The Pade approximant method w a s applied to Clapp's spinspin correlation formulation for the magnetic susceptibility in an Ising square lattice, in order to investigate the nature of the critical point.For the exact cluster formulation it is found that as the cluster size increases, the critical point, namely the point at which the susceptibility has a singularity, approaches Onsager's exact value of 0.414, and that ~( t ) -?/( tt c ) .For the selfconsistent formulation, twenty Pade approximants were calculated.The value of the critical point converged very rapidly to 0.41862770 and it was found that X ( t ) l/(tt c ) .
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.
We use the Pad\'e approximant method to investigate the nature of the singularity in the specific heat for some three-dimensional Ising model lattices and to investigate the nature and location of the singularity in the magnetic susceptibility for some three-dimensional Heisenberg model lattices. We find that the three-dimensional Ising model specific heat becomes singular like $log|T\ensuremath{-}{T}_{c}|$ with a different coefficient above and below the singular point. For the Heisenberg model we find that the magnetic susceptibility does not behave like the Curie-Weiss law, but tends to infinity more rapidly than $\frac{1}{(T\ensuremath{-}{T}_{c})}$.