## Developing a Hartree-Fock program, separation energies

We can use these results to attempt our first link with experimental data, namely to compute the shell gap or the separation energies. The shell gap for neutrons is given by $$\Delta S_n= 2BE(N,Z)-BE(N-1,Z)-BE(N+1,Z).$$ For $${}^{16}\mbox{O}$$ we have an experimental value for the shell gap of $$11.51$$ MeV for neutrons, while our Hartree-Fock calculations result in $$25.65$$ MeV. This means that correlations beyond a simple Hartree-Fock calculation with a two-body force play an important role in nuclear physics. The splitting between the $$0p_{3/2}^{\nu}$$ and the $$0p_{1/2}^{\nu}$$ state is 4.88 MeV, while the experimental value for the gap between the ground state $$1/2^{-}$$ and the first excited $$3/2^{-}$$ states is 6.08 MeV. The two-nucleon spin-orbit force plays a central role here. In our discussion of nuclear forces we will see how the spin-orbit force comes into play here.