The Ising model with long-range interactions

The theory of closed-form approximations in crystal statistics developed by Yvon, Rushbrooke and Scoins, Fournet, and Domb and Hiley is extended to include long-range forces. This extension is used to obtain successive approximations for the entropy, specific heat and the susceptibility in zero field for the disordered state, for arbitrary long-range forces. The general forms of the approximations are studied to determine the shape dependence of the exact properties when the specimen is ellipsoidal and the interactions are dipolar. It is found that, in the disordered state, the susceptibility is shape-dependent while the specific heat and entropy are not. A numerical investigation of the susceptibility using the first two closed forms (a Bethe-type approximation) with an interaction potential of type r-p shows that, as the range of the force increases, the critical point depends predominantly on the dimensionality and becomes insensitive to the detailed structure of the lattice. As p → d (the dimensionality of the lattice), it is found that this approximation gives the exact behaviour in the limit. This conclusion, however, is not valid for dipolar forces.