Hard-Sphere Lattice Gases. I. Plane-Square Lattice

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Abstract
A plane‐square lattice gas of hard ``squares'' which exclude the occupation of nearest‐neighbor sites is studied by deriving 13 terms of the activity and the virial series and nine terms of appropriate high‐density expansions. Using the ratio and Padé approximant extrapolation techniques it is found that the gas undergoes a continuous (or ``second‐order'') transition to an ordered state at an activity zt=3.80±2 and a density ρt=(0.740±0.008)ρmax. The ordered state is characterized by a difference of the sublattice occupation probabilities, R(z), which vanishes at the |transition as (z—zt)β with β≈⅛. The pressure at the transition is given by pa2/kBT=0.792±5. The compressibility exhibits a maximum at or near the transition point but probably remains finite and continuous through the transition. A suitably defined ``staggered compressibility,'' which measures the tendency towards sublattice ordering, diverges sharply as the transition is approached from either side. A double expansion which converges at all densities is derived and examined numerically and the exact behavior of finite lattices of N=4, 16, 20, and 24 sites is discussed.