Occupation numbers of spherical orbits in self-consistent beyond-mean-field methods

We present a method to compute the number of particles occupying spherical single-particle (SSP) levels within the energy density functional (EDF) framework. These SSP levels are defined for each nucleus by performing self-consistent mean-field calculations. The nuclear many-body states, in which the occupation numbers are evaluated, are obtained with a symmetry conserving configuration mixing (SCCM) method based on the Gogny EDF. The method allows a closer comparison between EDF and shell model with configuration mixing in large valence spaces (SM-CI) results, and can serve as a guidance to define physically sound valence spaces for SM-CI calculations. As a first application of the method, we analyze the onset of deformation in neutron-rich $N=40$ isotones and the role of the SSP levels around this harmonic oscillator magic number, with particular emphasis in the structure of ${}^{64}\mathrm{Cr}$.

Figure 1

(a) HFB-Potential energy surface and HFB single-particle energies for (b) protons and (c) neutrons as a function of the axial quadrupole deformation calculated for ${}^{64}\mathrm{Cr}$ with the Gogny D1S interaction.

Figure 2

(a) Potential energy surface as a function of the axial quadrupole deformation calculated for the nucleus ${}^{64}\mathrm{Cr}$ with HFB (thin-dotted), PN-VAP (dashed), and particle-number and angular momentum, $J=0$, projection (continuous) approaches. With the thick-dashed line, the ground state collective wave-function is shown. (b) and (c) Occupation numbers of spherical orbits as a function of the axial quadrupole deformation for protons and neutrons respectively.

Figure 3

(a) Excitation energies and (b) collective wave functions for states belonging to the ground state band for ${}^{64}\mathrm{Cr}$. (c) $2_1^{+}$ and $4_1^{+}$ excitation energies for the chromium isotopic chain computed with the present SCCM method. Experimental values are taken from Ref. [65] (and references therein) and empty symbols represent spin assignments within brackets.

Figure 4

Difference between the occupation numbers of spherical orbits for the ground state of ${}^{64}\mathrm{Cr}$ calculated with the SCCM method and the occupation numbers of those orbits in the spherical HFB configuration (filled bars) and in the spherical HF—normal filling—approximation (empty bars).

Figure 5

HFB-spherical single-particle energies for (a) proton and (b) neutron levels calculated for neutron rich $N=40$ isotones with the Gogny D1S interaction.

Figure 6

(a) Occupation numbers of the neutron $1g_{9/2}$ and $2d_{5/2}$ spherical orbits for the ground states of the $N=40$ isotones. (b) Mean values of the quadrupole deformation for the same states. Filled and open symbols refer to SCCM and SM-CI results respectively. Asterisks represents the quasi+pseudo SU(3) regime.