Normal Modes of a Semi-Infinite Ionic Crystal

The normal modes of a semi-infinite ionic crystal bounded by a pair of (100) faces normal to the $z$ direction but infinite in the $x$ and $y$ directions have been determined by a combination of analytical and numerical methods. Cyclic boundary conditions are imposed on the displacements along the $x$ and $y$ directions, but the presence of a pair of free surfaces is correctly incorporated into both the short-range and the long-range Coulomb contributions to the dynamical matrix. The latter contribution is made rapidly convergent by a modified Bessel-function transformation. The $6L\times 6L$ ($L=\mathrm{number}\mathrm{of}\mathrm{atomic}\mathrm{planes}\mathrm{for}\mathrm{the}\mathrm{slab}$) eigenvalue equation for the normal-mode frequencies is solved numerically for general values of the wave vector throughout the two-dimensional first Brilluion zone. The two lowest-frequency modes are Rayleigh waves, whose degeneracy is slightly split by the presence of a pair of free surfaces. Optical surface modes are found whose limiting frequencies at infinite wavelength differ from those of the bulk LO and TO modes. The contribution to infrared absorption at infinite wavelength of the optical surface modes have been calculated and the effects of relaxing the intraplanar lattice parameter and the interplanar separations to minimize the potential energy of the slab have also been determined.