Spin-Lattice Relaxation in Some Rare-Earth Salts at Helium Temperatures; Observation of the Phonon Bottleneck

Abstract

By observing the transient recovery of microwave paramagnetic resonance signals at $\ensuremath{\nu}\ensuremath{\approx}9.3$ kMc/sec and $\ensuremath{\nu}\ensuremath{\approx}34$ kMc/sec, we measure the spin-lattice relaxation rate ${{T}_{1}}^{\ensuremath{-}1}$ for the rare earth ions Nd, Pr, and Sm in the double nitrate [${\mathrm{La}}_{2}$${\mathrm{Mg}}_{3}$${(\mathrm{N}{\mathrm{O}}_{3})}_{12}$\ifmmode\cdot\else\textperiodcentered\fi{}24${\mathrm{H}}_{2}$O] and for Ce and Nd in the ethyl sulfate [La${({\mathrm{C}}_{2}{\mathrm{H}}_{5}\mathrm{S}{\mathrm{O}}_{4})}_{3}$\ifmmode\cdot\else\textperiodcentered\fi{}9${\mathrm{H}}_{2}$O] in the temperature range $1.4\ifmmode^\circ\else\textdegree\fi{}<T<5\ifmmode^\circ\else\textdegree\fi{}$K. We observe the direct process, ${{T}_{1}}^{\ensuremath{-}1}\ensuremath{\propto}T$; the Orbach process, ${{T}_{1}}^{\ensuremath{-}1}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\Delta}}{\mathrm{kT}})$; and the Raman process, ${{T}_{1}}^{\ensuremath{-}1}\ensuremath{\propto}{T}^{7}$ and ${T}^{9}$. The measured relaxation rates are in good agreement with simple theoretical estimates based on Orbach's phenomenological approach. For example, for 1% Nd in the ethyl sulfate with $z\ensuremath{\perp}H$ at $\ensuremath{\nu}=9.3$ kMc/sec we measure ${{T}_{1}}^{\ensuremath{-}1}=1.7T+3.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{T}^{9}$ ${\mathrm{sec}}^{\ensuremath{-}1}$, as compared to the theoretical estimate, ${{T}_{1}}^{\ensuremath{-}1}=1.4T+1.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{T}^{9}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The data, together with similar measurements by others, lead to the over-all conclusion that spin-lattice relaxation at low temperatures in rare earth salts is reasonably well understood.At the lowest temperatures, where the direct process dominates, we observe in the double nitrate several instances of a spin-bath relaxation rate ${{T}_{b}}^{\ensuremath{-}1}$ which is not the direct spin-lattice (i.e., spin-phonon) process, but rather a slower phonon-limited "bottle-neck" process, with a temperature dependence ${{T}_{b}}^{\ensuremath{-}1}\ensuremath{\propto}{T}^{2}$. This dependence along with that on crystal size and paramagnetic ion concentration is in good agreement with simple theoretical expectations. The data indicate that the hot phonon-bath relaxation time is the time taken by sound waves to traverse the crystal half-thickness. For 1% Pr in the double nitrate at 1.4\ifmmode^\circ\else\textdegree\fi{}K the bottleneck is severe, the observed rate ${{T}_{b}}^{\ensuremath{-}1}$ being \ensuremath{\sim}${10}^{3}$ times smaller than the true direct rate ${{T}_{1}}^{\ensuremath{-}1}$.