Manifestation of the internal symmetry of irreducible triatomic interaction in the nonlinear properties of low-symmetry metals
A theory of the elastic moduli of crystals and amorphous substances is constructed that considers the internal symmetry of the irreducible energy of triatomic clusters. Allowing for the internal symmetry of clusters allows us to build three invariants by which the potential energy of any free triatomic cluster depends on the atomic coordinates. Direct calculation of the elastic modulus using an analytic form of the dependence of invariants on atomic coordinates shows that the adopted model leads to the following relationship between independent components of the third-rank elasticity tensor: $${C_{xx,\;yy,\;zz\;}} - ({C_{xx,\;yz,yz\;}} + {C_{yy,\;xz,\;xz\;}} + {C_{zz,\;xy,\;xy}}) + 2{C_{xy,\;yz,\;zx\;}} = 0$$ This result is independent of the global symmetry of a substance and represents a generalization of the widely known Cauñhy discrepancies, which follow from models that consider only pair interactions. Fifteen laws of conservation that determine the movement of a point representing the atomic coordinates of cluster atoms in 9-dimensional configuration spaces of triatomic clusters are identified. The symmetry group of a triatomic cluster that determines these laws of conservation is identified.