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 Nd³⁺ and Er³⁺ in both lanthanum trichloride, LaCl₃, and lanthanum ethyl sulphate, La(C₂H₅OS₄)₃. 9H₂O, 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 T⁷, and a resonance Raman-(Orbach-) process transition probability proportional to e^-ΔkT. 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.