Generalized Shell Model and Phonon Dispersions

Among the most interesting and successful applications of the shell model has been its use in calculating o)(q) dispersion relations. For this purpose, COCHRAN [4.10,11,13,14] introduced a generalization of the shell model which, to use the picture developed in Sect.4.3.1, introduces additional interactions between the cores of the ions and between the core of one ion and the shell of another (Fig.4.11). 

We give the result without derivation. The dynamical matrix M(q) can be written in the form 

In this expression, R + ZCZ describes the interaction between rigid ions, see (4.29). C is the Coulomb matrix discussed in Sects.4.2.1,2. The third term in (4.118) takes into account the polarizability of the ions. T includes the intra-ionic core-shell interactions k. Z and Y are diagonal 3n x 3n matrices which designate the ion charges z e(z shell 

A more general shell model in which some of the above restrictions have been relaxed has subsequently been applied to Nal [4.39]. Good agreement with experiments is achieved but the model does contain nine parameters. 

An interesting extension of the shell model is the deformable shell model or breathing shell model introduced by SCHRODER [4.16]. This model allows for radial deformations of the shells in the course of lattice vibrations which leads to three-body interactions and, correspondingly, the model does not predict the Cauchy relations [4.46]. 

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