Temperature-Dependent Internal Field Distribution and Magnetic Susceptibility of a Dilute Ising Spin System

Abstract

The temperature-dependent probability distribution of internal exchange fields, $\overline{H}$, is obtained for a set of randomly distributed Ising-model spins using a modified form of the statistical model of Margenau. The impurities are assumed to interact via a convergent long-range potential which alternates in sign as a function of position. When spin correlations between the magnetic impurities are neglected and a mean-random-field (MRF) approximation is used, the probability distribution $P (\overline{H})$ is given by a nonlinear integral equation. For a $\frac{1}{{r}^{3}}$ potential, the self-consistent probability distribution is, in the MRF approximation, a Lorentzian with a temperature- and concentration-dependent width $\ensuremath{\Delta}(\ensuremath{\beta})$, where $\ensuremath{\beta}=\frac{1}{({k}_{B}T)}$ and $T$ is the temperature. The function $\ensuremath{\Delta}(\ensuremath{\beta})$ is also given by a nonlinear integral equation which is solved for very high and very low temperatures. Using $P(\overline{H})$ derived for a $\frac{1}{{r}^{3}}$ potential, the magnetic susceptibility $\ensuremath{\chi}(\ensuremath{\beta})$ and the specific heat ${C}_{v}(\ensuremath{\beta})$ are obtained for all temperatures. The model gives a magnetic susceptibility which exhibits a maximum as a function of temperature for all nonzero (but sufficiently small) impurity concentrations. The temperature of the maximum is proportional to the impurity concentration. Possible applications of the model to the temperature and concentration dependence of $\ensuremath{\chi}(\ensuremath{\beta})$ and ${C}_{v}(\ensuremath{\beta})$ of dilute magnetic alloys are discussed.