Displacement field and elastic constants in non-ideal crystals

In this work a periodic crystal with point defects is described in the framework of linear response theory for broken symmetry states using correlation functions and Zwanzig-Mori equations. The main results are microscopic expressions for the elastic constants and for the coarse-grained density, point-defect density, and displacement field, which are valid in real crystals, where vacancies and interstitials are present. The coarse-grained density field differs from the small wave vector limit of the microscopic density. In the long wavelength limit, we recover the phenomenological description of elasticity theory including the defect density.