Quantum-Mechanical Third Virial Coefficient and Three-Body Phase Shifts

A formalism is presented for the calculation of the quantum-mechanical third virial coefficient in terms of two- and three-body phase shifts. This is done for Boltzmann statistics and in the absence of bound states. The method is based on the expansion of three-body wave functions in terms of hyperspherical harmonics in six-dimensional space. Basic scattering equations are set down, including expressions for three-particle $S$, $T$, and $R$ matrices and for the three-body phase shifts. Connection is then made with statistical mechanics and an expression proposed for the evaluation of the Boltzmann three-body cluster in terms of these three-body phase shifts. Using this method, the behavior of the third virial coefficient of a gas subject to binary square-well interactions is studied in the limit when $T$ approaches zero.