Quantum-Mechanical Second Virial Coefficient of a Hard-Sphere Gas at High Temperatures

The high-temperature expansion of the quantum-mechanical second virial coefficient $B\left(T\right)\mathrm{}$ of a gas of particles interacting via a hard-core pair potential is determined to fourth order in the ratio of the thermal wavelength $\lambda ={\left(\frac{h^2}{2\pi \mathrm{mkT}}\right)}^{\frac{1}{2}}$ to the extension $d$ of the hard core. The result is $B\left(T\right)=\frac{2}{3}\pi d^3\left[1+\frac{3}{2\sqrt{2}}\frac{\lambda }{d}+\frac{1}{\pi }{\left(\frac{\lambda }{d}\right)}^2+\frac{1}{16\pi \sqrt{2}}{\left(\frac{\lambda }{d}\right)}^3-\frac{1}{105{\pi }^2}{\left(\frac{\lambda }{d}\right)}^4\right].$ The first term is the classical value. The second term was found by Uhlenbeck and Beth. The third term, apart from a missing factor of 2, was obtained by Mohling. The correct value $\frac{1}{\pi }$, together with the fourth term, was obtained by Handelsman and Keller. Our calculation is based upon the method of Handelsman and Keller, viz., an expansion of the thermal Green's function and its boundary conditions in powers of $\frac{\lambda }{d}$. Our exact value for the coefficient of ${\left(\frac{\lambda }{d}\right)}^4$ confirms a numerical estimate of -0.000965 obtained by Boyd, Larsen, and Kilpatrick.