Nuclear shape coexistence in Po isotopes: An interacting boson model study

Background: The lead region, Po, Pb, Hg, and Pt, shows up the presence of coexisting structures having different deformation and corresponding to different particle-hole configurations in the shell-model language.

Purpose: We intend to study the importance of configuration mixing in the understanding of the nuclear structure of even-even Po isotopes, where the shape coexistence phenomena are not clear enough.

Method: We study in detail a long chain of polonium isotopes, ${}^{190–208}\mathrm{Po}$, using the interacting boson model with configuration mixing (IBM-CM). We fix the parameters of the Hamiltonians through a least-squares fit to the known energies and absolute $B\left(E2\right)$ transition rates of states up to 3 MeV.

Results: We obtained the IBM-CM Hamiltonians and we calculate excitation energies, $B\left(E2\right)$'s, electric quadrupole moments, nuclear radii and isotopic shifts, quadrupole shape invariants, wave functions, and deformations.

Conclusions: We obtain a good agreement with the experimental data for all the studied observables and we conclude that shape coexistence phenomenon is hidden in Po isotopes, very much as in the case of the Pt isotopes.

Figure 1

Experimental energy-level systematics for the Po isotopes. Only levels up to $E_x\sim 3.0$ MeV are shown. The symbol $\uparrow$ indicates the beginning of the region where extra levels of low spin, indicated with spin, parity assignments between brackets in the Nuclear Data Sheet references, start appearing. The symbol $*$ for the $8^{+}$ level for mass $A=202$ indicates that the energy is within an interval $\le 40$ keV above the $6^{+}$ level. Dashed lines connect states that are supposed to have a similar structure.

Figure 2

Absolute energy of the lowest unperturbed regular and intruder $0_1^{+}$ states for ${}^{190–208}\mathrm{Po}$. The arrows correspond to the correlation energies in the $N$ and $N+2$ subspaces (see also the text for a more detailed discussion).

Figure 3

Experimental excitation energies (up to $E_x\approx 3.0$ MeV) (a) and the theoretical results (b), obtained from the IBM-CM.

Figure 4

Comparison of the absolute $B\left(E2\right)$ reduced transition probabilities along the yrast band, given in W.u. Panel (a) corresponds to known experimental data and panel (b) to the theoretical IBM-CM results.

Figure 5

Comparison of the few nonyrast intraband absolute $B\left(E2\right)$ reduced transition probabilities, given in W.u. Panel (a) corresponds to the few known experimental data, panel (b) to the theoretical IBM-CM results.

Figure 6

IBM-CM values of the quadrupole moments for the $2_1^{+}$ and $2_2^{+}$ states in ${}^{190–208}\mathrm{Po}$. The quadrupole moments are given in units $e\text{b}$. The dash-dotted and dotted lines indicate the quadrupole moments when the mixing Hamiltonian is switched off, corresponding to the unperturbed $2_1^{+}, 2_2^{+}$, and $2_3^{+}$ states.

Figure 7

Experimental excitation energies and absolute $B\left(E2\right)$ transition rates for selected states in ${}^{190–198}\mathrm{Po}$.

Figure 8

Theoretical excitation energies and absolute $B\left(E2\right)$ transition rates for selected states in ${}^{190–198}\mathrm{Po}$.

Figure 9

(a) Experimental and theoretical quadrupole moment for the $8_1^{+}$ state (note that the theoretical values for $A=206$ and 208 are not shown owing to the reduced number of valence bosons). (b) Gyromagnetic factor for the $8_1^{+}$ state (experimental data and theoretical results).

Figure 10

Regular content of the two lowest-lying states for each $J$ value (solid lines with solid symbols correspond with the first state, while dashed lines with open symbols correspond with the second state) resulting from the IBM-CM calculation, as presented in Fig. 3.

Figure 11

Energy spectra for the IBM-CM Hamiltonian presented in Table 2, switching off the mixing term. The two lowest-lying regular states and the lowest-lying intruder state for each of the angular momenta are shown (solid lines with solid symbols for the regular states while dashed lines with open symbols are used for the intruder ones).

Figure 12

Overlap of the wave functions of Eq. (7), with the wave functions describing the unperturbed basis Eqs. (9) and (10). (a) Overlaps for first $0^{+},2^{+},3^{+},4^{+};5^{+},6^{+},7^{+},8^{+}$ state; (b) overlaps for the corresponding second state (see also text).

Figure 13

Schematic view of $\alpha$ decay from the Rn nuclei into the Po nuclei and from the Po nuclei into the Pb nuclei.

Figure 14

(a) Charge mean-square radii for the Po nuclei. (b) Isotopic shift for the Po nuclei. The data are taken from Ref. [118].

Figure 15

Matrix coherent-state calculation for ${}^{190–194}\mathrm{Po}$, corresponding with the present IBM-CM Hamiltonian (Table 2). The energy spacing between adjacent contour lines equals 100 keV and the deepest energy minimum is set to zero, corresponding to the red color.

Figure 16

Comparison of the value of $\beta$ extracted from the theoretical quadrupole shape invariants (a) and the ones extracted from experimental and theoretical $B\left(E2\right)$ values (b), corresponding to the $0_1^{+}$ and $0_2^{+}$ states.

Figure 17

Comparison of the experimental (a) and theoretical (b) kinematic moment of inertia for selected Po isotopes with $106\le N\le 114$.

Figure 18

Comparison of the experimental moment of inertia in Po (a), Pb (b), Hg (c), and Pt (d) for selected isotones with $106\le N\le 114$.