Deformation and mixing of coexisting shapes in neutron-deficient polonium isotopes

Coulomb-excitation experiments are performed with postaccelerated beams of neutron-deficient ${}^{196,198,200,202}\mathrm{Po}$ isotopes at the REX-ISOLDE facility. A set of matrix elements, coupling the low-lying states in these isotopes, is extracted. In the two heaviest isotopes, ${}^{200,202}\mathrm{Po}$, the transitional and diagonal matrix elements of the $2_1^{+}$ state are determined. In ${}^{196,198}\mathrm{Po}$ multistep Coulomb excitation is observed, populating the $4_1^{+},0_2^{+}$, and $2_2^{+}$ states. The experimental results are compared to the results from the measurement of mean-square charge radii in polonium isotopes, confirming the onset of deformation from ${}^{196}\mathrm{Po}$ onwards. Three model descriptions are used to compare to the data. Calculations with the beyond-mean-field model, the interacting boson model, and the general Bohr Hamiltonian model show partial agreement with the experimental data. Finally, calculations with a phenomenological two-level mixing model hint at the mixing of a spherical structure with a weakly deformed rotational structure.

Figure 1

(a) Relative $\delta \langle r^2\rangle$ for the even-$A{}_{80}\mathrm{Hg}$, ${}_{82}\mathrm{Pb}$, and ${}_{84}\mathrm{Po}$ isotopes [8]. The changes in charge radii, relative to $N=126$, are normalized to the difference in charge radius between $N=122$ and $N=124$. (b) Energy level systematics of the positive-parity states for neutron-deficient even-mass polonium isotopes. Filled (red) symbols show yrast levels; open (blue) symbols, nonyrast levels. Data are taken from Nuclear Data Sheets.

Figure 2

Particle energy versus scattering angle in the laboratory frame of reference ${\theta }_{\mathrm{LAB}}$ for ${}^{200}\mathrm{Po}$ on ${}^{104}\mathrm{Pd}$. The color scale on the vertical axis represents the intensity in each bin. Only particles that are coincident with at least one $\gamma$ ray are shown. Gates chosen to select the ${}^{104}\mathrm{Pd}$ recoil are shown in black. The two innermost strips are not taken into account, as it is not possible to distinguish between the beam and the recoil particles in this angular range.

Figure 3

Time difference between particle and $\gamma$ ray, gated on the $2_1^{+}\to 0_1^{+}\gamma$-ray transitions (blue lines) and gated on polonium x rays (red lines). The prompt and random coincidence windows used are shown in black. (a) Sum of $p-\gamma$ timing information for all quadrants (2009 data of ${}^{200}\mathrm{Po}$ on ${}^{104}\mathrm{Pd}$). (b) $p-\gamma$ timing information for quadrants 1 and 2 where the downscaling was not properly set (2012 data of ${}^{198}\mathrm{Po}$ on ${}^{94}\mathrm{Mo}$).

Figure 4

The $\gamma$-ray energy spectra shown here display the 2009 data of ${}^{200}\mathrm{Po}$ on the ${}^{104}\mathrm{Pd}$ target. (a) All $\gamma$ rays that were detected. (b) Prompt $\gamma$ rays (black line) and random $\gamma$ rays [gray (red) line] that fulfill the kinematic conditions. (c) Subtracted prompt-minus-random $\gamma$ spectrum.

Figure 5

The ratio of the number of observed x rays to the number of (rescaled) expected x rays is plotted for all studied polonium isotopes. For ${}^{202}\mathrm{Po}$ two points are drawn: the circle represents the data on ${}^{104}\mathrm{Pd}$; the square, the data on ${}^{94}\mathrm{Mo}$ data. The solid horizontal black line represents the scaling factor with the associated uncertainty (dashed horizontal black lines) deduced from the ${}^{202,206}\mathrm{Po}$ data.

Figure 6

Level energies (in keV) and reduced transition probabilities $B\left(E2\right)\uparrow$ (in W.u.) of low-lying excited states in ${}^{94,95}\mathrm{Mo}$. Data are taken from Nuclear Data Sheets and Refs. [42] and [43].

Figure 7

Background-subtracted $\gamma$-ray energy spectrum for ${}^{198}\mathrm{Po}$ on ${}^{94}\mathrm{Mo}$ (black line) and Doppler corrected for the target [gray (red) line].

Figure 8

Background-subtracted and Doppler-corrected $\gamma$-ray energy spectrum following the Coulomb excitation of ${}^{202}\mathrm{Po}$, induced by the ${}^{202}\mathrm{Po}$ beam impinging (a) on the ${}^{104}\mathrm{Pd}$ and (b) on the ${}^{94}\mathrm{Mo}$ target. The gray (red) spectrum is Doppler corrected for the target; the black spectrum is Doppler corrected for the projectile. Observed transitions are highlighted.

Figure 9

Background-subtracted and Doppler-corrected $\gamma$-ray energy spectrum following the Coulomb excitation of ${}^{200}\mathrm{Po}$, induced by the ${}^{200}\mathrm{Po}$ beam impinging on the ${}^{104}\mathrm{Pd}$ target. The gray (red) spectrum is Doppler corrected for the target; the black spectrum is Doppler corrected for the projectile. Observed transitions are highlighted.

