Liquid He-4. II. Three-Body Correlations and Their Contribution to the Binding Energy and the Depletion

The three-body contribution to the binding energy of liquid He-4 is calculated. The methods applied are those developed mainly by Bethe and collaborators for nuclear-matter calculations, and later modified by the author jot liquid He-3. Various two-body wave functions and reaction-matrix elements are calculated, as is the effective interaction or representation of the reaction matrix in coordinate space. Three-body wave functions and three-body energy contributions are then calculated or estimated. The calculations are repeated for several values of the density and for several values of the intermediate-state relative momentum between the two interacting particles. The three-body contribution to the binding energy is found to be approximately -1.3 to -1.6 K per particle. This is, however, a rather uncertain value because the results are very sensitive to the values chosen for the average relative momentum in the calculations of the effective interaction off the energy shell. Also, in parts of the calculations only the S wave is included, and it is taken to represent the effect of the total wave function. The total binding energy of liquid 4He is estimated by adding together the contribution from separate calculations of the two-body and the three-body interaction. A self-consistent solution is obtained through the depletion of the zero-momentum state, which is also estimated. The calculations are done for two different two-body potentials, an Yntema-Schneider potential given by Brueckner and Gammel, and a Frost-Musulin potential given by Bruch and McGee. The theoretical results are -4.0 to -5.0 K for the binding energy and 44-50% for the depletion. The corresponding experimental results are -7 K and 83 %, respectively, i.e., the difference is generally within a factor of 1.5-2. The agreement with experimental results is probably reasonably good (or bad) since four-body and higher order cluster terms are not included.