T1 AND T2 FOR 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 (1T2), 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 T1 can be quite significant, but that the effect on T2 is slight in all cases. For Ce3+, Nd3+, and Sm3+ in the double nitrates, we find that T1=T2 within 12.5% for the magnetic field perpendicular to the symmetry axis for all conditions. For Ce3+ in LaMg double nitrate with the field perpendicular to the symmetry axis, measurements at 10 Gc/sec should give T1=T2 within 3%, whereas the experimental result reported by Stapleton and Brower is T1=13T2. 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 T1 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 T1 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. © 1969 The American Physical Society.