Computing lattice sums for calculating the elastic moduli of bcc metals via cluster decomposition

Abstract

The complete potential energy of a crystal \(E\left( {\vec r_{ik} } \right)\) is presented in the form of an expansion in irreducible interactions in clusters containing pairs, triplets, and quadruplets of atoms, situated on A2 lattice sites. The full set of invariants \(\left\{ {I_j \left( {\vec r_{ik} } \right)} \right\}\), on which \(\left\{ {I_j \left( {\vec r_{ik} } \right)} \right\}\) can depend is found. Vectors \(\vec r_{ik}\) are presented in the form of an expansion of the base of a Brave lattice. This allows us to present \(I_j \left( {\vec r_{ik} } \right)\) in the form of integers (lattice sums) multiplied by τ m , where τ is half of an elementary cell rib, and m = const is determined by the model. The sum of the Lenard-Jones potential and the potentials of tri- and tetra-atomic interactions was chosen as the model potential. Within this model, elastic moduli of the second and third order were calculated for crystals with A2-type structure.