A generalized Uhlenbeck and Beth formula for the third cluster coefficient

• Published in: Annals of physics Vol. 374; pp. 291 - 313
• Main Authors: Larsen, Sigurd Yves, Lassaut, Monique, Amaya-Tapia, Alejandro
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.11.2016; Elsevier BV; Elsevier Masson
• Subjects: ADIABATIC APPROXIMATION; BOUND STATE; Bound states; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Eigenphase shifts; NUCLEAR PHYSICS AND RADIATION PHYSICS; Nuclear Theory; Particle physics; Physics; Third quantum virial; THREE-BODY PROBLEM; Eigenphase shifts; Third quantum virial; Bound states
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2016.08.011

Relatively recently (Amaya-Tapia et al., 2011), we presented a formula for the evaluation of the third Bose fugacity coefficient–leading to the third virial coefficient–in terms of three-body eigenphase shifts, for particles subject to repulsive forces. An analytical calculation for a 1-dim. model, for which the result is known, confirmed the validity of this approach. We now extend the formalism to particles with attractive forces, and therefore must allow for the possibility that the particles have bound states. We thus obtain a true generalization of the famous formula of Uhlenbeck and Beth (Uhlenbeck and Beth, 1936; Beth and Uhlenbeck, 1937) and of Gropper (Gropper, 1936, 1937) for the second virial. We illustrate our formalism by a calculation, in an adiabatic approximation, of the third cluster in one dimension, using McGuire’s model as in our previous paper, but with attractive forces. The inclusion of three-body bound states is trivial; taking into account states having asymptotically two particles bound, and one free, is not.