Crystal symmetry the direct method

If a tensor T represents a physical property of a crystal, it must be invariant under the operations of the point group of the crystal. But if T is invariant under the generators of the point group, it is certainly invariant under any of the point group operators and so it will be sufficient to examine the effect of the group generators on T. 

Example 15.2-2 Determine the non-zero elements of the elasticity tensor ciJki for a crystal of D4 symmetry. The generalized form of Hooke s law is 

These symmetries in eq. (3) reduce the number of independent tensor components for a triclinic crystal from eighty-one to twenty-one, which in Voigt notation form a symmetric 

A similar procedure may be followed for other point groups and for tensors representing other physical properties of crystals. 

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