Details

The symmetry operations used in this Demonstration correspond to the symmetries of the 32 crystallographic point groups. Each element in the matrix corresponds to a material parameter for a crystal. The matrix dimensions correspond to common electrical, mechanical, magnetic, and optical interactions in crystalline materials; for example, 3×3 matrices are used to represent permittivity or the refractive index, 3×6 matrices correspond to the piezoelectric effect, and 6×3 matrixes correspond to the linear electro-optic effect. The same is true for 6×6 matrices; for these the matrix can be assumed to be symmetric (e.g. the element in row , column is the same as the one in row , column ), which is the case for elastic stiffness. A nonsymmetric 6×6 matrix would include the elasto-optic effect, electrostriction, and magnetostriction. The resulting matrices and method of derivation are nicely explained in [1].

There are a few situations where this Demonstration may disagree with conventions used in some books. For example, the 6×3 point group 4 uses the coefficient though could also be used; this is because and are equal, so the choice of one over the other is arbitrary. The transformations agree with those given in [1]. The matrices are transformed with and directional cosine matrices as follows:

,

,

,

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Reference

[1] R. E. Newnham, Properties of Materials, New York: Oxford University Press, 2005.