Calculation of the Anharmonic Damping of Rayleigh Surface Modes

The frequency and temperature dependence of the damping of Rayleigh surface modes by cubic anharmonic terms in the crystal potential energy has been calculated for a model of a simple cubic crystal with nearest- and next-nearest-neighbor harmonic forces and with nearest-neighbor cubic anharmonic forces. The damping mechanisms considered in this paper are the interactions between the Rayleigh wave and two bulk phonons which have not been perturbed by the presence of free crystalline surfaces—in particular, interactions in which the Rayleigh wave combines with one bulk phonon to create a second bulk phonon. In the limit that $\frac{\hslash {\omega }_s}{k_{B}T}\ll 1$, where ${\omega }_s$ is the frequency of the Rayleigh wave and $T$ is the absolute temperature, it is found that the Rayleigh-wave damping constant (half the inverse lifetime of the Rayleigh wave) is proportional to ${\omega }_s{\left(k_{B}T\right)}^4$. The damping of low-frequency bulk transverse waves by cubic anharmonic processes at low temperatures is also calculated on the basis of the same crystal model. It is found that when $\frac{\hslash {\omega }_t}{k_{B}T}\ll 1$, where ${\omega }_t$ is the frequency of the transverse wave, the damping constant is proportional to ${\omega }_t{\left(k_{B}T\right)}^4$. From a comparison of these two results, it appears that the damping of Rayleigh waves is larger than that of bulk transverse waves.