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Journal of Physics C Solid State Physics

Critical isotherm of the spin 1 Ising ferromagnet

P F Fox, David S. Gaunt

Published May 1, 1970• 12 citations
Publisher

Abstract

Series expansions are used to analyse the shape of the critical isotherm of the spin 1 Ising model for several two- and three- dimensional lattices. The results are consistent with the conjecture that the exponent delta is the same as for the spin 1 / 2 case.
DOI
10.1088/0022-3719/3/5/002

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The basic low temperature series expansion data for the spin one-half Ising model on the hydrogen peroxide lattice are used to obtain series in z=exp(-2J/k B T) along the coexistence curve for the specific heat C H , the magnetization M, and its first five derivatives delta 1 M/ delta mu 1 , where mu =exp(-2mH/k B T). The same data are used to derive series in mu along the critical isotherm for M and its first five derivatives delta l M/ delta z l . Ratio and Pade approximant analysis yield estimates of the critical exponents and critical amplitudes. On the whole the estimates of the critical exponents support scaling theory although a few of the exponent estimates are not in good agreement with scaling.

Journal of Physics A Mathematical Nuclear and General•Mar 21, 1974•5 citations
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.
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.
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 $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
The basic low temperature series expansion data for the spin one-half Ising model on the hydrogen peroxide lattice are used to obtain series in z=exp(-2J/k B T) along the coexistence curve for the specific heat C H , the magnetization M, and its first five derivatives delta 1 M/ delta mu 1 , where mu =exp(-2mH/k B T). The same data are used to derive series in mu along the critical isotherm for M and its first five derivatives delta l M/ delta z l . Ratio and Pade approximant analysis yield estimates of the critical exponents and critical amplitudes. On the whole the estimates of the critical exponents support scaling theory although a few of the exponent estimates are not in good agreement with scaling.