Shell-model study of Pb, Bi, Po, At, Rn, and Fr isotopes with masses from 210 to 217

Large-scale shell-model calculations are performed for even-even, odd-mass, and doubly odd nuclei of Pb, Bi, Po, At, Rn, and Fr isotopes in the neutron rich region ($N>126$) assuming ${}^{208}\mathrm{Pb}$ as a doubly magic core. Seven orbitals above the magic number 126, the $1g_{9/2}$, $0i_{11/2}$, $0j_{15/2}$, $2d_{5/2}$, $3s_{1/2}$, $1g_{7/2}$, and $2d_{3/2}$ orbitals, are considered for neutrons and all the six orbitals between the magic numbers 82 and 126, the $0h_{9/2}$, $1f_{7/2}$, $0i_{13/2}$, $2p_{3/2}$, $1f_{5/2}$, and $2p_{1/2}$ orbitals, are considered for protons. For a phenomenological effective two-body interaction, one set of the monopole-pairing, quadrupole-quadrupole, and multipole-pairing interactions is adopted for all the nuclei considered. The calculated energies and electromagnetic properties are compared with the experimental data. Many isomeric states are analyzed in terms of the shell-model configurations. The spins and parities of some experimentally ambiguous states are suggested.

Figure 1

The theoretical energy spectrum of ${}^{210}\mathrm{Pb}$ (Calc.) in comparison with experimental data (Expt.). The experimental data are taken from Refs. [15, 16]. The squares and diamonds represent experimental positive and negative parity states, respectively. The $\times$ marks and + marks represent theoretical positive and negative parity states, respectively.

Figure 2

Same as Fig. 1, but for ${}^{212}\mathrm{Pb}$. The experimental data are taken from Refs. [15, 17].

Figure 3

Same as Fig. 1, but for ${}^{211}\mathrm{Pb}$. The experimental data are taken from Refs. [15, 18].

Figure 4

Same as Fig. 1, but for ${}^{211}\mathrm{Bi}$. The experimental data are taken from Refs. [15, 21]. Each ambiguous state with only one possible set of spin-parity in experiment is shown with parentheses, while each ambiguous state with more than two possible sets of spin-parity is shown separately with square bracket.

Figure 5

Same as Fig. 4, but for ${}^{213}\mathrm{Bi}$. The experimental data are taken from Refs. [15, 18].

Figure 6

Same as Fig. 1, but for ${}^{210}\mathrm{Bi}$. The experimental data are taken from Refs. [15, 16].

Figure 7

Comparison of the low-lying energy levels of ${}^{210}\mathrm{Bi}$ with those by the strength parameter ${\kappa }_{\nu \pi }=0.06\text{MeV}/b^4$, which are indicated by the filled circles. This value of ${\kappa }_{\nu \pi }=0.06\text{MeV}/b^4$ is the same in magnitude as used in Ref. [13].

Figure 8

Same as Fig. 4, but for ${}^{212}\mathrm{Bi}$. The experimental data are taken from Refs. [15, 17].

Figure 9

Same as Fig. 1, but for ${}^{212}\mathrm{Po}$. The state at 1.249 MeV is not shown in the figure since the spin-parity is not assigned in experiment. The experimental data are taken from Refs. [15, 17, 22, 23].

Figure 10

Same as Fig. 9, but with ${\varepsilon }_{\nu }\left(i_{11/2}\right)=0.1$ MeV and $G_{0\nu }=0.095$ MeV. The spin-parity of the state at 1.249 MeV indicated by a triangle is not assigned in experiment, but it is suggested to have a spin-parity of ${10}^{+}$ in this calculation.

Figure 11

Same as Fig. 1, but for ${}^{214}\mathrm{Po}$. The experimental data are taken from Refs. [15, 24].

Figure 12

Same as Fig. 4, but for ${}^{211}\mathrm{Po}$. The experimental data are taken from Refs. [15, 18].

Figure 13

Same as Fig. 4, but for ${}^{213}\mathrm{Po}$. The experimental data are taken from Refs. [2, 15, 21].

Figure 14

Same as Fig. 4, but for ${}^{213}\mathrm{At}$. The experimental data are taken from Refs. [15, 21].

Figure 15

Same as Fig. 1, but for ${}^{215}\mathrm{At}$. The experimental data are taken from Refs. [15, 30].

Figure 16

Same as Fig. 4, but for ${}^{212}\mathrm{At}$. The experimental data are taken from Refs. [15, 17].

Figure 17

Same as Fig. 1, but for ${}^{214}\mathrm{At}$. The experimental data are taken from Refs. [15, 24].

Figure 18

Same as Fig. 4, but for ${}^{214}\mathrm{Rn}$. The experimental data are taken from Refs. [15, 24].

Figure 19

Same as Fig. 4, but for ${}^{216}\mathrm{Rn}$. The experimental data are taken from Refs. [15, 32].

Figure 20

Same as Fig. 1, but for ${}^{213}\mathrm{Rn}$. The experimental data are taken from Refs. [15, 21].

Figure 21

Same as Fig. 4, but for ${}^{215}\mathrm{Rn}$. The experimental data are taken from Refs. [15, 30].

Figure 22

Same as Fig. 1, but for ${}^{215}\mathrm{Fr}$. The experimental data are taken from Refs. [15, 30].

Figure 23

Same as Fig. 1, but for ${}^{217}\mathrm{Fr}$. The experimental data are taken from Refs. [15, 33].

Figure 24

Same as Fig. 1, but for ${}^{214}\mathrm{Fr}$. The experimental data are taken from Refs. [15, 24].

Figure 25

Same as Fig. 4, but for ${}^{216}\mathrm{Fr}$. The experimental data are taken from Refs. [15, 32].

Figure 26

(a) The neutron expectation numbers of collective pairs for ${}^{212}\mathrm{Po}$. The S, D, and G indicate $S$ pair, $D$ pair, and $G$ pair, respectively. (b) The neutron expectation numbers of non-collective ${\left(g_{9/2}\right)}^2$ pair $\left[{\left(g_{9/2}\right)}^2\right]$ and $g_{9/2}{i}_{11/2}$ pair ($g_{9/2}{i}_{11/2}$). The definitions of noncollective pairs are given in the text. (c) The proton expectation numbers of collective pairs for ${}^{212}\mathrm{Po}$.

Figure 27

The same as Fig. 26, but for ${}^{214}\mathrm{Po}$.

Figure 28

The same as Fig. 26, but for ${}^{214}\mathrm{Rn}$.

Figure 29

The same as Fig. 26, but for ${}^{216}\mathrm{Rn}$.