Origin of the evolution of spin-orbit and pseudospin-orbit splittings in neutron drops

We found the same staggering pattern in the evolution of the spin-orbit (SO) and pseudospin-orbit (PSO) splittings in neutron drops obtained by the energy-density functional (EDF) theory using SAMi-T and SLy5 parametrizations, with and without the tensor terms respectively. This staggering evolution is similar to the results recently obtained by the relativistic Brueckner-Hartree-Fock (RBHF) theory with the Bonn A potential, which was claimed to be a tensor effect. In this work, we present an intuitive but essential explanation on the origin of this staggering evolution in the Skyrme EDF theory. We found that, for both the SO and PSO partners, the energy splittings mainly originate from the SO potential. The staggering evolution of both the SO and PSO splittings is mainly due to the drastic change of density while neutrons fill different single-particle orbits. The key to determine the specific staggering pattern is the strength of the neutron density gradient term and the SO density term in the SO potential, not necessarily the tensor force.

Figure 1

Total energies of neutron drops $N=6\text{–}50$ scaled by $\hslash \omega N^{4/3}$, calculated by EDF theory with SLy5, SAMi-T, and RBHF theory with Bonn A [14]. The average of several other ab initio results is taken from Ref. [16].

Figure 2

Spin-orbit (SO) splittings of $1p,1d$ partners (a) and pseudospin-orbit (PSO) splittings of $1\tilde{p},1\tilde{d}$ partners (b) in neutron drops $N=6\text{–}50$ obtained by EDF theory with SLy5, SAMi-T, and RBHF theory with Bonn A [15]. The labels $1p_{1/2},1d_{5/2},$ etc. at the bottom denote the single-particle states occupied in order.

Figure 3

SO form factor $W_n$ (a), first-order derivative of neutron density ${\rho }_n^{\prime }$ (b), SO density $J_n$ (c), and neutron density ${\rho }_n$ (d) for neutron drops $N=20,24,28,30$ calculated by SLy5.