Some Dynamical Properties of Surface Atoms

Expressions for the mean-square velocity and mean-square amplitude of an atom in the surface layers of a simple cubic crystal are obtained using double-time Green's functions. They are evaluated explicitly in the high-temperature limit. The Hamiltonian of the unperturbed crystal is taken in the harmonic approximation with nearest- and next-nearest-neighbor central force interactions between the atoms. The perturbation produces free surfaces by subtracting off all the interactions (or bonds) which cross a certain plane. The exponential in the Debye-Waller factor for a surface atom, calculated to lowest order in inverse temperature, is twice as large as its bulk value for $\gamma$-ray emission perpendicular to the surface and about 30% larger than the bulk value for emission parallel to the surface. These surface corrections decay very rapidly as one goes into the crystal from the surface. In the fifth atomic layer the exponential in the Debye-Waller factor is within 5% of the bulk value. In the second-order Doppler shift the effect of the surface is to reduce the second-order term in inverse temperature (first quantum correction) by about 30%. The lowest-order term is unaffected. If the crystal is made isotropic in the long wavelength, continuum limit surface waves which are the classical Rayleigh waves appear.