Separation energies and energy gaps

Using say the neutron separation energies (alternatively the proton separation energies) $$ S_n= -Q_n= BE(N,Z)-BE(N-1,Z), $$ we can define the so-called energy gap for neutrons (or protons) as $$ \Delta S_n= BE(N,Z)-BE(N-1,Z)-\left(BE(N+1,Z)-BE(N,Z)\right), $$ or $$ \Delta S_n= 2BE(N,Z)-BE(N-1,Z)-BE(N+1,Z). $$ This quantity can in turn be used to determine which nuclei are magic or not. For protons we would have $$ \Delta S_p= 2BE(N,Z)-BE(N,Z-1)-BE(N,Z+1). $$ We leave it as an exercise to the reader to define and interpret the two-neutron or two-proton gaps.