The equilibrium concentration of aggregates of cation vacancies and divalent impurities in NaCl-type lattices (Point-Charge Approximation)

A method is derived for the calculation of the free energy of a crystal with aggregates of divalent cation impurities and cation vacancies. The free energy is explicitly given for the case of isolated impurities and vacancies, dipoles, dimers, and trimers. The equilibrium condition for the free energy yields a simple formula for the equilibrium concentration of all aggregates. If the association energy for all aggregates is known an algebraic equation of sixth order has to be solved. As an example the concentrations in the case of the validity of the point charge approximation of the association energies are computed.