Figure 10

(a) Background-subtracted and Doppler-corrected $\gamma$-ray energy spectrum following the Coulomb excitation of ${}^{198}\mathrm{Po}$, induced by the ${}^{198}\mathrm{Po}$ beam on the ${}^{94}\mathrm{Mo}$ target. The $\gamma$-ray energies are Doppler corrected for ${}^{198}\mathrm{Po}$; the target Doppler correction is shown in Fig. 7. Observed transitions are highlighted. (b) Energy of $\gamma$ rays coincident with the $2_1^{+}\to 0_1^{+}\gamma$ ray at 605 keV in ${}^{198}\mathrm{Po}$. The gated spectrum is background subtracted and Doppler corrected for ${}^{198}\mathrm{Po}$. Observed transitions in ${}^{198}\mathrm{Po}$ are highlighted.

Figure 11

(a) Background-subtracted and Doppler-corrected $\gamma$-ray energy spectrum following the Coulomb excitation of ${}^{196}\mathrm{Po}$, induced by the ${}^{196}\mathrm{Po}$ beam on the ${}^{104}\mathrm{Pd}$ target during equally long laser-ON (black spectrum) and laser-OFF [gray (red) spectrum] windows. The $\gamma$ energies are Doppler corrected for mass 196 and observed transitions are highlighted. The broad structure around 550 keV is due to the ${}^{104}\mathrm{Pd}$ target excitation at 556 keV. (b) Energy of $\gamma$ rays coincident with the $2_1^{+}\to 0_1^{+}\gamma$ ray at 463 keV in ${}^{196}\mathrm{Po}$. The gated spectrum is background subtracted and Doppler corrected for ${}^{196}\mathrm{Po}$. Observed transitions in ${}^{196}\mathrm{Po}$ are highlighted.

Figure 12

${\chi }^2$ surface of the transitional and diagonal matrix element of the $2_1^{+}$ state in ${}^{202}\mathrm{Po}$. The ${\chi }^2$ is the sum of the ${\chi }^2$ resulting from the experiment on ${}^{104}\mathrm{Pd}$ and the ${\chi }^2$ extracted from the ${}^{94}\mathrm{Mo}$ experiment. Projection of the $1\sigma$ contour (dashed lines) gives the correlated uncertainties of the two parameters extracted (see Table 7).

Figure 13

Large-scale ${\chi }^2$ surface of the transitional and diagonal matrix elements of the $2_1^{+}$ state in ${}^{200}\mathrm{Po}$. Projection of the $1\sigma$ contour gives the correlated uncertainties of the two parameters extracted (see Table 7).

Figure 14

Result of the ${\chi }^2$-surface analysis for the Coulomb excitation of ${}^{198}\mathrm{Po}$ on ${}^{94}\mathrm{Mo}$ showing the $1\sigma$ contour constructed by letting ${\chi }^2$ increase to ${\chi }_{\min }^2+1=4.9$. Projection of the $1\sigma$ contour gives the correlated uncertainties of the two parameters that are extracted.

Figure 15

Experimental levels of the low-lying structures in ${}^{196,198,200,202}\mathrm{Po}$. Level energies (in keV) are taken from Nuclear Data Sheets. Transitional $|Q_t|$ values (in eb) are based on the experimentally determined matrix elements. The width of the arrows represents the relative size of the transitional quadrupole moments $|Q_t|$. Experimental level energies and $|Q_t|$ values are compared to the same information, predicted by the BMF [19], IBM [58], and GBH [59] models.

Figure 16

Deformation parameters of the ground state, extracted from the charge radii $\delta \langle r^2\rangle$ (triangles) [61] and sum of squared matrix elements according to Eq. (6) (squares). For odd isotopes the deformation parameter for the $3/2^{-}$ ground state is shown. The data point for ${}^{194}\mathrm{Po}$ is deduced from the lifetime measurement [15].

Figure 17

Experimental $|Q_t|$ values extracted from the measured matrix elements $\langle 0_1^{+}\left|\right|E2\left|\right|2_1^{+}\rangle$ and $\langle 0_1^{+}\left|\right|E2\left|\right|2_2^{+}\rangle$ in the even-even polonium nuclei as a function of the mass number. Data for $A=194$ are taken from [15]. Experimental values are compared to predictions from three theoretical model descriptions: the beyond-mean-field (BMF) model [19], the interacting boson model (IBM) [58], and the general Bohr Hamiltonian (GBH) model [59].

Figure 18

Experimentally determined values for the spectroscopic quadrupole moment $Q_s$ of the $2_1^{+}$ state as a function of the mass of the polonium isotope. In ${}^{196}\mathrm{Po}$ no model-independent distinction could be made between two solutions for the matrix elements, yielding two different results for $Q_s$. Both results are shown here with a small offset from integer $A$ for clarity. Experimental results are compared to predictions from the BMF model [19], IBM [58], and GBH model [59].

Figure 19

Measured $E2$ matrix elements determined in this work, compared to those extracted from two-level mixing calculations for ${}^{196}\mathrm{Po}$ (filled red symbols), ${}^{198}\mathrm{Po}$ (filled blue symbols), ${}^{200}\mathrm{Po}$ (filled green symbol), and ${}^{202}\mathrm{Po}$ (open red symbol). Measured $1\sigma$ error bars are shown. In ${}^{196}\mathrm{Po}$ solution 2 (see Table 9) is adopted. The experimental $E2$ matrix element $\langle 0_1^{+}\left|\right|E2\left|\right|2_1^{+}\rangle$ for ${}^{194}\mathrm{Po}$ (open blue symbol) is deduced from the lifetime measurement [15].