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(kTJ) 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
45 and
47 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.
The cluster-variation-Pade-approximant method is a recently proposed tool, based on the extrapolation of low- and high-temperature results obtained with the cluster-variation method, for the determination of critical parameters in Ising-like models. Here the method is applied to the three-dimensional simple cubic Ising model, and new results, obtained with an 18-site basic cluster, are reported. Other techniques for extracting nonclassical critical exponents are also applied and their results compared with those by the cluster-variation-Pade-approximant method.
A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.