Application of the Padé Approximant Method to the Investigation of Some Magnetic Properties of the Ising Model

Abstract

On the basis of the Pad\'e approximant method we deduce from the exact series expansions for the Ising model that the reduced magnetic susceptibility behaves at the critical point as ${\ensuremath{\chi}}_{\mathrm{fcc}}\ensuremath{\approx}{[\frac{0.09923}{(0.101767\ensuremath{-}w)}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{bcc}}\ensuremath{\approx}{[\frac{0.152773}{(0.1561789\ensuremath{-}w)}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{sc}}\ensuremath{\approx}{[\frac{0.22138}{(0.218156\ensuremath{-}w}]}^{\frac{5}{4}}$, ${\ensuremath{\chi}}_{\mathrm{t}}\ensuremath{\approx}{[\frac{0.2432}{(2\ensuremath{-}\sqrt{3}\ensuremath{-}w)}]}^{\frac{7}{4}}$, ${\ensuremath{\chi}}_{\mathrm{sq}}\ensuremath{\approx}{[\frac{0.35724}{(\sqrt{2}\ensuremath{-}1\ensuremath{-}w)}]}^{\frac{7}{4}}$, and ${\ensuremath{\chi}}_{\mathrm{h}}\ensuremath{\approx}{[\frac{0.4506}{(\frac{1}{\sqrt{3}}\ensuremath{-}w)}]}^{\frac{7}{4}}$, where $w=tanh(\frac{J}{\mathrm{kT}})$ and the last figure quoted is somewhat uncertain. The spontaneous magnetization is found to behave as ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{fcc}}\ensuremath{\approx}{[12.5(0.664658\ensuremath{-}{z}^{2})]}^{0.3}$, ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{bcc}}\ensuremath{\approx}{[10.4(0.5326607\ensuremath{-}{z}^{2})]}^{0.3}$, ${(\frac{{I}_{0}}{{I}_{\ensuremath{\infty}}})}_{\mathrm{sc}}\ensuremath{\approx}{[10.9(0.411940\ensuremath{-}{z}^{2})]}^{0.3}$, where $z=\mathrm{exp}(\ensuremath{-}\frac{2J}{\mathrm{kT}})$ and again the last place quoted is somewhat uncertain. The numbers $\frac{5}{4}$ and $\frac{7}{4}$ have an error of at most ${10}^{\ensuremath{-}3}$, and 0.3 of at most ${10}^{\ensuremath{-}2}$. The lattices referred to are fcc, face-centered cubic; bcc, body-centered cubic; sc, simple cubic; t, triangular; sq, simple quadratic; and h, honeycomb.