Spin-orbit densities

Figure 2: Radial components $J_n(r)$ of the neutron vector SO densities $\mbox{\boldmath${J}$\unboldmath}_n$, calculated for the $Z$=28 isotopes with $N$=28-38 (left panel) and $N$=40-50 (right panel).

In order to illustrate the influence of tensor densities on single-particle and global nuclear properties, in Fig. 2 are shown the radial components, $J_n(r)$= $\mbox{\boldmath${J}$\unboldmath }_n\cdot\mbox{\boldmath${r}$\unboldmath }/r$, of the neutron vector SO densities $\mbox{\boldmath${J}$\unboldmath }_n$, calculated for the nickel isotopes between $N$=28 and 50. Calculations have been performed for the Skyrme SLy4 interaction,[19] by using the Hartree-Fock-Bogolyubov (HFB) method with the spherical symmetry assumed.[20]

One can see that the SO densities are mostly positive and peaked near the surface. At $N$=28, the SO density is large and its major part comes from the occupied $\nu$f$_{7/2}$ orbital (SUS system). By adding neutrons in the shell above $N$=28, this part is gradually cancelled by an increasing in magnitude, negative contribution from the SO partner $\nu$f$_{5/2}$. At the same time contributions from the $\nu$p$_{3/2}$ and $\nu$p$_{1/2}$ orbitals appear. When both pairs of the SO partners are occupied around $N$=38, and when the $\nu$g$_{9/2}$ orbital is still empty (SS system), the SO density is rather small. Beyond $N$=40, it increases again until the $\nu$g$_{9/2}$ orbital becomes fully occupied at $N$=50. Note the shift of the SO densities to larger distances, which occurs at the point of the switch-over between the dominating 1$f$ and 1$g$ contributions.

A similar pattern of varying SO densities is valid for all shells. For SS systems, one obtains small SO densities, while for SUS systems, the SO densities are large. Therefore, the SO densities are small at magic shells $N,Z$=2, 8, and 20, and large at magic shells $N,Z$=28, 50, 82, and 126.

Since the $j_>$ partners are always occupied first, the SO densities are mostly positive. Note that the derivatives of particle densities are mostly negative, and also peaked at the surface; therefore, for positive coupling constants, the SO and tensor forces split the SO partners in opposite directions, cf. Eq. (4).