General spin Ising model with diluted and random crystal field in the pair approximation

The spin-σ Ising model with diluted and symmetric random crystal field is studied for σ⩾1 in the pair approximation based on Bogoliubov inequality for the free energy. Global phase diagrams are obtained in the temperature versus crystal field plane for the whole range of concentration in the diluted model. A lower critical concentration, above which there is no more a stable ferromagnetic phase at low temperatures for arbitrarily large values of the crystal field, is achieved from the present approach. This behavior, not reproduced by mean field approximation, is in agreement with more reliable results from effective field theory for σ=1. Some new phase diagrams are, however, topologically different from those obtained by mean field procedure and effective field theory. Models with a symmetric random crystal field distribution are also discussed.