Spin Hamiltonian of Two Equivalent Nuclei: Application to theI2−Centers

Abstract

The second-order hyperfine splittings observed in the ESR spectrum of an unpaired electron spin interacting with two equivalent nuclei depend upon the relative magnitude of the quadrupole interaction ${\mathcal{H}}_{Q}$ and the second-order effect ${{\mathcal{H}}_{\mathrm{hf}}}^{2}$ of the hyperfine interaction ${\mathcal{H}}_{\mathrm{hf}}$. If the first-order effect of ${\mathcal{H}}_{Q}$ is much smaller than ${{\mathcal{H}}_{\mathrm{hf}}}^{2}$ (case A), then a perturbation analysis of the spin Hamiltonian can be conveniently carried out in the coupled $|{I}_{1}{I}_{2}I{M}_{I}〉$ representation, and for the allowed ESR transitions the ${I}_{1}+{I}_{2}+1$ values of $I$ are good quantum numbers. If, however, ${\mathcal{H}}_{Q}$ is much larger than ${{\mathcal{H}}_{\mathrm{hf}}}^{2}$ (case B), then the perturbation analysis must be carried out in the $|{I}_{1}{I}_{2}, {{M}_{1}}^{2}+{{M}_{2}}^{2}, {M}_{I}〉$ representation, and ${{M}_{1}}^{2}+{{M}_{2}}^{2}$ replaces $I$ as a good quantum number. The second-order hyperfine patterns are qualitatively and quantitatively quite different for the two cases. As a specific example, the ESR spectra of the ${V}_{K}$-type ${\mathrm{I}}_{2}^{\ensuremath{-}}$ center in KI, and the ${V}_{K}$- and ${V}_{F}$-type ${\mathrm{I}}_{2}^{\ensuremath{-}}$ centers in ${\mathrm{Pb}}^{++}$-doped KCl: KI and KBr: KI are discussed. The ESR analysis is limited to the situation where the magnetic field is parallel to the internuclear axis ($\ensuremath{\theta}=0\ifmmode^\circ\else\textdegree\fi{}$), and it is shown that the ${\mathrm{I}}_{2}^{\ensuremath{-}}$ spectra are very well described by the case B perturbation solution. The $\ensuremath{\theta}=0\ifmmode^\circ\else\textdegree\fi{}$ spectrum shows weakly allowed transitions from which the quadrupole term can be accurately determined. In these transitions ${M}_{1}$ and ${M}_{2}$ change by one unit but in opposite senses, so that ${M}_{I}$ remains unchanged. A short discussion of the formation and decay of the ${\mathrm{I}}_{2}^{\ensuremath{-}}$ centers is also given.