Spin-Lattice Relaxation Due to Local Vibrations with Temperature-Independent Amplitudes

Abstract

In the usual treatment of the temperature dependence of spin-lattice relaxation of impurity ions, the vibrational properties of the paramagnetic ion are assumed to be those of a normal host ion. For impurities that do not affect the vibrational properties of the lattice, this is expected to be a resonable approximation. However, if a paramagnetic impurity is associated with a gross defect in a crystal, it should be expected that the temperature dependence of the spin-lattice relaxation time will be dominated by the vibrational properties of the defect. An ion could be trapped at a site about which the potential is highly anharmonic and such that the vibrational amplitude is very large and effectively independent of the state of excitation. Under these circumstances lattice vibrations whose amplitudes have the usual temperature dependence could beat with the vibrations of the defect, producing a difference frequency resulting in a spin transition. In the high-temperature limit one would expect a spin transition rate that is proportional to $T$ rather than to ${T}^{2}$ because only the lattice-vibration part of the interaction is temperature-dependent and at high temperatures this dependence is linear. In this paper two simple defect models are discussed. In the first, a paramagnetic ion tunnels between two stable positions, and in the second, the ion is trapped in a one-dimensional square well. In the tunneling model, the form of the temperature dependence is independent of the details of the spin-lattice interaction, whereas the square-well results are sensitive to these details. In the latter case, the effect of electric and magnetic interactions is discussed. Recent measurements by Feldman, Castle, and Wagner of the spin-lattice relaxation time of hydrogen centers in fused quartz exhibit the behavior described by either of these models, but their data can be fitted better by the square-well model.