The diluted Ising model on linear and Bethe lattices

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Abstract
Mixtures of Ising dipoles A and non-magnetic atoms B on Bethe lattices (Cayley trees) are considered. Random mixtures, with all A B distributions equally probable, and equilibrium mixtures are both treated. Equations of state for equilibrium mixtures are obtained by the Rushbrooke-Scions method (1955) and the zero-field susceptibility derived. The susceptibility is derived for the random mixture by a high-temperature series expansion. The linear lattice is treated as a special case and the relation of susceptibility to mean A-cluster size is discussed for both equilibrium and random mixtures. Curves of critical temperature against xA are considered for general coordination number, especially in the equilibrium case with wA less than the minimum value for the appearance of spontaneous magnetization. The critical index yT for variation of susceptibility with temperature at constant xA and the corresponding index Yx for variation with xA at constant temperature are found to be equal to unity except at isolated points. It is shown that an attempt to apply Bethe pair statistics to the random mixture gives instead results corresponding to an equilibrium mixture subject to constraints.