Hard-Square Solids at High Densities

An asymptotic expansion for the Helmholtz free energy FN of a system of N hard squares of side σ is obtained, with the result FN / NkBT ≃ lim r→1 2 ln(λ / σ) − 2 ln(τ − 1) + C + D(τ − 1) + E(τ − 1)2 + ···, where kB is Boltzmann's constant, λ is the mean thermal de Broglie wavelength, and τ is the ratio of the system area to the close-packed area. A formal expression is derived for the constant C, while D, E, ··· are the coefficients in the expansion of 2 ln(τ1/2 + 1) in powers of (τ − 1). The modified cell-cluster technique is used to evaluate C, with the result C = − 0.26042240.