheme6]

There are 2 independent components.








































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