Abstract
The orbit-lattice Hamiltonian for rare-earth ions is obtained using a new approach in which it is not necessary to calculate the normal modes of the cluster consisting of the central ion and its first neighbors. The determination of the parameters describing the first-order orbit-lattice coupling is greatly simplified, whatever the environment. The problem of relaxation of a ${\ensuremath{\Gamma}}_{8}$ ground quartet is then studied. The equations of evolution of the populations are solved for one-phonon and two-phonon processes, and the relaxation times are calculated. It is shown that, under certain initial conditions, two relaxation times are sufficient. The angular variation for the one-phonon process is established. The calculations are greatly simplified by noting relations between matrix elements which were obtained from local symmetry and time-reversal considerations. The experimental results on MgO: ${\mathrm{Er}}^{3+}$ verify the existence of two relaxation times and confirm their predicted angular variation. The discrepancy (a factor of 4) between experimental and theoretical values is discussed.
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