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. 20-24
Section 1.1.4.9. Reduction of the components of a tensor of rank 4^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
There is no reduction; all the components are independent. Their number is equal to 81. They are usually represented as a matrix, where components are replaced by ijkl, for brevity: This matrix can be represented symbolically by where the matrix has been subdivided for clarity in to nine submatrices.
The reduction is obtained by the method of direct inspection. For a twofold axis parallel to , one finds
There are 41 independent components.
For symmetric tensors such as those representing principal properties, one finds the following, representing the nonzero components for the leading diagonal and for one half of the others.