Core-polarization effects

In spite of the fact that the s.p. levels belong to basic building blocks of the MF methods, there is still a vivid debate concerning their physical reality. The question whether they constitute only a set of auxiliary quantities, or represent real physical entities that can be inferred from experimental data, was never of any concern for methods based on phenomenological one-body potentials. Indeed, these potentials were bluntly fitted directly to reproduce the s.p. levels deduced, in one way or another, from empirical information around doubly-magic nuclei, see, e.g., Refs. [39,9]. In turn, these potentials also appear to properly (satisfactorily) reproduce the one-quasiparticle band-heads in open-shell nuclei, see, e.g., Ref. [40]. This success seems to legitimate the physical reliability of the theoretical s.p. levels within the microscopic-macroscopic approaches.

The debate concerns mostly the self-consistent MF approaches based on the EDF methods or two-body effective interactions. The arguments typically brought forward in this context underline the fact that the self-consistent MFs are most often tailored to reproduce bulk nuclear properties like masses, densities, radii, and certain properties of nuclear matter. Consequently, the underlying interactions are non-local with effective masses $m^\star/m \sim 0.8$ [41], what in turn artificially lowers the density of s.p. levels around the Fermi energy. It was pointed out by several authors [5,6,42] that restitution of physical density of the s.p. levels around the Fermi energy can be achieved only after inclusion of particle-vibration coupling, i.e., by going beyond MF.

Within the effective theories, however, these arguments do not seem to be fully convincing. First of all, an effective theory with properly chosen data set used to fit its parameters should automatically select physical value of the effective mass and readjust other parameters (coupling constants) to this value. Examples of such implicit scaling are well known for the SHF theory including: (i) explicit effective mass scaling of the coupling constants $C^s$ and $C^T$ through the fit to the spectroscopic Landau parameters [21]; (ii) direct dependence of the isovector coupling constants $C_1^\tau$ and $C_1^\rho$ on the isoscalar effective mass through the fit to the (observable) symmetry energy strength [43]; (iii) numerical indications for the $m^\star$ scaling of the SO interaction inferred from the $f_{7/2}$-$d_{3/2}$ splittings in $A\sim 44$ nuclei [22]. The differences between various parameterizations of the Skyrme-force (or functional) parameters rather clearly suggest that such an implicit $m^\star$ dependence of functional coupling constants is a fact, which is, however, neither well recognized nor understood so far.

Secondly, the use of effective interaction with parameters fitted at the MF level is not well justified in beyond-MF approximation. Rather unavoidable double counting results in such a case, and quantitative estimates of level shifts resulting from such calculations need not be very reliable. It is, therefore, quite difficult to accept the viewpoint that unsatisfactory spectroscopic properties of, in particular, modern Skyrme forces can be cured solely by going beyond MF. On the contrary, the magnitude of discrepancies between the SHF and experimental s.p. levels (see below) rather clearly suggest that: (i) the data sets used to fit the force (or functional) parameters are incomplete and (ii) the interaction/functional should be extended. This is exactly the task undertaken in the present exploratory work. We extend the conventional EDFs based on Skyrme interactions by including tensor terms, and fit the corresponding coupling constants to the SO splittings rather than to masses. The preliminary goal is to improve spectroscopic properties of functionals, even at the expense of the quality in reproducing the binding energies.

Empirical energy of a given s.p. neutron orbital can be deduced from the difference between the ground-state energy of the doubly-magic core with $N$ neutrons, $E_0(N)$, and energies, $E_p(N+1)$ or $E_h(N-1)$, of its odd neighboring isotopes having a single-particle (p) or single-hole (h) occupying that orbital, i.e.,

$$\epsilon_p(N) = E_p(N+1) - E_0(N),$$ (21)

$$\epsilon_h(N) = E_0(N) - E_h(N-1),$$ (22)

see, for example, Refs. [7,8,9] and referenced cited therein. Note that the total energies above are negative numbers and decrease with increasing numbers of particles, $E_h(N-1)>E_0(N)>E_p(N+1)$. Similarly, we can define a measure of the neutron shell gap as the difference between the lowest single-particle and highest single-hole energy,

$$\Delta\epsilon_{\text{gap}}(N) = \epsilon_p(N) - \epsilon_h(N)$$ (23)

$$= E_p(N+1) + E_h(N-1) - 2E_0(N).$$

Single-particle energies of proton orbitals and proton shell gaps are defined in an analogous way.

Consistently with the empirical definitions, in the present paper, the same procedure is used on the theoretical level [4]. It means that we determine the total energies of doubly-magic cores and their odd neighbors by using the EDF method, and then we calculate the s.p. energies as the corresponding differences (21) or (22). In this way, we avoid all the ambiguities related to questions of what the s.p. energies really are and how they can be extracted from data. Our methodology simply amounts to a specific way of comparing measured and calculated masses of nuclei. In odd nuclei, one has to ensure that particular s.p. orbitals are occupied by odd particles, but still the calculated total energies correspond to masses of their ground and low-lying excited states. Since here we consider only doubly-magic nuclei and their odd neighbors, the pairing correlations are neglected.

In our calculations, both time-even and time-odd polarizations exerted by an odd particle/hole on the doubly-magic core are evaluated self-consistently. In order to discuss these polarizations, let us recall that the extreme s.p. model, in which ground-state energies are sums of fixed s.p. energies of occupied orbitals, is the model with no polarization of any kind included. In this model, the differences of ground-state energies, Eqs. (21) and (22), are, of course, exactly equal to the model s.p. energies. Having this background model in mind, we can distinguish three kinds of polarization effects: