Modern formulation of the nuclear mean-field theory is based on the energy density formalism,[1] which has been over the years developed for electronic systems. According to the formal Hohenberg-Kohn[2] and Kohn-Sham theorems,[3] exact ground-state energies of many-fermion systems can be obtained by minimizing certain exact functional of one-body density. These theorems do not provide any method to construct the exact functional in a systematic way; nevertheless one can build phenomenological functionals and test their performance against experimental data. Such an approach is also consistent with the ideas of the effective field theory, whereupon properties of composite objects at low energies can be described by Lagrangians which include high-energy dynamics in the form of the appropriate series of contact terms.
Within the energy density formalism, one treats the nucleus as a single composite object described by a set of one-body densities. At low energies, when the densities are varying slowly in the nuclear interior and then go smoothly to zero at the nuclear surface, one can consider only local densities that are built of the one-body density matrix and its derivatives up to the second-order. Systematics construction of the most general energy density functional (EDF) consistent with symmetries is then possible,[4] and gives a generalization of the extremely successful approach based on the Skyrme effective interaction.[5]
The origins of the spin-orbit (SO) splitting in nuclei can be attributed to the bare two-body SO and tensor interactions,[6] which contribute differently to the spin-saturated (SS) and spin-unsaturated (SUS) nuclei. An alternative explanation is also sought in the relativistic mean-field theories with meson couplings[7], where no distinction between the SS and SUS systems is obtained.
The Skyrme interaction was introduced into nuclear physics more then 30 years ago,[8] and shortly after it was supplemented by tensor forces.[9,10] However, after these ground-breaking studies, in most of the subsequent applications the tensor forces were not taken into account. Moreover, within many Skyrme-force parameterizations constructed to date, the tensor terms in the EDF that were coming from the central force, were quite arbitrarily set equal to zero.[5]
In the present paper, I discuss the form of the tensor terms in the
EDF and their influence on the single-particle energies and nuclear
masses. In Sec. 2, I recall the recent experimental
evidence on the changes of shell structure in neutron-rich
20 nuclei.[11,12] This is only one of
several such examples recently identified in light nuclei and
interpreted within the shell-model by introducing tensor
interactions.[13,14] Properties of the tensor terms
in the EDF are discussed in Sec. 3, and in Sec. 4,
I present results of calculations for single-particle energies and
masses obtained with tensor terms included in the mean-field
approach.