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. 16-17
Section 1.1.4.7. Reduction of the components of a tensor of rank 2^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France |
The reduction is given for each of the 11 Laue classes.
Groups , 1: no reduction, the tensor has 9 independent components. The result is represented in the following symbolic way (Nye, 1957, 1985): where the sign • represents a nonzero component.
Groups 2m, 2, m: it is sufficient to consider the twofold axis or the mirror. As the representative matrix is diagonal, the calculation is immediate. Taking the twofold axis to be parallel to , one has
The other components are not affected. The result is represented as
There are 5 independent components. If the twofold axis is taken along axis , which is the usual case in crystallography, the table of independent components becomes
Groups mmm, 2mm, 222: the reduction is obtained by considering two perpendicular twofold axes, parallel to and to , respectively. One obtains
There are 3 independent components.
We remarked in Section 1.1.4.6.2.3 that, in the case of tensors of rank 2, the reduction is the same for threefold, fourfold or sixfold axes. It suffices therefore to perform the reduction for the tetragonal groups. That for the other systems follows automatically.
If we consider a fourfold axis parallel to represented by the matrix given in (1.1.4.3), by applying the direct inspection method one finds where the symbol ⊖ means that the corresponding component is numerically equal to that to which it is linked, but of opposite sign. There are 3 independent components.
The cubic system is characterized by the presence of threefold axes along the directions. The action of a threefold axis along [111] on the components of a vector results in a permutation of these components, which become, respectively, and then . One deduces that the components of a tensor of rank 2 satisfy the relations
The cubic groups all include as a subgroup the group 23 of which the generating elements are a twofold axis along and a threefold axis along [111]. If one combines the corresponding results, one deduces that which can be summarized by
There is a single independent component and the medium behaves like a property represented by a tensor of rank 2, like an isotropic medium.
If the tensor is symmetric, the number of independent components is still reduced. One obtains the following, representing the nonzero components for the leading diagonal and for one half of the others.
There are 6 independent components. It is possible to interpret the number of independent components of a tensor of rank 2 by considering the associated quadric, for instance the optical indicatrix. In the triclinic system, the quadric is any quadric. It is characterized by six parameters: the lengths of the three axes and the orientation of these axes relative to the crystallographic axes.
There are 4 independent components. The quadric is still any quadric, but one of its axes coincides with the twofold axis of the monoclinic lattice. Four parameters are required: the lengths of the axes and one angle.
There are 3 independent components. The quadric is any quadric, the axes of which coincide with the crystallographic axes. Only three parameters are required.
There are 2 independent components. The quadric is of revolution. It is characterized by two parameters: the lengths of its two axes.
References
Nye, J. F. (1957). Physical Properties of Crystals, 1st ed. Oxford: Clarendon Press.Nye, J. F. (1985). Physical Properties of Crystals, revised ed. Oxford University Press.