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

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 $I_0{\mathrm{(T)\sim |\rho }}_{\mathrm{liq}}{\mathrm{-\rho }}_{\mathrm{ga5}}{\mathrm{|\sim |T}}_c{\mathrm{-T|}}^{\beta }$ 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 ${\chi }_0{\mathrm{(T)\sim \kappa }}_0{\mathrm{(T)\sim |T}}_c{\mathrm{-T|}}^{{\mathrm{-\gamma }}^{\prime }}$ 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 T_c. 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.