Neutron $2p$ and $1f$ spin-orbit splittings were recently measured in the isotones ${}^{37}\mathrm{S}$ and ${}^{35}\mathrm{Si}$ by $(d,p)$ transfer reactions. Values were reported by using the major fragments of the states. An important reduction of the $p$ splitting was observed, from ${}^{37}\mathrm{S}$ to ${}^{35}\mathrm{Si}$, associated with a strong modification of the spin-orbit potential in the central region of the nucleus ${}^{35}\mathrm{Si}$. We analyze $2p$ and $1f$ neutron spin-orbit splittings in the $N=20$ isotones ${}^{40}\mathrm{Ca},{}^{36}\mathrm{S}$, and ${}^{34}\mathrm{Si}$. We employ several Skyrme and Gogny interactions to reliably isolate pure spin-orbit and tensor-induced contributions, within the mean-field approximation. We use interactions (i) without the tensor force, (ii) with the tensor force and with tensor parameters adjusted on top of existing parametrizations, nd (iii) with the tensor force and with tensor and spin-orbit parameters adjusted simultaneously on top of existing parametrizations. We predict in cases (ii) and (iii) a non-negligible reduction of both $p$ and $f$ splittings, associated with neutron-proton tensor effects, from ${}^{40}\mathrm{Ca}$ to ${}^{36}\mathrm{S}$. The two splittings are further decreased for the three types of interactions, going from ${}^{36}\mathrm{S}$ to ${}^{34}\mathrm{Si}$. This reduction is produced by the spin-orbit force and is not affected by tensor-induced contributions. For both reductions, from ${}^{40}\mathrm{Ca}$ to ${}^{36}\mathrm{S}$ and from ${}^{36}\mathrm{S}$ to ${}^{34}\mathrm{Si}$, we predict in all cases that the modification is more pronounced for $p$ than for $f$ splittings. The measurement of the centroids for neutron $2p$ and $1f$ states in the nuclei ${}^{36}\mathrm{S}$ and ${}^{34}\mathrm{Si}$ would be interesting to validate this prediction experimentally. We show the importance of using interactions of type (iii), because they provide $p$ and $f$ splittings in the nucleus ${}^{40}\mathrm{Ca}$ which are in agreement with the corresponding experimental values.

• Received 14 September 2015

DOI:https://doi.org/10.1103/PhysRevC.92.054316