Transformation operators for symmetry elements

Abstract

Since the physical property tensors depend on symmetry, mathematical methods for determining the influence of symmetry are needed. More specifically, transformation matrices are required for the symmetry elements that generate the crystal class. As explained earlier, some classes require only one symmetry element while others may require two or three. None require more than three. This chapter also includes a brief discussion of the seven Curie groups used for textured solids and liquids. As explained in Chapter 2, transformations from one coordinate system to another can be specified by a set of nine direction cosines, aij where i, j = 1, 2, 3. The first index, i, refers to the “new” or transformed axis, the second index j to the “old” or reference axis. There are four types of symmetry elements that require discussion: rotation axes, mirror planes, inversion centers, and inversion axes in which rotation is accompanied by inversion. All rotations are assumed to be in the counterclockwise direction. Other symmetry elements such as rotoreflection axes are not needed. The fourteen symmetry elements in Table 4.1 generate the 32 crystal classes. No proof is offered here but this statement can be verified geometrically using the stereographic projections in Chapter 3. There are two final points concerning these transformation matrices. First, keep in mind that they must obey the orthogonality conditions described in Chapter 2. This provides a useful way of avoiding mistakes. The second point concerns handedness. Symmetry elements involving reflection or inversion reverses the handedness of the coordinate system. This of course includes inversion axes such as 1 ̅̅, 2 ̅ = m, 3 ̅, 4 ̅, and 6 ̅. The handedness change can be verified by showing that the determinant of the transformation matrix is −1. For ordinary rotation axes such as 2, 3, 4, and 6, there is no change in handedness, and the determinant is +1. In Table 4.2 we list the minimum symmetry requirements for each of the 32 crystal classes. These are the transformation operations needed to develop the physical property matrices for single crystals. These techniques will be described in the next chapter.