Competition between pairing correlations and deformation from the odd-even mass staggering of francium and radium isotopes

The masses of ${}^{222,224,226\text{–}233}\mathrm{Fr}$ and ${}^{233,234}\mathrm{Ra}$ have been determined with the Penning-trap mass spectrometer ISOLTRAP at the ISOLDE facility at CERN, including the previously unknown mass and half-life of ${}^{233}\mathrm{Fr}$. We study the evolution of the odd-even staggering of binding energies along the francium and radium isotopic chains and of its lowest-order estimator, ${\Delta }^3\left(N\right)$. An enhancement of the staggering of ${\Delta }^3\left(N\right)$ is observed towards neutron number $N=146$, which points to contributions beyond pairing correlations. These contributions are investigated in the Hartree-Fock and Hartree-Fock-Bogoliubov approaches, emphasizing the connections to the single-particle level density and nuclear deformation.

Figure 1

Schematic view of the ISOLTRAP setup. The inset shows a time-of-flight resonance of ${}^{233}\mathrm{Fr}{}^{+}$. For details, see text.

Figure 2

Beta-decay curve for the $A=233$ ISOLDE beam. Data from 50 separate acquisitions are added, and the average background is subtracted. A double exponential fit with daughter feeding is represented by the solid red curve.

Figure 3

Two-neutron separation energy $S_{2N}$ in the francium and radium isotopic chains. Experimental data from the AME [39] (open symbols) and ISOLTRAP (full symbols). The data are compared to spherical (statically deformed) HFB calculations depicted by dashed (solid) red lines and HF calculations with static deformation (solid blue lines). The thick green lines represent the HFB calculations with and without octupole correlations included (continuous and dashed, respectively) [55]. The thick orange lines represent BCS calculations [54] including only static deformation (dotted line) or also dynamic quadrupole correlations (continuous line).

Figure 4

Neutron Nilsson levels of the radium isotopes between $N=128$ and $N=148$ from the HF calculation. For each $N$, the Nilsson levels of the corresponding isotope are presented. The Fermi level of each isotope is also shown.

Figure 5

Three-point estimator of the pairing gap ${\Delta }^3\left(N\right)$ as a function of $N$ for the francium and radium isotopic chains. Data are compared to HFB calculations with quasiparticle blocking (for the radium isotopes) in the spherical and prolate-deformed approximations (blue dashed and red dashed lines, respectively). The new data show an enhancement of the staggering of ${\Delta }^3\left(N\right)$ towards neutron number $N=146$.

Figure 6

Three-point estimator of the pairing gap ${\Delta }^3\left(N\right)$ calculated for the francium and radium isotopes in the absence of pairing correlations (HF approach, left panel) and with pairing correlations but no quasiparticle blocking (HFB approach, right panel).

Figure 7

Drop between even and odd values of the empirical pairing gap ${\tilde{\Delta }}^3\left(N_{\mathrm{even}}\right)$ for experimental values and for the calculated values of Fig. 6. The left panel shows calculations in the HF approximation, while the right panel shows the same comparison for HFB calculations without quasiparticle blocking.