Spin-orbit and tensor energy densities, mean fields, and interactions

We begin by recalling the form of the EDF, which will be used in the present study. In the notation defined in Ref. [10] (see Ref. [11] for more details and extensions), the EDF reads

$${\mathcal E} = \int {\rm d}^3{\mathbf r}\, {\mathcal H}({\mathbf r}),$$ (1)

where the local energy density ${\mathcal H}({\mathbf r})$ is a sum of the kinetic energy, and of the potential-energy isoscalar ($t=0$) and isovector ($t=1$) terms,

$${\mathcal H}({\mathbf r}) = \frac{\hbar^2}{2m}\tau_0 + {\mathcal H}_0({\mathbf r}) + {\mathcal H}_1({\mathbf r}) ,$$ (2)

with

$${\mathcal H}_t({\mathbf r}) = {\mathcal H}^{\text{even}}_t({\mathbf r}) + {\mathcal H}^{\text{odd}}_t ({\mathbf r}) ,$$ (3)

and

$$\mathcal{H}_t^{\text{even}} = C^{\rho}_t \rho^2_t + C^{\Delta \rho}_t \rho_t\Delta\rho_t +$$ (4)

$$\quad C^{\tau}_t \rho_t\tau_t + C^J_t {\mathbb{J}}^2_t + C^{\nabla J}_t \rho_t {\mathbf \nabla}\cdot{\mathbf J}_t,$$

$$\mathcal{H}_t^{\text{odd}} = C^{s}_t {\mathbf s}^2_t + C^{\Delta s}_t {\mathbf s}_t\cdot\Delta {\mathbf s}_t +$$ (5)

$$\quad C^{T}_t{\mathbf s}_t \cdot {\mathbf T}_t + C^j_t {\mathbf j^2_t} + C^{\nabla J}_t {\mathbf s}_t \cdot ({\mathbf \nabla}\times {\mathbf j}_t).$$

For the time-even, $\rho_t$, $\tau_t$, and ${\mathbb{J}}_t$, and time-odd, ${\mathbf s}_t$, ${\mathbf T}_t$, and ${\mathbf j}_t$, local densities we follow the convention introduced in Ref. [12], see also Refs. [3,11] and references cited therein.

In particular, the SO density ${\mathbf J}$ is the vector part of the spin-current tensor density ${\mathbb{J}}$, i.e.,

$${\mathbb{J}}_{\mu \nu} = \tfrac{1}{3} J^{(0)} \delta_{\mu \nu} + \tfrac{1}{2} \varepsilon_{\mu \nu \eta} {J}_\eta + J^{(2)}_{\mu \nu},$$ (6)

with

$${\mathbb{J}}^2 \equiv \sum_{\mu \nu} {\mathbb{J}}_{\mu \nu}^2 = \... ...{(0)})^2 + \tfrac{1}{2} {\mathbf J}^2 + \sum_{\mu \nu } (J^{(2)}_{\mu \nu})^2 .$$ (7)

In the context of the present study, the time-even tensor and SO parts of the EDF,

$$\mathcal{H}_T = C^J_0 {\mathbb{J}}^2_0 + C^J_1 {\mathbb{J}}^2_1 ,$$ (8)

$$\mathcal{H}_{SO} = C^{\nabla J}_0 \rho_0 {\mathbf \nabla}\cdot{\mathbf J}_0 + C^{\nabla J}_1 \rho_1 {\mathbf \nabla}\cdot{\mathbf J}_1 ,$$ (9)

are of particular interest.

In the spherical-symmetry limit, the scalar $J^{(0)}$ and symmetric-tensor $J^{(2)}_{\mu \nu}$ parts of the spin-current tensor vanish, and thus

$$\mathcal{H}_T = \tfrac{1}{2} C^J_0 J^2_0(r) + \tfrac{1}{2} C^J_1 J^2_1(r) ,$$ (10)

$$\mathcal{H}_{SO} = -C^{\nabla J}_0 J_0(r)\frac{d\rho_0}{dr} -C^{\nabla J}_1 J_1(r)\frac{d\rho_1}{dr} ,$$ (11)

where the SO density has only the radial component, ${\mathbf J}_t=\frac{{\mathbf r}}{r}J_t(r)$. Variation of the tensor and SO parts of the EDF over the radial SO densities $J(r)$ gives the spherical isoscalar ($t=0$) and isovector ($t=1$) SO MFs,

$$W_t^{SO} = \frac{1}{2r}\left( C^J_t J_t(r) - C^{\nabla J}_t \frac{d\rho_t}{dr}\right) {\mathbf L} \cdot {\mathbf S},$$ (12)

which can be easily translated into the neutron ($q=n$) and proton ($q=p$) SO MFs,

$$W_q^{SO} = \frac{1}{2r}\bigg\{(C^J_0-C^J_1) J_0(r) + 2C^J_1 J_q(r)$$ (13)

$$-(C^{\nabla J}_0-C^{\nabla J}_1) \frac{d\rho_0}{dr} - 2C^{\nabla J}_1 \frac{d\rho_q}{dr}\bigg\} {\mathbf L} \cdot {\mathbf S}.$$

Although below we perform calculations without assuming spherical and time-reversal symmetries, here we do not repeat general expressions for the SO mean-fields, which can be found in Refs. [12,11]. We also note that, in principle, in this general case, one could use different coupling constant multiplying each of the three terms in Eq. (7). In the present exploratory study, we do not implement this possible extension of the EDF, and we use unique tensor coupling constants $C^J_t$, as defined in Eq. (4).

