A General Method for Determining Crystal Field Effects on Magnetic Ions

Abstract

There is no tabulation or easy method of analysis for the crystalline electric field (CEF) in systems of low or even intermediate symmetry. Consequently these systems have received little attention despite their wide occurrence in nature. The basic theory is well established¹, and tabulations exist for crystal fields whose symmetry is cubic² or hexagonal with the ideal c/a ratio³. These tabulations are useful when the symmetry is cubic, but there are few systems whose c/a is sufficiently close to $\sqrt {8/3}$ to make this anything better than a first approximation. The general Hamiltonian for a hexagonal CEF can be written, in the notation of Hutchings¹, as

$$ {\rm{H = B}}_{\rm{2}}^{\rm{0}}{\rm{O}}_{\rm{2}}^{\rm{0}}{\rm{ + B}}_{\rm{4}}^{\rm{0}}{\rm{O}}_{\rm{4}}^{\rm{0}}{\rm{ + B}}_{\rm{6}}^{\rm{0}}{\rm{O}}_6^{\rm{0}}{\rm{ + B}}_{\rm{6}}^{\rm{6}}{\rm{O}}_{\rm{6}}^{\rm{6}} $$

This can be simplified for systems with the ideal axial ratio by using the result that B ⁰₂ and B ⁰₆ /B ⁶₆ =77/8 . When c/a departs from ideal, rapidly becomes non zero, and the ratio of the sixth order terms departs significantly from 77/8. The ordering of the resultant energy level scheme often bears little resemblance to that for ideal c/a.