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.
Exact low-density and high-density series are derived for the plane-triangular (pt) lattice gas, and in three dimensions, for the simple cubic (sc) and body-centered cubic (bcc) lattice gases, with first-neighbor exclusions. The series are studied by the ratio and Padé approximant extrapolation techniques. In all cases we find a continuous transition to an ordered state at the activities zt=11.05±15 (pt), 1.09±7 (sc) and 0.77±5 (bcc), and the densities ρt/ρ0=0.832±8 (pt), 0.427±20 (sc) and 0.355±20 (bcc). The grand potentials at these points are (pv0/kBT) = Γt=0.839±5 (pt), 0.380±10 (sc) and 0.290±15 (bcc). In all cases the uncertainties refer to the last decimal places.
The ordered state is characterized by a long-range order parameter which vanishes continuously and rapidly at the transition point. The transition is signed from above and below by the rapid divergence (faster than a simple pole) of the ``staggered compressibility.'' Apparently, the isothermal compressiibility also diverges to infinity from either side but with considerably less rapidity.
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