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. 17-20
Section 1.1.4.8. Reduction of the components of a tensor of rank 3^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
Choosing the twofold axis parallel to and applying the direct inspection method, one finds
There are 13 independent components. If the twofold axis is parallel to , one finds
One obtains the matrix representing the operator m by multiplying by −1 the coefficients of the matrix representing a twofold axis. The result of the reduction will then be exactly complementary: the components of the tensor which include an odd number of 3's are now equal to zero. One writes the result as follows:
There are 14 independent components. If the mirror axis is normal to , one finds
There are three orthonormal twofold axes. The reduction is obtained by combining the results associated with two twofold axes, parallel to and , respectively.
There are 6 independent components.
The method of direct inspection can be applied for a fourfold axis. One finds
There are 7 independent components.
The matrix corresponding to axis is and the form of the matrix is
There are 6 independent components.
One combines either the reductions for groups and 222, or the reductions for groups and 2mm.
The number of independent components is of course the same, 6.
It was shown in Section 1.1.4.6.2.3 that, in the case of tensors of rank 3, the reduction is the same for axes of order 4, 6 or higher. The reduction will then be the same as for the tetragonal system.
One combines the reductions for the groups corresponding to a threefold axis parallel to and to a mirror perpendicular to :
There are 2 independent components.
One combines the reductions corresponding to a twofold axis parallel to and to a threefold axis parallel to [111]:
There are 2 independent components.