Quantum Corrections to the Square-Well Classical Second Virial Coefficient

Abstract

The first quantum correction to the classical value of the second virial coefficient for the square-well potential is calculated. The result is $B\left(T\right)=\frac{2}{3}\pi d^3[{\eta }^3+(1-{\eta }^3\left)e^{-\beta u}\right]+2^{-\frac{1}{2}}\pi \lambda d^2\{e^{-\beta u}+{\eta }^2[1+e^{-\beta u}-2\mathrm{}{e}^{-\frac{\beta u}{2}}{I}_0\left(\frac{1}{2}\beta u\right)\left]\right\}$, with $\lambda ={\left(\frac{h^2}{2\pi \mathrm{mkT}}\right)}^{\frac{1}{2}}$ and $\beta ={\mathrm{}\left(\mathrm{kT}\right)\mathrm{}}^{-1\mathrm{}}$. Here, $u$ denotes the depth of the well, $d$ is the diameter of the hard core, and $\eta d$ is the range of the potential.