Low-Temperature Expansion of the Third-Cluster Coefficient of a Quantum Gas

The low-temperature behavior of the third-quantum-cluster coefficient is investigated using the multiple-scattering form of the binary-collision expansion. For hard spheres and Boltzmann statistics we find $b_3=2\mathrm{}{\left(\frac{a}{\lambda }\right)}^2-\frac{4}{3}\sqrt{2}\pi {\left(\frac{a}{\lambda }\right)}^3-\frac{16}{3}(4\mathrm{}\pi -3\mathrm{}\sqrt{3}){\left(\frac{a}{\lambda }\right)}^4\ln \left(\frac{a}{\lambda }\right)+O\left({\left(\frac{a}{\lambda }\right)}^4\right)$ where $a$ is the sphere diameter and $\lambda$ is the thermal wavelength. The first two terms were obtained some time ago by Lee and Yang and by Pais and Uhlenbeck. The occurrence of a term of the form ${\lambda }^{-4\mathrm{}}\ln \lambda$ was predicted recently by Adhikari and Amado. The expansion is also given for Bose-Einstein and Fermi-Dirac statistics, and for the case of an intermolecular potential without bound states. The limitations of such low-temperature expansions are discussed.