Anomalously long lifetimes of high-energy surface acoustic phonons

The lifetime of Rayleigh surface phonons as opposed to decay via anharmonic three-phonon processes is calculated in the regime $\hslash {\omega }_R\gg k_{B}T$. The surface phonon decays predominantly by combining with a bulk transverse phonon to create another bulk transverse phonon. The energymomentum conservation of the processes imposes the condition that those bulk phonons which may interact with the surface phonons should possess frequencies of the same order of ${\omega }_R$. However, such high-frequency phonons are rarely excited thermally under the condition considered here, leading to an anomalously long lifetime of the surface phonons in proportion to ${\omega }_R^{-4.5\mathrm{}}{T}^{-0.5\mathrm{}}\exp \left(\frac{A\hslash {\omega }_R}{k_{B}T}\right)$, where $A$ is a constant of the order of 0.1. We also estimate numerically the magnitude of the lifetime.