Tensor and spin-orbit parts of the local nuclear energy density functional

In this work we consider the local EDF, ${\mathcal H}({\mathbf r})$, of the Skyrme-type. It consists of a kinetic energy and a sum of isoscalar ($t$=0) and isovector ($t$=1) potential energy terms:

$${\mathcal H}({\mathbf r}) = \frac{\hbar^2}{2m}\tau_0 + \sum_{t=0,... ...mathbf r})^{\text{even}} + {\mathcal H}_t ({\mathbf r})^{\text{odd}} \biggr\} ,$$ (1)

where

$$\mathcal{H}_t^{\text{even}} = C^{\rho}_t[\rho_0] \rho^2_t + C^{\Delta \rho}_t \rho_t\Delta\rho_t +$$ (2)

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

$$\mathcal{H}_t^{\text{odd}} = C^{s}_t [\rho_0 ] {\mathbf s}^2_t + C^{\Delta s}_t {\mathbf s}_t\cdot\Delta {\mathbf s}_t +$$ (3)

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

with the density-dependent primary coupling constants $C^{\rho}_t[\rho_0]$ and $C^{s}_t [\rho_0 ]$. The potential energy terms are bilinear forms of either time-even ($\rho$, $\tau$, ${\mathbb{J}}$) or time-odd ( ${\mathbf s}$, ${\mathbf j}$, ${\mathbf T}$) densities and their derivatives, see, e.g., Ref. [10] for details. The ${\mathbf J}_{t}$ density denotes the vector part of the spin-current tensor, ${\mathbf J}_{t,\lambda} = \sum_{\mu\nu} \epsilon_{\lambda\mu\nu}{\mathbb{J}}_{t,\mu\nu}$.

In this work we focus on the tensor,

$$\mathcal{H}^T = C^J_0 {\mathbb{J}}^2_0 + C^J_1 {\mathbb{J}}^2_1 ,$$ (4)

and the SO terms,

$$\mathcal{H}^{SO} = C^{\nabla J}_0 \rho_0 {\mathbf \nabla}\cdot{\mathbf J}_0 + C^{\nabla J}_1 \rho_1 {\mathbf \nabla}\cdot{\mathbf J}_1 .$$ (5)

In the limit of spherical symmetry, the vector part ${\mathbf J}_{t} \equiv J_t (r) {\mathbf e}_{r}$ is the only non-vanishing part of the tensor density ${\mathbb{J}}_{\mu\nu}$. Hence, in this limit, the tensor part of the functional (4) reduces to:

$$\mathcal{H}^T = \frac{1}{2}C^J_0 {\mathbf J}^2_0 + \frac{1}{2}C^J_1 {\mathbf J}^2_1.$$ (6)

By performing variation of the functional with respect to $J_t (r)$ one obtains the one-body SO potential:

$$W_t^{SO} = - \frac{1}{2r}\left( C^{\nabla J}_t \frac{d\rho_t}{dr} - C^J_t J_t(r)\right) {\mathbf L} \cdot {\mathbf S},$$ (7)

which is composed of two terms. The first term is coming form the SO term in the functional, Eq. (5). It is proportional to the radial derivative of the particle density and is relatively slowly varying with $N$ and $Z$. The second component is due to the tensor term (4). It is proportional to the SO density $J_t (r)$ which is strongly shell-filling dependent. Indeed, in the spherical symmetry limit, the SO vector density can be written as [26]:

$$J(r) = \frac{1}{4\pi r^3}\sum_{n,j,l}(2j+1)v^2_{njl}$$

$$\times \left[ j(j+1)-l(l+1)-\frac{3}{4} \right] \psi^2_{njl}(r),$$ (8)

where $v^2_{njl}$ and $\psi^2_{njl}(r)$ are occupation probabilities and radial wave functions, respectively, of states with given quantum numbers. If both SO partners $j_\gtrless=l \pm 1/2$ are fully occupied, i.e., when the system is spin-saturated (SS) the $J(r)$ density vanishes. Examples of the SS systems include $^{16}$O, $^{40}$Ca, or $^{80}$Zr at spherical shape. Most of the nuclei are spin-unsaturated (SUS). The SO vector density reaches its maximum when one (or more) of the SO partners is fully occupied while the other one is completely empty.