Lattice Absorption in Finite Crystals

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Abstract
Certain size and shape effects in the infrared lattice absorption spectra of crystals of finite size have been predicted theoretically by Rosenstock using a simple nearest-neighbor model in the harmonic approximation with free end boundary conditions. Atoms of equal mass, but of opposite sign, were assumed to lie on adjacent sites. In the present investigations, these results were extended to consider the case where the adjacent atoms are of unequal mass. In the case of the one-dimensional chain, this inequality can introduce surface modes and critical points at the zone boundary. Absorption of radiation propagating along the chain can occur because of interaction with the transverse vibrations over the entire range of permitted lattice frequencies and that of the surface mode. However, the amount of energy absorbed (neglecting beam-attenuation effects) can be shown to be directly proportional to the chain length in the frequency region of the longwavelength optical branch modes, but is independent of chain length elsewhere. In the three-dimensional case, the absorption for the cases of adjacent atoms of equal mass and that for adjacent atoms of unequal mass are expected to be similar, and calculations have been carried out only for the former situation. Subsidiary absorption bands in the frequency region associated with some, but not all, critical points can occur, and this can be readily explained in terms of the atomic amplitudes. A size dependence of the absorption corresponding to the one-dimensional chain can be expected. The question of infrared absorption in one-dimensional chains whose length can be either smaller or larger than the wavelength of the interacting radiation has been considered from a simple point of view. In this approach, the nature of the modes interacting with the radiation can be regarded as different for large crystals and small crystals. However, the total absorption in the fundamental frequency region is directly proportional to the chain length, regardless of the ratio of chain length to the wavelength of interacting radiation. The possibility of experimental observation of these various effects is discussed.