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Proceedings of the Physical Society

On the critical behaviour of ferromagnets

Cyril Domb, D L Hunter

Published Nov 1, 1965• 183 citations
Publisher

Abstract

Using high-temperature series expansions, an attempt is made to characterize the behaviour of the Ising model of a ferromagnet in a non-zero magnetic field. The functional form derived can be used to pass to the low-temperature side of the Curie point, and hence to determine the low-temperature critical behaviour. As a result, it is suggested that there are two basic parameters, in terms of which critical indices for all thermodynamic properties can be expressed.

DOI
10.1088/0370-1328/86/5/127

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Journal of Physics C Solid State Physics•Oct 14, 1977•0 citations
Using high-temperature series expansions, an attempt is made to characterize the behaviour of the Ising model of a ferromagnet in a non-zero magnetic field. The functional form derived can be used to pass to the low-temperature side of the Curie point, and hence to determine the low-temperature critical behaviour. As a result, it is suggested that there are two basic parameters, in terms of which critical indices for all thermodynamic properties can be expressed.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
For the ferromagnetic Ising model the low-temperature series expansion with temperature grouping polynomials is studied. We show that certain roots of these polynomials converge to the critical field Hc{H}_{c}Hc​, and in favorable cases we can determine the critical field quite accurately. Knowing the critical field Hc{H}_{c}Hc​, one can determine the asymptotic behavior of the temperature grouping polynomials numerically. The essential feature is a power-law behavior. Hence, the low-temperature critical indices ${\ensuremath{\alpha}}^{\ensuremath{'}}$, $\ensuremath{\beta}$, and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ can be determined. The values are in general agreement with those found by Pad\'e analysis. A critique of the accuracy of the method and its possibilities is given.
Two properties of the Ising model are considered. Firstly, by extrapolating appropriate series expansions, the spontaneous magnetization of a ferromagnet for three-dimensional lattices is obtained. From the results accurate values for the critical temperature are found. Secondly, the high temperature susceptibility of an antiferromagnet is derived from the approximation of Kikuchi. The resulting susceptibility curves show that for loose-packed lattices a maximum value occurs at a temperature slightly above the critical point, while for the close-packed lattices no such effect is obtained.
A high temperature expansion is obtained for the zero field susceptibility of a generalized Ising model, which can be used to describe the critical behaviour of various multicomponent systems. The first seven terms of the series are obtained for the face centred cubic lattice. Estimates of the critical temperature obtained from this series are considerably more precise than those obtained previously from the zero field free energy. Estimates are also obtained for the critical exponents of the susceptibility and the derivatives of the zero field free energy. The results are compared with the predictions of mean field theory.
A method is described for obtaining perturbation expansions of the pair correlation function about the ferromagnetically ordered state of the S=1/2 Ising model. The expansions take the form of a double power series in temperature and field variables and may be used in any region of the T-H plane of the ferromagnetic and in the disordered region of the antiferromagnet. Coefficients are calculated for the face-centred cubic lattice which allow the low-temperature expansions in zero field to be obtained through u31 and the high-field expansions along the critical isotherm to be obtained through mu 6. Agreement is obtained with existing high-temperature expansions.