Spin-Lattice Relaxation at Zero Magnetic Field. I. Phenomenological Theory of Relaxation among Hyperfine Levels

Spin-lattice relaxation at zero external magnetic field is studied for the Kramers ions ${\mathrm{Nd}}^{3+}$ and ${\mathrm{Er}}^{3+}$ in both lanthanum trichloride, La${\mathrm{Cl}}_3$, and lanthanum ethyl sulphate, La${\left({\mathrm{C}}_2{\mathrm{H}}_5\mathrm{O}{\mathrm{S}}_4\right)}_3$. 9${\mathrm{H}}_2$O, in the 1-15-K temperature range. Employing a phenomenological theory which treats the crystal lattice-ion interaction as the first term in a Taylor-series expansion of the crystal field potential, we find a negligible direct- (one-phonon) process transition probability proportional to $T$, a Raman-process transition probability proportional to $T^7$, and a resonance Raman-(Orbach-) process transition probability proportional to $e^{-\frac{\Delta }{\mathrm{kT}}}$. The multilevel zerofield problem is solved using matrix techniques. It is found that some levels return to equilibrium in an almost-single-exponential fashion, while others can be discussed in terms of a sum of up to seven exponentials. Both long- and short-pulse experiments are discussed.