Perturbation Methods in the Calculation of Zeeman Interactions and Magnetic Dipole Line Strengths ford3Trigonal-Crystal Spectra

Abstract

Using a perturbation expansion based on a strong cubic-field zeroth-order approximation, we have obtained analytical expressions for the $g$ values of the ${t_2}^3$, ${}_{}{}^4{}_{}{}^{}A_2^{}{}_{}{}^{}$, and ${}_{}{}^2{}_{}{}^{}F$ terms for $d^3$ impurity ions in trigonal crystal fields. We have compared these expressions whth the results of a numerical calculation in which the magnetic dipole operator was transformed to the basis of eigenvectors of the zero-field Hamiltonian computed within the complete $d^3$ configuration, and we find them to be a very good approximation. This is not true of the published $g$-value expressions which are currently available. Absolute magnetic dipole absorption cross sections and nonlinear $g$ values are also calculated. This the first time such a detailed calculation of these quantities has been made for transition-ion impurity systems. For levels derived from cubic terms other than ${t_2}^3$, ${}_{}{}^4{}_{}{}^{}A_2^{}{}_{}{}^{}$, and ${}_{}{}^2{}_{}{}^{}E$, the analytical-perturbation techniques are not satisfactory and numerical methods must be used. We present an analysis of the $g$ values of the nominally ${t_2}^3$ terms, ${}_{}{}^4{}_{}{}^{}A_2^{}{}_{}{}^{}$, ${}_{}{}^2{}_{}{}^{}E$, ${}_{}{}^2{}_{}{}^{}T_1^{}{}_{}{}^{}$, and ${}_{}{}^2{}_{}{}^{}T_2^{}{}_{}{}^{}$, for emerald, ruby, Zn${\mathrm{Al}}_2$ ${\mathrm{O}}_4$: ${\mathrm{Cr}}^{3+}$, MgO: ${\mathrm{Cr}}^{3+}$, and ZnO: ${\mathrm{Co}}^{2+}$, and of magnetic dipole absorption cross sections for centrosymmetric Zn${\mathrm{Al}}_2$ ${\mathrm{O}}_4$: ${\mathrm{Cr}}^{3+}$ and MgO:

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