Lattice Statistics-A Review and an Exact Isotherm for a Plane Lattice Gas

Progress in the theory of lattice statistics is reviewed with emphasis on the use of series expansions to study the critical behavior and on the exact results obtained by transformation theory. Recent work is reported which indicates that the magnetization of the three‐dimensional Ising model vanishes as (T_c − T)^β with $\mathrm{\beta \simeq }\frac{5}{16}$ , while the low‐temperature susceptibility diverges as (T_c − T)^−γ with $\mathrm{\gamma =}\frac{5}{4}\mathrm{(\gamma =}\frac{7}{4}$ in two dimensions). An Appendix is devoted to a detailed tabulation of the exact numerical values and best estimates for the critical temperatures, energies, specific heats, entropies, magnetizations, and the ferro‐ and antiferromagnetic susceptibilities of four plane lattices and the simple, body‐centered, and face‐centered cubic lattices. A square lattice gas with infinite nearest‐neighbor repulsions and weak second‐neighbor attractions (across alternate squares) is solved exactly for one particular temperature by transformation. The gas undergoes a transition at a density ${\rho }_t=\frac{1}{4}\sqrt{2}$ , the corresponding form of the isotherm being $$\frac{{\mathrm{p-p}}_t}{\mathrm{kT}}=\frac{{\mathrm{\alpha (\rho }}_t\mathrm{-\rho )}}{\ln {\mathrm{\hspace{0.17em}\alpha \hspace{0.17em}|\rho }}_t\mathrm{-\rho |}}$$ , where $\text{α=π(1+}\frac{3}{4}\sqrt{2})$ , so that the isothermal compressibility becomes logarithmically infinite.