Proxy-SU(3) symmetry in heavy deformed nuclei

Background: Microscopic calculations of heavy nuclei face considerable difficulties due to the sizes of the matrices that need to be solved. Various approximation schemes have been invoked, for example by truncating the spaces, imposing seniority limits, or appealing to various symmetry schemes such as pseudo-SU(3). This paper proposes a new symmetry scheme also based on SU(3). This proxy-SU(3) can be applied to well-deformed nuclei, is simple to use, and can yield analytic predictions.

Purpose: To present the new scheme and its microscopic motivation, and to test it using a Nilsson model calculation with the original shell model orbits and with the new proxy set.

Method: We invoke an approximate, analytic, treatment of the Nilsson model, that allows the above vetting and yet is also transparent in understanding the approximations involved in the new proxy-SU(3).

Results: It is found that the new scheme yields a Nilsson diagram for well-deformed nuclei that is very close to the original Nilsson diagram. The specific levels of approximation in the new scheme are also shown, for each major shell.

Conclusions: The new proxy-SU(3) scheme is a good approximation to the full set of orbits in a major shell. Being able to replace a complex shell model calculation with a symmetry-based description now opens up the possibility to predict many properties of nuclei analytically and often in a parameter-free way. The new scheme works best for heavier nuclei, precisely where full microscopic calculations are most challenged. Some cases in which the new scheme can be used, often analytically, to make specific predictions, are shown in a subsequent paper.

Figure 1

Schematic representation of the 50–82 shell and the replacement leading to the proxy $sdg$ shell. See Sec. 2 for further discussion.

Figure 2

Energies (in units of $\hslash {\omega }_0)$ of the Nilsson Hamiltonian as functions of the deformation parameter $\epsilon$. Note that, in these panels, the energies shown are not the usual solutions to the Nilsson Hamiltonian, but use matrix elements obtained analytically in the asymptotic deformed basis. Therefore they are not valid for small deformations and would not go to the usual degeneracies at $\epsilon =0$. They are, however, quite accurate for $\epsilon >0.15$. The panels in this figure therefore also start at $\epsilon =0.15$. The Nilsson parameters are taken from Table 1. The $1h_{11/2}$ orbitals in (a) and (b), as well as the $1g_{9/2}$ orbitals in (c), are indicated by dashed lines. The $1h_{11/2}$ orbital labels in (a) and (b), as well as the $1g_{9/2}$ orbital labels in (c) and (d), appear in boldface. Orbitals are grouped in color only to facilitate visualizing the patterns of orbital evolution. Note that the Nilsson labels at the right are always in the same order as the energies of the orbitals as they appear at the right as well (largest deformation shown). Therefore, in some cases the order of the Nilsson orbitals changes slightly from panel to panel. (a) Energies (diagonal matrix elements) for the 50–82 proton shell, including the contributions from both the $H_{\mathrm{osc}}$ term and the small perturbations (the $\mathbf{l}·\mathbf{s}$ and ${\mathbf{l}}^2$ terms). Therefore the Nilsson trajectories are straight and exhibit crossings. (b) Results of the full diagonalization for the 50–82 proton shell, in which the nondiagonal matrix elements are taken into account. Hence the Nilsson trajectories show the usual curvatures and avoided crossings. (c) Energies (diagonal matrix elements) for the $sdg$ proton shell, resulting after the replacement of the $1h_{11/2}$ orbitals of the 50–82 shell by their 0[110] $1g_{9/2}$ counterparts. Therefore the Nilsson trajectories are straight and exhibit crossings. (d) Results of the full diagonalization for the $sdg$ proton shell. Hence the Nilsson trajectories again show the curvatures [enhanced relative to Fig. 2 by the mixing related to the spurious off-diagonal matrix elements of the $1g_{9/2}$ orbit] and avoided crossings. See Sec. 4b for further discussion.

Figure 3

Same as Fig. 2, but for the diagonal matrix elements (in units of $\hslash {\omega }_0)$ of the Nilsson Hamiltonian for the 82–126 (a) and $pfh$ (c) neutron shells compared to the results of the full diagonalization for the 82–126 (b) and $pfh$ (d) neutron shells, as functions of the deformation parameter $\epsilon$. The Nilsson parameters are taken from Table 1. The $1i_{13/2}$ orbitals in (a) and (b), as well as the $1h_{11/2}$ orbitals in (c), are indicated by dashed lines. The $1i_{13/2}$ orbital labels in (a) and (b), as well as the $1h_{11/2}$ orbital labels in (c) and (d), appear in boldface. Orbitals are grouped in color only to facilitate visualizing the patterns of orbital evolution. Note that the Nilsson labels at the right are always in the same order as the energies of the orbitals as they appear at the right as well (largest deformation shown). Therefore, in some cases the order of the Nilsson orbitals changes slightly from panel to panel.