Empirical pairing gaps, shell effects, and di-neutron spatial correlation in neutron-rich nuclei

The empirical pairing gaps derived from four different odd-even mass staggering formulas are compared. By performing single-$j$ shell and multi-shell seniority model calculations as well as by using the standard HFB approach with Skyrme force we show that the simplest three-point formula $\Delta_C^ {(3)}(N)=\frac{1}{2}\left[B(N,Z)+B(N-2,Z)-2B(N-1,Z)\right]$ can provide a good measure of the neutron pairing gap in even-$N$ nuclei. It removes to a large extent the contribution from the nuclear mean field as well as contributions from shell structure details. It is also less contaminated by the Wigner effect for nuclei around $N=Z$. We also show that the strength of $\Delta^ {(3)}_C(N)$ can serve as a good indication of the two-particle spatial correlation in the nucleus of concern and that the weakening of $\Delta^ {(3)}_C(N)$ in some neutron-rich nuclei indicates that the di-neutron correlation itself is weak in these nuclei.