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The normal modes of vibration in an ionic crystal of finite thickness are found both by using lattice dynamics and by using electrodynamics, and neglecting retardation. If the wavelength is much larger than the lattice parameter, both methods give coupled integral equations involving the ionic displacements and the normal-mode frequencies. There are two classes of normal modes; those with an oscillatory spatial dependence and frequencies equal to ₓ₎ and ₋₎, the usual transverse optical (TO) and longitudinal optical (LO) frequencies at k0 in an infinite crystal, and those with an exponential dependence on distance across the slab and frequencies between ₓ₎ and ₋₎. A qualitative connection between the normal modes and optical absorption in a slab is presented.
Fuchs et al. (1965) studied this question.