Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperature. II

As the continuation of a preceding paper, an expansion for the quantum-mechanical free energy $F$ of a hard-sphere gas at high temperature is extended up to the second order in the thermal wavelength $\lambda ={\left(\frac{2\pi {\hslash }^2}{\mathrm{mkT}}\right)}^{\frac{1}{2}}$. To reach this order, one must study the three-body problem in a lowest-order approximation, in which adjacent sphere surfaces can be regarded as parallel planes. Coefficients of the $\lambda$ series for $F$ are given in terms of classical correlation functions. Using known density expansions for these correlation functions, one can obtain $\lambda$ expansions for the virial coefficients; the third virial coefficient is $B_3=\left(\frac{5{\pi }^2{a}^6}{18}\right)[1+(\frac{3}{\sqrt{2}}\left)\right(\frac{\lambda }{a})+1.707660\mathrm{}{\left(\frac{\lambda }{a}\right)}^2+\cdots ]$, where $a$ is the hard-sphere diameter (only the last term is a new result).