Spin-Lattice Relaxation of Rare-Earth Ions in LaF3

Abstract

The spin-bath relaxation rate ${T_b}^{-1\mathrm{}}$ has been measured in the temperature range $1.3<T<\mathrm{}5\mathrm{}°$K for each of the trivalent Kramers ions: ${\mathrm{Ce}}^{3+}$, ${\mathrm{Nd}}^{3+}$, ${\mathrm{Sm}}^{3+}$, ${\mathrm{Gd}}^{3+}$, ${\mathrm{Dy}}^{3+}$, ${\mathrm{Er}}^{3+}$, and ${\mathrm{Yb}}^{3+}$, magnetically diluted in La${\mathrm{F}}_3$. In three cases, ${\mathrm{Nd}}^{3+}$, ${\mathrm{Er}}^{3+}$, and ${\mathrm{Yb}}^{3+}$, the temperature range was extended to $0.2<T<\mathrm{}5\mathrm{}°$K. All measurements were made by observing the transient recovery of the microwave paramagnetic resonance at the frequency $\nu \approx 9.3$ Gc/sec. The temperature dependence of the Orbach process ${T_{10}}^{-1\mathrm{}}\propto \exp (-\frac{\Delta }{T})$ is displayed for Nd, Sm, and Er; and the Raman process ${T_{1R}}^{-1\mathrm{}}\propto T^9$ is displayed for Nd, Er, and Yb. These do not depend on crystal orientation nor on the concentration of paramagnetic ions. The values $\Delta \left(\mathrm{Nd}\right)=57\mathrm{}°$K and $\Delta \left(\mathrm{Er}\right)=72\mathrm{}°$K are found to agree reasonably with the energy of the first crystal-field state, previously determined optically; we find $\Delta \left(\mathrm{Sm}\right)=50\mathrm{}°$K. The direct process, ${T_{1d}}^{-1\mathrm{}}\propto T$, is observed only for the lowest concentrations of paramagnetic ions: 0.1% Nd and 0.05% Er. For other ions and at higher concentrations the direct process was usually obscured by cross relaxation, probably to excited states of exchange coupled pairs or larger clusters of magnetic ions as evidenced by a temperature dependence ${T_b}^{-1\mathrm{}}\propto csch\left(\frac{{\Delta }^{\prime }}{\mathrm{kT}}\right)$, where ${\Delta }^{\prime }$ is of order of the exchange energy. No clear-cut example of a phonon bottleneck, ${T_b}^{-1\mathrm{}}\propto T^2$, is found. From optical data for Nd and Er, wave functions and crystal-field parameters are obtained, from which simple theoretical estimates of the relaxation rates are found to be in moderate agreement with the data.