Simple Calculation of the Mean-Square Displacements of Bulk and Surface Atoms of Diamond-Structure Crystals

We use a method based on the expansion of the dynamical matrix to determine the thermal dependence of the mean-square displacement of the atoms $〈u^2〉$. The method is applied to evaluate $〈u^2〉$ for bulk atoms and (100) and (111) surface atoms of a diamond-structure crystal. The dynamical matrix is derived from the elastic energy. That elastic energy, built up from symmetry requirements, depends only on two constants $\alpha$ and $\beta$ in a nearest-neighbor approximation. Numerical values of $\alpha$ and $\beta$ are deduced from experimental data on elastic constants $C_{11}$, $C_{12}$, and $C_{44}$. We plot our result as a function of the temperature for the case of silicon, germanium, and diamond.