Neutron $2p$ and $1f$ spin--orbit splittings in ${}^{40}$Ca, ${}^{36}$S, and ${}^{34}$Si $N=20$ isotones: tensor--induced and pure spin--orbit effects

Abstract

Neutron $2p$ and $1f$ spin--orbit splittings were recently measured in the isotones ${}^{37}$S and ${}^{35}$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 ${}^{37}$S to ${}^{35}$Si, associated to a strong modification of the spin--orbit potential in the central region of the nucleus ${}^{35}$Si. We analyze $2p$ and $1f$ neutron spin--orbit splittings in the $N=20$ isotones ${}^{40}$Ca, ${}^{36}$S, and ${}^{34}$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; (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 to neutron--proton tensor effects, from ${}^{40}$Ca to ${}^{36}$S. The two splittings are further decreased for the three types of interactions, going from ${}^{36}$S to ${}^{34}$Si. This reduction is produced by the spin--orbit force and is not affected by tensor--induced contributions. For both reductions, from ${}^{40}$Ca to ${}^{36}$S and from ${}^{36}$S to ${}^{34}$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 ${}^{36}$S and ${}^{34}$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 ${}^{40}$Ca which are in agreement with the corresponding experimental values.