Direct Spin-Lattice Relaxation Processes

A spin-lattice Hamiltonian for direct processes (the dynamic spin Hamiltonian) is defined which contains the effective spin operator and nuclear spin operator used in the static spin Hamiltonian. From the form of the dynamic Hamiltonian, certain combinations of the lattice operators can be identified as the dynamic analog of the static spin Hamiltonian parameters. The direct relaxation rates can be expressed in terms of spectral densities of the products of the dynamic spin Hamiltonian parameters. Assuming that the interaction of the ion with the crystal is adequately described by a crystal field, the sources of which are characterized by a symmetry group, it is possible to express the spectral densities involved in the spin-lattice relaxation in terms of the spectral densities of the normal modes of the complex. We use the symmetry of the complex and the crystal to limit the number of independent spectral densities of the normal modes without making any detailed assumptions about the lattice phonons nor the way that the sources of the crystal field participate in the lattice vibrations. Particular attention is given to the case of a Kramers doublet with hyperfine structure, and the matrix elements of the spin-lattice Hamiltonian between the eigenstates of the static spin Hamiltonian are given explicitly. These results are applied to the case of divalent cobalt in a nearly cubic field, and it is found that all twenty two of the relaxation rates between levels for which the energy separation is $\sim g\beta H$ for any direction of the applied field may be expressed in terms of four constants if the nuclear quadrupole interaction is ignored. The effect of the dynamic hyperfine interaction is found to be surprisingly large. Since all of the relaxation rates depend on the direction of the applied magnetic field, the four constants can be vastly overdetermined by experimental measurements of the direct relaxation processes.