Transforming Crystal Field Parameters by Rotation of the Coordinate System

There is a special program cf2mcphas for rotating crystal field parameters (Stevens Notation) given in the orientation $abc\vert\vert$xyz to the McPhase notation $abc\vert\vert$yzx. The original crystal field parameters have to be given in a file of the above format of a single ion parameter file. Output is written to stdout. Note, there is also a program mcphas2cf for transforming parameters back. For more general rotations of crystal field parameters you can use the program rotateBlm(see chapter 19).

As an example we consider a tetragonal system, where the crystal field parameters are usually given such that $abc\vert\vert$xyz - we denote these parameters as $B_l^m$. For tetragonal symmetry only $B_2^0, B_4^0, B_4^4, B_6^0$ and $B_6^6$ are not zero. In order to use these parameters in mocule cfield, they have to be transformed to a coordinate system $xyz$ with $abc\vert\vert yzx$. Denoting the new crystal field parameters which are to be used as $Blm$ the transformation is given by

$$B20 = -\frac{1}{2} B_2^0$$

$$B22 = +\frac{3}{2} B_2^0$$

$$B40 = +\frac{3}{8} B_4^0+\frac{1}{8} B_4^4$$

$$B42 = -\frac{5}{2} B_4^0+\frac{1}{2} B_4^4$$

$$B44 = \frac{35}{8} B_4^0+\frac{1}{8} B_4^4$$

$$B60 = -\frac{5}{16} B_6^0-\frac{1}{16} B_6^4$$

$$B62 = +\frac{105}{32} B_6^0+\frac{5}{32} B_6^4$$

$$B64 = -\frac{63}{16} B_6^0+\frac{13}{16} B_6^4$$

$$B66 = \frac{231}{32} B_6^0+\frac{11}{32} B_6^4$$

As a second example we consider an orthorhombic system, where the crystal field parameters are given such that $abc\vert\vert$xyz - we denote these parameters as $B_l^m$. For orthorhombic symmetry only $B_2^0,B_2^2, B_4^0,B_4^2, B_4^4, B_6^0, B_6^2, % B_6^4$ and $B_6^6$ are not zero. In order to use these parameters in mocule cfield, they have to be transformed to a coordinate system $xyz$ with $abc\vert\vert yzx$. Denoting the new crystal field parameters which are to be used as $Blm$ the transformation is given by

$$B20 = -\frac{1}{2} B_2^0 -\frac{1}{2} B_2^2$$

$$B22 = +\frac{3}{2} B_2^0 -\frac{1}{2} B_2^2$$

$$B40 = +\frac{3}{8} B_4^0 +\frac{1}{8} B_4^2 +\frac{1}{8} B_4^4$$

$$B42 = -\frac{5}{2} B_4^0 -\frac{1}{2} B_4^2 +\frac{1}{2} B_4^4$$

$$B44 = \frac{35}{8} B_4^0 -\frac{7}{8} B_4^2 +\frac{1}{8} B_4^4$$

$$B60 = -\frac{5}{16} B_6^0-\frac{1}{16} B_6^2-\frac{1}{16} B_6^4-\frac{1}{16} B_6^6 <tex2html_comment_mark>$$

$$B62 = +\frac{105}{32} B_6^0+\frac{5}{32} B_6^4+...$$

$$B64 = -\frac{63}{16} B_6^0+\frac{13}{16} B_6^4+...$$

$$B66 = \frac{231}{32} B_6^0+\frac{11}{32} B_6^4+...$$

As a second example we consider an dhcp system with quasi cubic sites, where the crystal field parameters are given such that $abc\vert\vert$xyz - we denote these parameters as $B_l^m$. Only $B_2^0,B_2^{-1},B_2^2, B_4^0,B_4^{-1},B_4^2,B_4^{-3}, B_4^4, % B_6^0,B_6^{-1}, B_6^2,B_6^{-3}, B_6^4$ and $B_6^6$ are not zero. In order to use these parameters in mocule cfield, they have to be transformed to a coordinate system $xyz$ with $abc\vert\vert yzx$. Denoting the new crystal field parameters which are to be used as $Blm$ the transformation is given by

$$B21 = -2B_2^{-1}$$

$$B20 = -\frac{1}{2} B_2^0 -\frac{1}{2} B_2^2$$

$$B22 = +\frac{3}{2} B_2^0 -\frac{1}{2} B_2^2$$

$$B4-3 = -\frac{7}{4} B_4^{-1} -\frac{3}{4} B_4^{-3}$$

$$B4-1 = +\frac{3}{4} B_4^{-1} -\frac{1}{4} B_4^{-3}$$

$$B40 = +\frac{3}{8} B_4^0 +\frac{1}{8} B_4^2 +\frac{1}{8} B_4^4$$

$$B42 = -\frac{5}{2} B_4^0 -\frac{1}{2} B_4^2 +\frac{1}{2} B_4^4$$

$$B44 = \frac{35}{8} B_4^0 -\frac{7}{8} B_4^2 +\frac{1}{8} B_4^4$$

$$B60 = -\frac{5}{16} B_6^0-\frac{1}{16} B_6^4+...$$

$$B62 = +\frac{105}{32} B_6^0+\frac{5}{32} B_6^4+...$$

$$B64 = -\frac{63}{16} B_6^0+\frac{13}{16} B_6^4+...$$

$$B66 = \frac{231}{32} B_6^0+\frac{11}{32} B_6^4+...$$

The general transformation matrices are given by