DOI: 10.1063/1.1733766. Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point

Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point

10.1063/1.1733766

Crossref journal-article

AIP Publishing

The Journal of Chemical Physics (317)

Abstract

The Padé approximant procedure is used to study the low-temperature series for the Ising problem. The critical behavior of the spontaneous magnetization of an Ising ferromagnet and of the liquid-vapor coexistence curve of the corresponding lattice gas is found to be I0(T)∼|ρliq−ρga5|∼|Tc−T|β where (in three dimensions) 0.303≤β≤0.318. It is conjectured that β=5/16=0.31250. The low-temperature initial susceptibility and the compressibility of the lattice gas at condensation are found to diverge as χ0(T)∼κ0(T)∼|Tc−T|−γ′ where in two dimensions γ′=1.75±0.01 while in three dimensions γ′=1.25 (+0.07, —0.02). Consideration of a heuristic model partition function suggests the identity α′+2β+γ′=2 where α′ is the index of divergence of the specific heat below Tc. The amplitudes of the singularities are derived and extrapolation formulas are presented for the plane triangular, square, and honeycomb lattices and the three-dimensional face-centered, body-centered, and simple cubic lattices. The results are compared briefly with experimental evidence.

Bibliography

Essam, J. W., & Fisher, M. E. (1963). PadÃ© Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point. The Journal of Chemical Physics, 38(4), 802â812.

Authors 2

John W. Essam (first)
Michael E. Fisher (additional)

References 33 Referenced 316

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Dates

Type	When
Created	20 years, 7 months ago (Jan. 9, 2005, 1:59 a.m.)
Deposited	1 year, 6 months ago (Feb. 8, 2024, 4:24 p.m.)
Indexed	1 month, 3 weeks ago (July 1, 2025, 9:31 a.m.)
Issued	62 years, 6 months ago (Feb. 15, 1963)
Published	62 years, 6 months ago (Feb. 15, 1963)
Published Print	62 years, 6 months ago (Feb. 15, 1963)

Funders 0

None

BibTeX

@article{Essam_1963, title={Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point}, volume={38}, ISSN={1089-7690}, url={http://dx.doi.org/10.1063/1.1733766}, DOI={10.1063/1.1733766}, number={4}, journal={The Journal of Chemical Physics}, publisher={AIP Publishing}, author={Essam, John W. and Fisher, Michael E.}, year={1963}, month=feb, pages={802–812} }

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  "abstract": "<jats:p>The Pad\u00e9 approximant procedure is used to study the low-temperature series for the Ising problem. The critical behavior of the spontaneous magnetization of an Ising ferromagnet and of the liquid-vapor coexistence curve of the corresponding lattice gas is found to be I0(T)\u223c|\u03c1liq\u2212\u03c1ga5|\u223c|Tc\u2212T|\u03b2 where (in three dimensions) 0.303\u2264\u03b2\u22640.318. It is conjectured that \u03b2=5/16=0.31250. The low-temperature initial susceptibility and the compressibility of the lattice gas at condensation are found to diverge as \u03c70(T)\u223c\u03ba0(T)\u223c|Tc\u2212T|\u2212\u03b3\u2032 where in two dimensions \u03b3\u2032=1.75\u00b10.01 while in three dimensions \u03b3\u2032=1.25 (+0.07, \u20140.02). Consideration of a heuristic model partition function suggests the identity \u03b1\u2032+2\u03b2+\u03b3\u2032=2 where \u03b1\u2032 is the index of divergence of the specific heat below Tc. The amplitudes of the singularities are derived and extrapolation formulas are presented for the plane triangular, square, and honeycomb lattices and the three-dimensional face-centered, body-centered, and simple cubic lattices. The results are compared briefly with experimental evidence.</jats:p>",
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