Quantum Corrections to the Second Virial Coefficient for a Square-Well Potential

Abstract

The first two quantum corrections to the classical second virial coefficient are evaluated in the case that the intermolecular potential is $$\begin{array}{cc}\mathrm{V(}\mathbf{r}\mathrm{)=\infty ,}& \text{0<r≤a,}\\ \text{=−U<0,}& \mathrm{a<r\le b,}\\ \text{=0,}& \mathrm{r>b}\end{array}$$ . The first terms of the expansion of B(T) in powers of λ = (h²/2πmkT)^½ are $$\mathrm{B(T)=}\frac{2}{3}{\mathrm{\pi Na}}^3{e}^{\mathrm{U\beta }}\text{[1+(√}\frac{9}{8}{\mathrm{)\lambda /a+(\pi )}}^{\text{−1}}{\mathrm{(\lambda /a)}}^2\mathrm{]+}\frac{2}{3}{\mathrm{\pi Nb}}^3{\text{(1−e}}^{\mathrm{U\beta }}\mathrm{+(\surd }\frac{9}{8}{\text{)(λ/b)[1+e}}^{\mathrm{U\beta }}{\text{−2e}}^{\frac{1}{2}\mathrm{U\beta }}{I}_0(\frac{1}{2}{\mathrm{U\beta )]+(\pi )}}^{\text{−1}}{\mathrm{(\lambda /b)}}^2\{\frac{1}{2}{\mathrm{(e}}^{\mathrm{U\beta }}\text{−1)−}\frac{3}{8}{\mathrm{;(\pi U\beta )}}^{\frac{1}{2}}{e}^{\frac{1}{2}\mathrm{U\beta }}{\text{[3I}}_{\frac{1}{2}}(\frac{1}{2}{\mathrm{U\beta )+I}}_{\frac{5}{2}}(\frac{1}{2}\text{Uβ)]+[1+}\frac{1}{2}{\mathrm{(U\beta )}}^{\text{−1}}{\text{]3e}}^{\mathrm{U\beta }}{\mathrm{(U\beta )}}^{-\frac{1}{2}}\hspace{0.17em}\mathrm{Erf}{\mathrm{[(U\beta )}}^{\frac{1}{2}}\text{]−[1−}\frac{1}{2}{\mathrm{(U\beta )}}^{\text{−1}}{\text{]3(Uβ)}}^{-\frac{1}{2}}\hspace{0.17em}\mathrm{Erfi}{\mathrm{\hspace{0.17em}[(U\beta )}}^{\frac{1}{2}}\mathrm{]-}\frac{3}{2}{\mathrm{(U\beta )}}^{\text{−1}}{\text{×(1+e}}^{\mathrm{U\beta }}{\mathrm{)\})+O(\lambda }}^3\mathrm{).}$$ The contribution of the bound states to the first quantum correction is evaluated and the behavior of $B_1^{(\mathrm{b)}}{\mathrm{/B}}_1\hspace{0.17em}\mathrm{vs}\mathrm{\hspace{0.17em}T}$ is plotted.