The Self-Consistent Isotropic Einstein Model

In order to illustrate the SCHA, we consider the simplest possible model, the self-consistent isotropic Einstein model [5.461. In the Einstein model, each atom is bound by a spring with force constant (j to its equilibrium position. Starting from the general relation (3.12), we disregard all coupling coefficients other than the diagonal term (00) write for the force acting on the atom 0 in the direction a 

In a real crystal, the force constant (j = (00) is, of course, due to the interactions of atom 0 with its neighbouring atoms and this is expressed by the relation (3.17b) which gives 

In order to avoid computational difficulties, we choose as the interatomic potential a simple form of Gaussian type [5.46], namely, 

This is an explicit self-consistent equation for the force constant for 

5 Response Function and Perturbation Theory of Phonon-Phonon Interactions 

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