Realistic Mean-Field Hamiltonian and Method

An ideal tool to study the point-group symmetries and their impact on the implied properties of the nuclear single-particle spectra are non-selfconsistent realizations of the mean-field approach with parametrized potentials, as it will be illustrated in Sect. 5 below. One of the very successful approaches of this kind is provided by the Hamiltonians with the deformed Woods-Saxon type potentials with its universal¹ parametrization that has been in use over many years by now. The corresponding mean-field Hamiltonian, $\hat{H}_{\textrm{\small mf}}$, has the form,

$$\hat{H}_{\textrm{\small mf}} = \hat{t} + V_c(\vec{r}\,)+ V_{so}(\vec{r},\vec{p},\vec{s}),$$ (1)

with the central potential defined by

$$V_c(\vec{r}) \equiv \frac{\bar{V}}{1+\exp[{{\rm dist}_\Si... ...}\,)/a}]} + \frac{1}{2}(1+\tau_3)\,V_{Coulomb}(\vec{r}\,).$$ (2)

Above, $\bar{V}\equiv V_0\,[1\pm\kappa(N-Z)/(N+Z)]$ (signs: "+" for protons, "-" for neutrons); $V_0$, $\kappa$, and $a$ are adjustable parameters and $\tau_3$ denotes the third component of the isospin. The symbol ${\rm dist}_\Sigma(\vec{r}\,)$ denotes the distance between the current point position $\vec{r}$ and the nuclear surface $\Sigma$ defined in terms of the spherical harmonics $Y_{\lambda\mu}(\vartheta,\varphi)$ by

$$\Sigma:\quad R(\vartheta,\varphi) = R_0 c(\{\alpha\}) \... ...^\star_{\lambda\mu} Y_{\lambda\mu}(\vartheta,\varphi)\bigg].$$ (3)

Above, $R_0\equiv r_0\,A^{1/3}$ is the nuclear radius parameter and the function $c(\{\alpha\})$ takes care of the nuclear constant volume that is kept independent of the nuclear deformation. Note that there are effectively three parameters of the central potential for protons and three for neutrons; they are denoted as $\bar{V}, r_0$, and $a$ - the central potential depth, radius, and diffusivity parameters, respectively. It is sometimes convenient to introduce an alternative representation that replaces $\bar{V}$ for the protons and a similar parameter for the neutrons by a suitably chosen set of two parameters. In our case these are $V_0$ and $\kappa$ as introduced just below Eq. (2).

The spin-orbit potential has the usual form,

$$V_{so}(\vec{r},\vec{p},\vec{s}) \equiv \lambda_{so} [\nabla V(\vec{r})\wedge\vec{p}\,]\cdot\vec{s},$$ (4)

where $V(\vec{r}\,)$ is another Woods-Saxon type deformed potential that differs from the analogous term in the central potential by the numerical values of the adjustable constants: here $\lambda_{so},\;r_0^{(so)}$, and $a_{so}$.

Beginning with the single-particle spectra obtained by diagonalization of the above Hamiltonian within the Cartesian harmonic oscillator basis we apply the macroscopic-microscopic method of Strutinsky using the macroscopic Yukawa-plus-exponential approach. The formalism that we use here was presented in details in Refs.[12,13].