Identical potential-energy terms of the EDF, Eqs. (4) and (5), are obtained by averaging the Skyrme effective interaction within the Skyrme-Hartree-Fock (SHF) approximation [3]. By this procedure, the EDF coupling constants $C_t$ can be expressed through the Skyrme-force parameters, and one can use parameterizations existing in the literature. It is clear that one can study tensor and SO effects entirely within the EDF formalism, i.e., by considering the corresponding tensor and SO parts of the EDF, Eqs. (8) and (9), and coupling constants $C^J_t$ and $C^{\nabla J}_t$, respectively. However, in order to link this approach to those based on the Skyrme interactions, we recall here expressions based on averaging the zero-range tensor and SO forces [13,14], see also Refs. [11,15,16] for recent analyses. Namely, in the spherical-symmetry limit, one has

$${\mathcal H}_T = \tfrac{5}{8} \left[ t_{\text{e}} J_n(r) J_p(r) + t_{\text{o}} (J_0^2(r) - J_n(r) J_p(r) ) \right] ,$$ (14)

$${\mathcal H}_{SO} = \frac{1}{4}\left[3W_0J_0(r)\frac{d\rho_0}{dr} +W_1J_1(r)\frac{d\rho_1}{dr}\right],$$ (15)

where in Eq. (15) two different coupling constants, $W_0$ and $W_1$, were introduced following Ref. [17].

The corresponding SO MFs read

$$W_q^{SO} = \frac{1}{2r}\bigg\{\tfrac{5}{8} \left( (t_{\text{e}} + t_{\text{o}}) J_0(r) - (t_{\text{e}} - t_{\text{o}}) J_q(r) \right)$$ (16)

$$+ \frac{1}{4}\left( (3W_0-W_1) \frac{d\rho_0}{dr} - 2W_1 \frac{d\rho_q}{dr}\right) \bigg\} {\mathbf L} \cdot {\mathbf S} ,$$

[note that in Ref. [15], the factor of $\tfrac{1}{2}$ was missing at the $W_0$ term of Eq. (4)]. By comparing Eqs. (13) and (16), one obtains the following relations between the coupling constants:

$$C_0^J = \tfrac{5}{16}( 3t_{\text{o}} + t_{\text{e}} ) ,$$ (17)

$$C_1^J = \tfrac{5}{16}( t_{\text{o}} - t_{\text{e}} ) ,$$ (18)

$$C^{\nabla J}_0 = -\tfrac{3}{4} W_0 ,$$ (19)

$$C^{\nabla J}_1 = -\tfrac{1}{4} W_1.$$ (20)

For further discussion of the Skyrme forces and their relation to tensor components we refer the reader to an extensive and complete recent discussion presented in Ref. [16].

In this exploratory work, we base our considerations on the EDF method and deliberately break the connection between the functional (4), and the Skyrme central, tensor, and SO forces. Nevertheless, in the time-even sector, our starting point is the conventional Skyrme-force-inspired functional with coupling constants fixed at the values characteristic for either SkP [18], SLy4 [19], or SkO [20] Skyrme parameterizations. However, poorly known coupling constants in the time-odd sector (those which are not related to the time-even ones through the local-gauge invariance [10]) are fixed independently of their Skyrme-force values. For this purpose, the spin coupling constants $C^{s}_t$ are readjusted to reproduce empirical values of the Landau parameters, according to the prescription given in Refs. [21,22], and $C^{\Delta s}_t$ are set equal to zero. These variants of the standard functionals are below denoted by SkP$_{L}$, SLy4$_{L}$, and SkO$_{L}$.

Strictly pragmatic reasons, like technical complexity and lack of firm experimental benchmarks, made the majority of older Skyrme parameterizations simply disregard the tensor terms, by setting $C^J_t\equiv 0$. However, suggestions to study tensor effects on a one-body level were already made long time ago [13,14,23,24]. Recent experimental discoveries of new magic shell-openings in neutron-reach light nuclei, e.g., around $N=32$ [25,26], and their subsequent interpretation in terms of tensor interaction within the shell-model [27,28,29], caused a revival of interest in tensor terms within the MF approach [30,15,31,32,34,35,33,16,,37,38], which is naturally tailored to study s.p. levels. Indeed, as shown in Ref. [15], the tensor terms mark clear and unique fingerprints in isotonic and isotopic evolution of s.p. levels and, in particular, in the SO splittings.

Below we consider one example of the Skyrme force (SkP), which does contain the usual $C^J_t$ terms in the energy functional, and two examples of forces (SLy4 and SkO) that set these terms equal to zero. For the latter forces, there exist also variants (SLy5 and SkO') that do include the $C^J_t$ terms. We have checked that conclusions of our study do not depend on which of these variants are considered.

In this paper, we perform systematic study of the SO splittings. The goal is to resolve contributions to the SO MF (12) due to the tensor and SO parts of the EDF, and to readjust the corresponding coupling constants $C^J_t$ and $C^{\nabla J}_t$. It is shown that this goal can be essentially achieved by studying the $f_{7/2} - f_{5/2}$ SO splittings in three key nuclei: $^{40}$Ca, $^{48}$Ca, and $^{56}$Ni. Before we present in Sec. 4 details of the fitting procedure and values of the obtained coupling constants, first we discuss effects of the core polarization and its influence on the calculated and SO splittings.