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Register a new userOn the difference between proton and neutron spin-orbit splittings in nuclei
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The latest experimental data on nuclei at $^{132}$Sn permit us for the first time to determine the spin-orbit splittings of neutrons and protons in identical orbits in this neutron-rich doubly-magic region and compare the case to that of $^{208}$Pb. Using the new results, which are now consistent for the two neutron-rich doubly magic regions, a theoretical analysis defines the isotopic dependence of the mean field spin-orbit potential and leads to a simple explicit expression for the difference between the spin-orbit splittings of neutrons and protons. The isotopic dependence is explained in the framework of different theoretical approaches.
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We present expressions for the matrix elements of the spin--spin operator $vec S_{rm n}cdotvec S_{rm p}$ in a variety of coupling schemes. These results are then applied to calculate the expectation value $langlevec S_{rm n}cdotvec S_{rm p}rangle$ in eigenstates of a schematic Hamiltonian describing neutrons and protons interacting in a single-$l$ shell through a Surface Delta Interaction. The model allows us to trace $langlevec S_{rm n}cdotvec S_{rm p}rangle$ as a function of the competition between the isovector and isoscalar interaction strengths and the spin--orbit splitting of the $j=lpm frac{1}{2}$ shells. We find negative $langlevec S_{rm n}cdotvec S_{rm p}rangle$ values in the ground state of all even--even $N=Z$ nuclei, contrary to what has been observed in hadronic inelastic scattering at medium energies. We discuss the possible origin of this discrepancy and indicate directions for future theoretical and experimental studies related to neutron--proton spin--spin correlations.
Spin-orbit splitting is an essential ingredient for our understanding of the shell structure in nuclei. One of the most important advantages of relativistic mean-field (RMF) models in nuclear physics is the fact that the large spin-orbit (SO) potential emerges automatically from the inclusion of Lorentz-scalar and -vector potentials in the Dirac equation. It is therefore of great importance to compare the results of such models with experimental data. We investigate the size of $2p$ and $1f$ splittings for the isotone chain $^{40}$Ca, $^{38}$Ar, $^{36}$S, and $^{34}$Si in the framework of various relativistic and nonrelativistic density functionals. They are compared with the results of nonrelativistic models and with recent experimental data.
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.
It is shown that the renormalized nuclear deformations in different mass regions can be globally scaled by two probability partition factors of Boltzmann-like distribution, which are derived from the competing valence $np$ and like-nucleon interactions. The partition factors are simply related to the probabilities of anti-parallel and fully-aligned orientations of the angular momenta of the neutrons and protons in the valence $np$ pairs, responsible for spherical- and deformed-shape phases, respectively. The partition factors derived from the renormalized deformations are also present in the new scaling law for the energies of the first $2^+$ states. A striking concordance between the distributions of the renormalized deformations and of the newly introduced parameter for the energies of the first $2^+$ states over the extended mass region from Ge to Cf is achieved, giving strong support to the existence of two phases: anti-aligned and fully-aligned subsets of $np$ pairs.
We propose a particle number conserving formalism for the treatment of isovector-isoscalar pairing in nuclei with $N>Z$. The ground state of the pairing Hamiltonian is described by a quartet condensate to which is appended a pair condensate formed by the neutrons in excess. The quartets are built by two isovector pairs coupled to the total isospin $T=0$ and two collective isoscalar proton-neutron pairs. To probe this ansatz for the ground state we performed calculations for $N>Z$ nuclei with the valence nucleons moving above the cores $^{16}$O, $^{40}$Ca and $^{100}$Sn. The calculations are done with two pairing interactions, one state-independent and the other of zero range, which are supposed to scatter pairs in time-revered orbits. It is proven that the ground state correlation energies calculated within this approach are very close to the exact results provided by the diagonalization of the pairing Hamiltonian. Based on this formalism we have shown that moving away of N=Z line, both the isoscalar and the isovector proton-neutron pairing correlations remain significant and that they cannot be treated accurately by models based on a proton-neutron pair condensate.
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