Description of magnetic moments within the Gogny Hartree-Fock-Bogolyubov framework: Application to Hg isotopes

While the ground state configuration is classically determined by the variational principle minimizing the binding energy of the system, we propose here a different procedure to identify the configuration of the ground state in odd-$A$ nuclei. This procedure is based on the Hartree-Fock-Bogolyubov (HFB) framework with a self-consistent blocking of the unpaired nucleon and identifies the ground state as the blocked quasiparticle configuration compatible with the observed spin and parity and, most importantly, the measured magnetic moment. The magnetic moments are calculated within the HFB framework for all odd Hg isotopes for which experimental data is available. To validate the method, a systematic comparison between the predicted and measured electric quadrupole moments and isotopic shifts is performed. For even-even isotopes, we show that the ground state deformation, and more particularly the competition between the prolate and oblate shapes, can hardly be determined through a comparison of the QRPA $\beta$-decay half-life with experiment, though this approach also calls for an observable similar to the one used for odd-$A$ isotopes, namely the spin-flip component of the isovector part of the magnetic moment operator. For even isotopes with shape coexistence, no adequate constraint could be identified, except though the charge radius. Assuming light even-even Hg isotopes to have an oblate shape, the resulting charge radii staggering observed in the Hg chain by laser spectroscopy can through this identification procedure be reproduced.

Figure 1

${}^{177}\mathrm{Hg}$ axial potential energy curves for different $K^{\pi }$ blockings as a function of the quadrupole deformation parameter $\beta$. Calculations are performed with the D1M interaction and a 15 major oscillator shell basis.

Figure 2

D1M-HFB magnetic moments as a function of the spectroscopic quadrupole moment for the different energy minima for each $K^{\pi }$ state shown in Fig. 1 for ${}^{177}\mathrm{Hg}$. The red circles correspond to the bare magnetic moments and the blue squares to the effective ones. Both are joined by a solid line for the same $K^{\pi }$ blocked state. The bare magnetic moment for the $J^{\pi }=5/2^{-}$ and the effective magnetic moment for the $J^{\pi }=7/2^{-}$ are compatible with the experimental data shown by the black diamond [1, 14]. For the $3/2^{+}$, $5/2^{-}$, and $7/2^{+}$, the moments for the second minima observed in Fig. 1 are also shown.

Figure 3

Comparison between experimental [1, 12, 14] and HFB magnetic moments for the various Hg isotopes. The full symbols correspond to the GS and the open symbols to the isomeric states. The corresponding spins are given. Details on the theoretical error bars can be found in the text.

Figure 4

Comparison between experimental [1, 12] and HFB isotopic shifts $\delta {\langle r^2\rangle }^{A\text{–}198}$ for the various Hg isotopes. For the even $A=178–186$ isotopes, the isotopic shift for the prolate HFB configurations are shown with open circles. The corresponding full circles for even $A=178–186$ correspond to the oblate configuration.

Figure 5

Comparison between experimental [12, 14] and HFB spectroscopic quadrupole moments for the odd-$N$ Hg isotopes.