Matrix Transformations in Crystallography
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This Demonstration shows how various symmetry operations are used to simplify matrices; this process is called the direct inspection method. Select a crystallographic point group, the dimension of the matrix representation (number of rows × number of columns), and whether the original matrix should be assumed symmetric (only square matrices can be symmetric). The top row is the original untransformed matrix and the next row down displays the symmetry operations needed for the selected point group with the associated transformation matrix. The bottom row shows the result of each successive transformation step from left to right. It is in the last row that the matrix is progressively simplified, thus reducing the number of independent matrix elements.
Contributed by: Robert McIntosh (March 2014)
Open content licensed under CC BY-NC-SA
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 .
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 . The matrices are transformed with and directional cosine matrices as follows:
 R. E. Newnham, Properties of Materials, New York: Oxford University Press, 2005.