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. 13-14
Section 1.1.4.5.3. Antisymmetric tensors – axial tensors^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
A bilinear form is said to be antisymmetric if Its components satisfy the relations The associated matrix, T, is therefore also antisymmetric: The number of independent components is equal to , where n is the number of dimensions of the space. It is equal to 3 in a three-dimensional space, and one can consider these components as those of a pseudovector or axial vector. It must never be forgotten that under a change of basis the components of an axial vector transform like those of a tensor of rank 2.
Every tensor can be decomposed into the sum of two tensors, one symmetric and the other one antisymmetric: with and .
Example. As shown in Section 1.1.3.7.2, the components of the vector product of two vectors, x and y, are really the independent components of an antisymmetric tensor of rank 2. The magnetic quantities, B, H (Section 1.1.4.3.2), the tensor representing the pyromagnetic effect (Section 1.1.1.3) etc. are axial tensors.
If the rank of the tensor is higher than 2, the tensor may be antisymmetric with respect to the indices of one or several couples of indices.
Examples
The two preceding sections have shown examples of axial tensors of ranks 0 (pseudoscalar), 1 (pseudovector) and 2. They have in common that all their components change sign when the sign of the basis is changed, and this can be taken as the definition of an axial tensor. Their components are the components of an antisymmetric tensor of higher rank. It is important to bear in mind that in order to obtain their behaviour in a change of basis, one should first determine the behaviour of the components of this antisymmetric tensor.
References
Kumaraswamy, K. & Krishnamurthy, N. (1980). The acoustic gyrotropic tensor in crystals. Acta Cryst. A36, 760–762.