Lattice statics Green's function for a semi-infinite crystal

The lattice statics Green's function describes the linear response of displacement to the application of a force on a crystal lattice. From a macroscopic point of view, the crystal lattice behaves like an anisotropic elastic body. Therefore, the lattice statics Green's function must approach the elastic Green's function for large distances from the point at which the force is applied. The lattice used in calculating the lattice Green's function should have the elastic constants corresponding to the crystal and a symmetrical stress tensor in the continuum elastic limit. In order to satisfy these conditions, three-body forces are introduced. The Green's function for the infinite lattice with short-range atomic interaction is calculated immediately by translational symmetry. The Green's function for the defective lattice is derived from the Dyson equation. The definitive Green's function for the semi-infinite lattice is presented.