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

The total potential energy of a crystal $U(\{\mathbf{r}_{ik}\})$ as a function of the vectors $\mathbf{r}_{ik}$ connecting centers of equilibrium positions of the $i$th and $k$th 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 $A2$: $U(\{\mathbf{r}_{ik}\})\equiv\sum_{N=1}^4 E_N(\{\mathbf{r}_{ik}\})$. The curly brackets denote the “entire set”. A complete set of invariants $\{I_j(\{\mathbf{r}_{ik}\})\}_N$, which determine the energy of each individual cluster as a function of the vectors connecting centers of equilibrium positions of atoms in the cluster $E_N(\{\mathbf{r}_{ik}\})\equiv E_N(\{I_j(\{\mathbf{r}_{ik}\}_N)$, is obtained from symmetry considerations. The vectors $\mathbf{r}_{ik}$ are represented in the form of an expansion in the basis of the Bravais lattice. This makes it possible to represent the invariants $\{I_j(\{\mathbf{r}_{ik}\})\}_N$ in the form of polynomials of integers multiplied by $\tau_2^m$. Here, $\tau_2$ 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.