Coordinate system

Most of the part inside code, the coordinate system is represented by reduced coordinates but Cartesian coordinates. The reduced and Cartesian coordinates are related by cell parameters. Force constants are treated in Cartesian coordinates. Therefore the eigenvectors are given in Cartesian.

Rotations are based on a linear vector space. It is not always Cartesian or non-orthogonal. The determinants are 1 and each matrix element is 0, 1 or -1 in crystal.

Second class tensor can be rotated by similarity transformation, e.g., in Cartesian coordinate. Prior to use the reduced rotations, they are transformed to those in Cartesian coordinate and the transformations are written as,

$$\mathbf{S}^{\mathrm{Cartesian}} = \mathbf{R \cdot S} \cdot \mathbf{R}^{-1},$$ (3.23)

where, $\mathbf{R}$ is a cell axes matrix and $\mathbf{S}$ is represented in oblique coordinates. Then the force constants matrix $\mathbf{P}$ is transfered. In the case of similarity transformation, the transformation is written as,

$$\mathbf{P}^{\mathrm{tranform}} = \mathbf{R \cdot S} \cdot \mathbf... ...thbf{P}^{\mathrm{original}} \cdot \mathbf{R \cdot S^{-1}} \cdot \mathbf{R}^{-1}$$ (3.24)