Many-atom interactions in the theory of higher order elastic moduli: A general theory

The total potential energy of a crystal U({r _ik }) as a function of the vectors r _ik connecting centers of equilibrium positions of the ith and kth atoms is assumed to be represented as a sum of irreducible interaction energies in clusters containing pairs, triples, and quadruples of atoms located in sites of the crystal lattice A2: U({r _ik }) ≡ Σ _N=1 ⁴ E _N ({r _ik }). The curly brackets denote the “entire set.” A complete set of invariants {I _j ({r _ik })} _N , which determine the energy of each individual cluster as a function of the vectors connecting centers of equilibrium positions of atoms in the cluster E _N ({r _ik }) ≡ E _N ({I _j ({r _ik })} _N ), is obtained from symmetry considerations. The vectors r _ik are represented in the form of an expansion in the basis of the Bravais lattice. This makes it possible to represent the invariants {I _j ({r _ik })} _N in the form of polynomials of integers multiplied by τ ₂ ^m . Here, τ₂ is one-half of the edge of the unit cell in the A2 structure and m is a constant determined by the model of interaction energy in pairs, triples, and quadruples of atoms. The model interaction potential between atoms in the form of a sum of the Lennard-Jones interaction potential and similarly constructed interaction potentials of triples and quadruples of atoms is considered as an example. Within this model, analytical expressions for second-order and third-order elastic moduli of crystals with the A2 structure are obtained.