Heat-Pulse Propagation in Dielectric Solids

Abstract

The usual phonon Boltzmann equation is solved by using two mean relaxation times, ${\ensuremath{\tau}}_{N}$ for normal and ${\ensuremath{\tau}}_{R}$ for resistive processes. For a Debye solid with three polarizations, an explicit expression for the Fourier transform of the local temperature in a heat-pulse experiment is calculated. It describes hydrodynamic phenomena for $\ensuremath{\Omega}\ensuremath{\tau}\ensuremath{\ll}1$, such as second sound and diffusive heat conduction, and heat transport by ballistic phonons for $\ensuremath{\Omega}\ensuremath{\tau}\ensuremath{\gg}1$. In the intermediate regime, $\ensuremath{\Omega}\ensuremath{\tau}\ensuremath{\approx}1$, we find the following results: a second-sound wave with wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ can only propagate if $Q{\ensuremath{\tau}}_{N}$ and $Q{\ensuremath{\tau}}_{R}$ are smaller than certain critical values, ${(Q{\ensuremath{\tau}}_{N})}_{c}$ and ${(Q{\ensuremath{\tau}}_{R})}_{c}$, i.e., for $T\ensuremath{\ge}{T}_{c}$, assuming the usual monotonic $T$ dependence of ${\ensuremath{\tau}}_{N}$ and ${\ensuremath{\tau}}_{R}$. The velocity ${C}_{2}$ of second sound strongly depends on these relaxation times. Its maximum value, occurring at $T={T}_{c}$, is the larger the smaller the ratio $\frac{{({\ensuremath{\tau}}_{N})}_{c}}{{({\ensuremath{\tau}}_{R})}_{c}}$. Then ${C}_{2}$ decreases with rising $T$ and finally goes to zero for $\ensuremath{\Omega}{\ensuremath{\tau}}_{R}\ensuremath{\lesssim}1$.