Mean-Square Amplitudes of Vibration at a Surface

The ratios of the mean-square amplitudes of vibration at a surface to those in the bulk have been studied in detail for model crystals whose particles interact through a Lennard-Jones potential. Within a quasiharmonic approximation, the ratio for a particular surface and direction of vibration is approximately a universal function of $\frac{T}{{\Theta }_D}$, where $T$ is the temperature and ${\Theta }_D$ the bulk Debye temperature. The ratio increases rapidly with $T$ for $T<\mathrm{}\frac{1}{2}{\Theta }_D$ and then is nearly constant. Three factors increase the ratio above the value it would have in a simple force-constant model: (i) a temperature-independent decrease in the force constants at the surface relative to those in the bulk, (ii) a further temperature-dependent decrease, and (iii) anharmonicity. These factors may account for the tendency of experimental values of this ratio to be larger than values calculated with force-constant models.