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. 15-16
Section 1.1.4.6.2. The operator A is in diagonal form^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
If one takes as the system of axes the eigenvectors of the operator A, the matrix is written in the form where θ is the rotation angle, is taken parallel to the rotation axis and coefficient 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) can then be simplified and reduce to (without any summation).
If the product (without summation) is equal to unity, equation (1.1.4.2) is trivial and there is significance in the component . On the contrary, if it is different from 1, the only solution for (1.1.4.2) is that . One then finds immediately that certain components of the tensor are zero and that others are unchanged.
All the diagonal components are in this case equal to −1. One thus has:
By replacing the matrix coefficients by their expression, (1.1.4.2) becomes, for a proper rotation, where r is the number of indices equal to 1, s is the number of indices equal to 2, t is the number of indices equal to 3 and is the rank of the tensor. The component is not affected by the symmetry operation if where K is an integer, and is equal to zero if
The angle of rotation θ can be put into the form , where q is the order of the axis. The condition for the component not to be zero is then
The condition is fulfilled differently depending on the rank of the tensor, p, and the order of the axis, q. Indeed, we have and
It follows that:
The inconvenience of the diagonalization method is that the vectors and eigenvalues are, in general, complex, so in practice one uses another method. For instance, we may note that equation (1.1.4.1) can be written in the case of by associating with the tensor a matrix T: where B is the symmetry operation. Through identification of homologous coefficients in matrices T and , one obtains relations between components that enable the determination of the independent components.