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

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 χfcc≈[0.09923(0.101767-w)]5/4, χbcc≈[0.152773(0.1561789-w)]5/4, χsc≈[0.22138(0.218156-w]5/4, χₜ≈[0.2432(2-√3-w)]7/4, χsq≈[0.35724(√2-1-w)]7/4, and χₕ≈[0.4506(1√3-w)]7/4, where w=tanh(J/kT) and the last figure quoted is somewhat uncertain. The spontaneous magnetization is found to behave as (I₀I_∞)fcc≈[12.5(0.664658-z²)]0.3, (I₀I_∞)bcc≈[10.4(0.5326607-z²)]0.3, (I₀I_∞)sc≈[10.9(0.411940-z²)]0.3, where z=exp(-2J/kT) and again the last place quoted is somewhat uncertain. The numbers 5/4 and 7/4 have an error of at most 10^-3, and 0.3 of at most 10^-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.