Mariani, Mario; (1991) Many-body interactions and lattice dynamics of atomic crystals. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The aim of this work is to establish whether the addition of three body interactions are by themselves sufficient to ensure mechanical stability of a particular crystal structure or whether higher multibody interactions are needed also. The criterion chosen to establish such stability is namely the positive definiteness of the elastic energy, which can be calculated from the elastic constants. The elastic constants can in turn be calculated from values of the atomic force constants which are obtained by fitting theoretical dispersion curves to experimental data. A summary of the general theory of lattice dynamics as well as atomic elasticity theory is presented with additional comments on the various types of elastic constants often found in the literature. A parallel account of elasticity theory by including a general three body potential function instead of the usual two body (central force) potential is given where most of the simplifications found in previous works vanished so that some of the consequential results such as the Cauchy relations where found not to be valid. A new set of three body parameters are developed and used to work out expressions for the elastic constants and also to test whether in a cubic lattice a three body potential confers stability. We test the usefulness of such parameters when we use them to write down the elastic constants for the diamond crystal lattice. The expressions are then compared to those arrived at by the use of valence bond potentials. Calculations using general force constant parameters to describe atomic interactions show that we need to go to fifth nearest neighbours (at least) before the crystal stability conditions are fulfilled. This problem of stability is readdressed here by the inclusion of possible three body interactions amongst atoms. We also use an analytical expression based on a Morse potential function which describes the interactions amongst three carbon atoms. This potential comes from considering the spectroscopic data of the C₃ molecule. We use such a potential as an example of how the elastic constants of diamond might be calculated.

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