Calculating V_eff from the Free Inter-Nucleon Interaction

Figure 1.
The Brueckner G matrix is an iterated version of V, similar to the transtion matrix of scattering theory. However, the intermediate states (red) are restricted to exclude states that are already occupied in the closed shell core.

Consider the effective interaction V_eff appropriate for shell-model calculations on a system of two valence nucleons outside a closed-shell core. The nuclear medium of the core profoundly modifies the interaction of the valence nucleons, so that V_eff is very different from V, the bare interaction between free nucleons. There are two main effects that "renormalize" V to make the effective interaction V_eff. The first effect is that the Pauli principle forbids nucleons to be scattered into states that are already occupied in the core. The Brueckner G-matrix is a well-defined first approximation to V_eff that takes the Pauli exclusion effect into account, and also has the very desirable property that its matrix elements between independent-particle shell matrix elements are finite and of reasonable magnitude. In contrast, the bare interaction V would have extremely large positive shell-model matrix elements, because of the presence of a short-ranged "hard core" in V. The hard core is required in order for V to reproduce the behavior of two-nucleon s-wave phase shifts at high energies.

The second effect of the medium is core polarization, which contributes to V_eff through processes in which one nucleon emits a core phonon which is then absorbed by the other nucleon. Thus theoretical calculations of V_eff have generally been based on the G-matrix with core-polarization corrections.

Figure 2.
A valence particle in state p₁ excites the core C (thick line) to a state C* (green), and a second valence particle, in the process of de-exciting the core to state C, is scattered from state p₂ to state p₂'.

Figure 3.
Here the excited state of the core is a particle-hole state. The downward arrow represents a hole in the core. The ground state of the core is not shown in this type of diagram.