Tunneling and rattling in clathrate crystal

Abstract

We present tunneling and rattling motions of an off-center guest atom in a cage referring to a clathrate crystal ${\mathrm{La}}_{3}{\mathrm{Pd}}_{20}{\mathrm{Ge}}_{6}$. The elastic constant ${C}_{44}$ of ${\mathrm{La}}_{3}{\mathrm{Pd}}_{20}{\mathrm{Ge}}_{6}$ shows a Debye-type dispersion of around $20\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ obeying a relaxation time $\ensuremath{\tau}={\ensuremath{\tau}}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(E∕{k}_{\mathrm{B}}T)$ with an attempt time of ${\ensuremath{\tau}}_{0}=2.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\phantom{\rule{0.3em}{0ex}}\mathrm{s}$ and an activation energy $E=197\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. At low temperatures below $3\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ down to $20\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$, the ${C}_{44}$ shows a softening of ${C}_{44}={C}_{44}^{0}(T\ensuremath{-}{T}_{C}^{0})∕(T\ensuremath{-}\ensuremath{\Theta})$ with ${T}_{C}^{0}=\ensuremath{-}337.970\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$ and $\ensuremath{\Theta}=\ensuremath{-}338.044\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$. These facts are attributed to two different types of off-center motions with ${\ensuremath{\Gamma}}_{5}$ symmetry in the $4a$-site cage of ${\mathrm{La}}_{3}{\mathrm{Pd}}_{20}{\mathrm{Ge}}_{6}$, a thermally activated rattling motion over the potential hill, and a tunneling motion through the potential hill at low temperatures.