One-Particle Self-Energy and the Virial Coefficients

The relation between the virial coefficients and the self-energy of the one-particle propagator is discussed for the "impurity" or Lorentz model. Expressions are derived for the equilibrium virial coefficients in terms of the self-energy parts describing the scattering of a particle due to clusters of one, two, three, etc., isolated impurities. In particular the second virial coefficient is expressed in terms of the t matrix describing the scattering due to one impurity. This expression is identical in form to that derived by Watson for a real gas. The expression is then reduced to a form involving the phase shifts. (If a bound state contribution exists, this can easily be included.) No assumption is made about the symmetry of the potential, and the result is a generalization of the Beth-Uhlenbeck-Gropper result for local spherically symmetric potentials. The model is discussed classically.