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Journal of Physics C Solid State Physics

Critical isotherm of the Ising ferromagnet with spin S>1/2

P F Fox, David S. Gaunt

Published Oct 31, 1972• 26 citations
Publisher

Abstract

Series expansions are used to analyse the shape of the critical isotherm of the spin 1 and 3 / 2 Ising ferromagnet for several two and three dimensional lattices. No compelling reason is found to supposing that the exponent delta differs from its spin 1 / 2 value. Extensive series expansions appropriate for high magnetic fields or low temperatures are given in the field grouping.

DOI
10.1088/0022-3719/5/21/013

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The critical isotherm of the mixed spin Ising model

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Journal of Physics A Mathematical and General•Oct 1, 1983•1 citation
Series expansions are used to analyse the shape of the critical isotherm of the spin 1 and 3 / 2 Ising ferromagnet for several two and three dimensional lattices. No compelling reason is found to supposing that the exponent delta differs from its spin 1 / 2 value. Extensive series expansions appropriate for high magnetic fields or low temperatures are given in the field grouping.
Using extended high-field series expansions and more precise estimates of the critical temperature the critical isotherm of the spin 1 / 2 Ising ferromagnet is re-analysed. The conjectures delta =15 (d=2) and delta =5 (d=3) are confirmed to within 1 / 2 % and 1%, respectively. Good agreement is found, particularly in two dimensions, between series estimates of the critical amplitudes and the predictions of the lattice-lattice scaling hypothesis. Some tentative conclusions regarding the coincident and nonphysical singularities are drawn.
With the aid of exact seriesexpansions the shape of the critical isotherm of an Ising ferromagnet (or fluid) is re-analysed and it is conjectured that δ = 5 for all three-dimensional lattices. (The analysis strengthens previous conjectures that δ = 15 for all two-dimensional lattices.) Estimates are given for the amplitude of the singularity for the various lattices. Evidence is presented to support the conjecture that α' = α = 1/8, β = 5/16 and γ' = γ = 1¼ for all three-dimensional lattices.
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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.