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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 15

Section 1.1.4.6.2.1. Introduction

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.4.6.2.1. Introduction

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If one takes as the system of axes the eigenvectors of the operator A, the matrix is written in the form [\pmatrix { \exp {\rm i} \theta & 0 & 0\cr 0 & \exp -{\rm i} \theta & 0 \cr 0 & 0 & \pm 1 \cr },]where θ is the rotation angle, [Ox_{3}] is taken parallel to the rotation axis and coefficient [A_{3}] is equal to +1 or −1 depending on whether the rotation axis is direct or inverse (proper or improper operator).

The equations (1.1.4.1)[link] can then be simplified and reduce to [t^{ij}_{kl} = t^{ij}_{kl} A^i_{i}A^j_{j}B^k_{k}B^l_{l } \eqno{(1.1.4.2)}](without any summation).

If the product [A^{i}_{i}A\hskip1pt^{j}_{j}B^{k}_{k}B^{l}_{l}] (without summation) is equal to unity, equation (1.1.4.2)[link] is trivial and there is significance in the component [t_{kl}]. On the contrary, if it is different from 1, the only solution for (1.1.4.2)[link] is that [t^{ij}_{kl } = 0]. One then finds immediately that certain components of the tensor are zero and that others are unchanged.








































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