Critical temperatures and critical concentrations in diluted magnetic systems: General solution of the Ising model in effective-field theory approach

Using the single-site effective-field theory approximation the site-diluted spin-1∕2 Ising model is investigated simultaneously on lattices with arbitrary values of the coordination number. The polynomial equation is derived that drives the positions of the critical temperature of the model simultaneously for arbitrary values of the concentration p of the magnetic atoms as well as for all values of the coordination number z. This equation is used for detailed investigation of the dependence of the critical temperature values on the model parameters. It is shown that the critical temperatures of the model can be approximated with high precision by simple function zp−1 especially for large values of the coordination number. The general polynomial equation for the determination of the critical concentration below which the second order phase transition do not exist at all is obtained and analyzed and the explicit analytical expressions for the critical concentrations are found for coordination numbers z=3 and 4. Moreover, algebraic polynomial equations for the magnetization of the model as well as for the saturated magnetization are also derived and discussed.