T1andT2for Orbach Relaxation Processes

The master equation for the density matrix of the ionic levels involved in the relaxation process described by Orbach is developed in a form valid for low temperatures. In the absence of degeneracies, the equations for the diagonal elements of the density matrix are the rate equations for the populations which have been solved before; and the equations for the off-diagonal elements show that ($\frac{1}{T_2}$), the linewidth for a transition between two levels, is given by one-half the sum of the phonon-induced rates out of those levels. Cases in which the inverse lifetimes of the excited states are not small compared with the splitting of the doublets are investigated, and we find that the modification of $T_1$ can be quite significant, but that the effect on $T_2$ is slight in all cases. For ${\mathrm{Ce}}^{3+}$, ${\mathrm{Nd}}^{3+}$, and ${\mathrm{Sm}}^{3+}$ in the double nitrates, we find that $T_1=T_2$ within 12.5% for the magnetic field perpendicular to the symmetry axis for all conditions. For ${\mathrm{Ce}}^{3+}$ in LaMg double nitrate with the field perpendicular to the symmetry axis, measurements at 10 Gc/sec should give $T_1=T_2$ within 3%, whereas the experimental result reported by Stapleton and Brower is $T_1=13\mathrm{}{T}_2$. It is found that spin-spin interactions between the ground doublet and the excited doublet cannot be the cause of the discrepancy, but that a near-universal bottlenecking of a part of the phonon modes would lead to an explanation of the linewidth measurements and not be at variance with the size and concentration dependence of $T_1$ thus far reported for the double-nitrate crystals. This hypothesis is based on the calculations of Orbach and Vrederoe, which show that the lifetime of the transverse phonon modes may be much longer than that for the longitudinal modes. Further, the transverse phonons should produce the greater part of the orbitlattice interaction. This hypothesis suggests that a true spin-lattice $T_1$ would be measured only at very low concentrations, and that with increasing concentrations a bottleneck would develop so that the apparent relaxation time would increase with concentration until the bottlenecked relaxation became comparable with the rate due to other unbottlenecked phonons acting in parallel with the bottle-necked phonons. The experiments of Adde and Geschwind appear to show some aspects of this behavior.