Symmetry considerations for Crystal Field Parameters

Symmetry considerations allow to compile table 8, where the nonvanishing CEF parameters for all crystallographic point groups are given [71].

Table 8: Possible local symmetries and corresponding non zero CEF parameters. In the case when $m>0$ and both parameters $B_l^m$ and $B_l^{-m}$ are nonzero, one of these $B_l^m$ with $m>0$ can by made zero by a rotation of the coordinate system. However, the appropriate orientation of the coordinate system in these cases is not known a priori. It requires the knowledge of the CEF parameters. Note, that for cubic symmetry additionally $B_4^4=5 B_4^0$, $B_6^4=-21 B_6^0$ (for $xyz\vert\vert abc$), $L_4^4=5/\sqrt {70} L_4^0$, $L_6^4=-\sqrt {7/2} L_6^0$ (for $xyz\vert\vert abc$), $B_4^{3_{111}}=-20\sqrt {2} B_4^{0_{111}}$, $B_6^{3_{111}}=35\sqrt {2}/4 B_6^{0_{111}}$ $B_6^{6_{111}}=77/8 B_6^{0_{111}}$ for $z\vert\vert[111]$ $L_4^{3_{111}}=-\sqrt {10/7} L_4^{0_{111}}$, $L_6^{3_{111}}=\frac {1}{8}\sqrt {70/3} L_6^{0_{111}}$ $L_6^{6_{111}}=\frac {1}{8}\sqrt {77/3} L_6^{0_{111}}$ for $z\vert\vert[111]$ with $B_4^{0_{111}}=(-2/3)B_4^0$, $B_6^{0_{111}}=16/9B_6^0$. For the icosahedral case (not compatible with translational symmetry) B$_6^5$=-42B$_6^0$ [72].

symmetry

B$_2^0$

B$_2^{\pm 1}$

B$_2^{\pm 2}$

B$_4^0$

B$_4^{\pm 1}$

B$_4^{\pm 2}$

B$_4^{\pm 3}$

B$_4^{\pm 4}$

B$_6^0$

B$_6^{\pm 1}$

B$_6^{\pm 2}$

B$_6^{\pm 3}$

B$_6^{\pm 4}$

B$_6^{\pm 5}$

B$_6^{\pm 6}$

triclinic

C$_i$ C$_1$

+

$\pm$

$\pm$

+

$\pm$

$\pm$

$\pm$

$\pm$

+

$\pm$

$\pm$

$\pm$

$\pm$

$\pm$

$\pm$

monoclinic

C$_2$ C$_s$ C$_{2h}$

+

$\pm$

+

$\pm$

$\pm$

+

$\pm$

$\pm$

$\pm$

rhombic

C$_{2v}$ D$_{2}$ D$_{2h}$

+

+

+

+

+

+

+

+

+

tetragonal

C$_{4}$ S$_{4}$ C$_{4h}$

+

+

$\pm$

+

$\pm$

tetragonal

D$_{4}$ C$_{4v}$ D$_{2d}$ D$_{4h}$

+

+

+

+

+

trigonal

C$_{3}$ S$_{6}$

+

+

$\pm$

+

$\pm$

$\pm$

trigonal

D$_{3}$ C$_{3v}$ D$_{3d}$

+

+

+

+

+

+

hexagonal

C$_{6}$ C$_{3h}$ C$_{6h}$

+

+

+

$\pm$

hexagonal

D$_{6}$ C$_{6v}$ D$_{3h}$ D$_{6h}$

+

+

+

+

cubic

T T$_{h}$

+

+

+

+

+

+

cubic

T$_{d}$ O O$_{h}$

+

+

+

+

icosahedral

I$_h$

+

+