Shell model plus cluster description of negative parity states in ${}^{\mathbf{212}}$Po

The intraband electromagnetic transitions in ${}^{210}$Po and ${}^{210}$Pb are well described within the shell model approach. In contrast, similar transitions in ${}^{212}$Po are one order of magnitude smaller than the experimental values, suggesting the existence of an $\alpha$-cluster component in the structure of this nucleus. To probe this assumption we introduced Gaussian-like components in the single-particle orbitals. We thus obtained an enhancement of intraband transitions, as well as a proper description of the absolute $\alpha$-decay width in ${}^{212}$Po. We analyzed the recently measured unnatural parity states $I^{-}$ in ${}^{212}$Po in terms of the collective octupole excitation in ${}^{208}$Pb coupled to positive parity states in ${}^{210}$Pb. They are connected by relatively large dipole transitions to yrast positive natural parity states. We described $E1$ transitions by using the same $\alpha$-cluster component and an effective neutron dipole charge $e_{\nu }=-eZ/A$. $B\left(E2\right)$ values and absolute $\alpha$-decay width in ${}^{212}$Po are simultaneously described within the shell model plus a cluster component depending upon one free strength parameter.

Figure 1

(a) A Gaussian centered on the nuclear surface (solid line), given by Eq. (2.22), and its spherical harmonic oscillator components (dot-dashed lines). (b) The expansion coefficiens (2.21) of the Gaussian (2.22) for different angular momenta $l$.

Figure 2

Energies of the positive parity states in ${}^{210}$Pb (circles), ${}^{210}$Po (squares), and ${}^{212}$Po (diamonds), as a function of angular momentum. Experimental values are drawn with filled symbols and theoretical results [18] are drawn with open symbols.

Figure 3

Energies of the negative parity states in ${}^{212}$Po as a function of angular momentum. Experimental values [14, 15] are drawn with filled symbols and theoretical predictions (this work) are drawn with open symbols.

Figure 4

(a) The $\alpha$-particle preformation amplitude squared (2.23) (solid line) and its cluster part (dashed line).(b) The logarithm of the ratio between theoretical and experimental $\alpha$-decay widths versus cm radius (solid line). The dashed and dot-dashed line are the two terms of Eq. (2.25), respectively.

Figure 5

(a) $B\left(E1\right)$ values for transitions $I^{-}\to I^{+}$ in Table  versus the proton-neutron coupling strength $F_{3^{-}}\left(\pi \nu \right)$ for a vanishing $\alpha$-clustering amplitude ${\mathcal{N}}^{\left(\mathrm{clus}\right)}=0$ (solid lines). The transitions between natural parity states ${(I+1)}^{-}\to I^{+}$ are given by dashed lines. (b) Same as in (a) but for ${\mathcal{N}}^{\left(\mathrm{clus}\right)}=0.3$.

Figure 6

(a) The largest octupole TDA amplitude, given by the neutron pair ${(\nu g{\frac{9}{2}}^{+}\otimes \nu p{\frac{3}{2}}^{-})}_{3^{-}}$, versus the proton-neutron coupling strength $F_{3^{-}}\left(\pi \nu \right)$ is drawn by a dashed line. Other important amplitudes are given by dotted lines. The TDA amplitude giving the leading contribution in $E1$ transitions, corresponding to the neutron p-h pair ${(\nu g{\frac{9}{2}}^{+}\otimes \nu f{\frac{7}{2}}^{-})}_{3^{-}}$, is plotted by a solid line. The $\alpha$-clustering amplitude is ${\mathcal{N}}^{\left(\mathrm{clus}\right)}=0$. (b) Same as in (a), but for ${\mathcal{N}}^{\left(\mathrm{clus}\right)}=0.3$.

Figure 7

The distribution of different terms in the $E1$ transition operator (A6) divided by the electric charge (in fm) versus the neutron state number for the strongest transition $8_1^{-}\to 8^{+}$ in ${}^{212}$Po, by considering ${\mathcal{N}}^{\left(\mathrm{clus}\right)}=0$ (open circles connected by a dashed line). The maximum corresponds to the neutron dipole pair ${(\nu g{\frac{9}{2}}^{+}\otimes \nu f{\frac{7}{2}}^{-})}_{1^{-}}$. The dark circles connected with a solid line show the same dependence, but for ${\mathcal{N}}^{\left(\mathrm{clus}\right)}=0.3$.