Hard-Square Solids at High Densities

Abstract

An asymptotic expansion for the Helmholtz free energy $F_N$ of a system of $N$ hard squares of side $\sigma$ is obtained, with the result $$F_N{\mathrm{\hspace{0.17em}/\hspace{0.17em}Nk}}_B\mathrm{T\hspace{0.17em}\simeq }{\displaystyle {lim}_{\text{r→1}}}\text{\hspace{0.17em}2}\ln \text{(λ\hspace{0.17em}/\hspace{0.17em}σ)\hspace{0.17em}−\hspace{0.17em}2}\ln {\text{(τ\hspace{0.17em}−\hspace{0.17em}1)\hspace{0.17em}+\hspace{0.17em}C\hspace{0.17em}+\hspace{0.17em}D(τ\hspace{0.17em}−\hspace{0.17em}1)\hspace{0.17em}+\hspace{0.17em}E(τ\hspace{0.17em}−\hspace{0.17em}1)}}^2\mathrm{\hspace{0.17em}+\hspace{0.17em}···,}$$ where $k_B$ is Boltzmann's constant, $\lambda$ is the mean thermal de Broglie wavelength, and $\tau$ 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 ${\text{2 ln(τ}}^{\text{1/2}}\text{\hspace{0.17em}+\hspace{0.17em}1)}$ in powers of $\text{(τ\hspace{0.17em}−\hspace{0.17em}1)}$ . The modified cell‐cluster technique is used to evaluate $C$ , with the result $\text{C\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}0.26042240}$ .