Solution of the Bethe-Goldstone equation

Abstract

Various methods for solving the Bethe-Goldstone equation are critically reviewed from the theoretical standpoint, and are then tested in detailed calculations. The methods considered are (i) the integral equation method of Brueckner and Gammel, (ii) the separation method of Moszkowski and Scott, (iii) the reference spectrum method of Bethe et al., (iv) Köhler's “improved separation method”, and (v) the “modified Moszkowski-Scott” method of Bethe et al. A variety of input parameters are considered. Some of the sets of parameters are appropriate for infinite nuclear matter, or equivalently, for the high-density region of the nuclear interior. Other parameter sets correspond to interactions in the nuclear surface. In all cases the intermediate-state potential energies are assumed to vanish. The quantities studied are the G-matrix elements, the two-body wave functions, Fourier transforms of the latter, and the volume of the correlation hole (the “wound integral”). Hard-core potentials have been assumed: the “standard hard core potential” of Moszkowski and Scott, and the Reid 3S 1 potential. The latter contains a strong tensor component. The higher-order Pauli corrections of the reference-spectrum method are found to behave as a geometric series, even for the 3S 1 state. However, the geometric series approximation is seriously in error for the 3D 1 state. Also, separation methods are found to be quite unreliable for the 3S 1 state. The Brueckner-Gammel method is found to be rapid and convenient when (i) the complete Green function is expressed in terms of the reference-spectrum Green function, and (ii) the matrix inversion problem is replaced by iteration of the wave equation. Approximation methods are developed for the Fourier transforms and wound integrals.