International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 15-16
Section 1.1.4.6.2.3. General case^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
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.