A new method is presented for the calculation of the reaction matrix $G$ of the Brueckner-Goldstone theory. The spectrum of the intermediate states is replaced by a "reference spectrum" of the form $A+B{k}^2$ where the constants $A$ and $B$ are chosen so as to approximate, as closely as possible, the actual particle energies for $k$ between 3 and 6 ${\mathrm{F}}^{-1\mathrm{}}$. The reason for this choice is explained. With the reference spectrum, the Brueckner integral equation reduces to a differential equation which is easily solved. The case of a repulsive core can be solved explicitly, and can be summed over angular momentum, taking into account the correct statistical weights. If an attractive potential is added to the repulsive core, a simple "modified Born approximation" can be developed. Noncentral forces, such as tensor forces, are considered.

The actual $G$ matrix, $G^N$, is calculated from the reference matrix $G^R$. It is shown that this can be done to sufficient accuracy (0.1 to 0.2 MeV per nucleon) by a simple quadrature. The difference $G^N-G^R$ arises mainly from the Pauli principle which is not taken into account in $G^R$. A small correction, less than 1 MeV per nucleon, arises from the inaccuracy of the reference spectrum. This shows that the details of the particle energy spectrum are not important for the calculation of the nuclear binding energy.

The particle energy spectrum is carefully investigated. In agreement with Brueckner and Goldman, the $G$ matrices determining the potential energy of states in the Fermi sea are calculated "on the energy shell," and a more detailed justification is given for this procedure. Those for states above the Fermi sea are calculated "off the energy shell." This, in combination with the repulsive core, has the consequence of making the potential energy very large and positive for large $k$, corresponding to an effective mass between 0.8 and 0.9 for highly excited states. In addition, there is an energy gap at the Fermi momentum, a feature which helps to justify the reference spectrum.

A modified Moszkowski-Scott separation into short- and long-range potentials is developed and gives, in second order, results accurate to better than 0.1 MeV per particle. The wave functions of interacting particles are calculated in the reference spectrum approximation for central and tensor forces.

• Received 27 June 1962

DOI:https://doi.org/10.1103/PhysRev.129.225