Experimental evidence of exact E(5) symmetry in ${}^{82}\mathrm{Kr}$

The low-lying positive-parity structures of ${}^{82}\mathrm{Kr}$ have been investigated following their population in the ${}^{76}\mathrm{Ge}({}^9\mathrm{Be},3n)$ reaction at ${\mathrm{E}}_{\mathrm{lab}}\approx 31\mathrm{MeV}$ and using a large array of Compton suppressed HPGe clovers as the detection setup. Apart from other spectroscopic measurements, level lifetimes of the states have been extracted using the Doppler shift attenuation method. The level energies and the associated transition probabilities exhibit superlative overlap with the calculations in the framework of the interaction boson approximation and the E(5) model. The ${}^{82}\mathrm{Kr}$ nucleus, thus, exemplifies the critical point of phase (shape) transition from the U(5) (spherical vibrator) to the O(6) ($\gamma$-soft rotor) paradigms of nuclear structure.

Figure 1

The partial level scheme of ${}^{82}\mathrm{Kr}$ obtained in the present Letter. The level energies (represented by the magenta-colored numbers) are rounded off to the nearest keV. The newly observed transitions have been red colored (marked by “$*$'s”) whereas green-colored transitions have been first observed in heavy-ion reactions and were seen previously in $\beta$ decay of ${}^{82}\mathrm{Rb}$ [33, 34]. Here, the indexing is just for the states observed in this experiment, and there are additional $0^{+}, 2^{+}$, and $4^{+}$ levels that were observed previously in ${}^{82}\mathrm{Kr}$.

Figure 2

Experimental (symbols with $X$ and $Y$ error bars) and calculated (represented by $\times ,•,⧫$, and $▿$ symbols) DCO ratio and linear polarization values for different multipole mixing ratio ($\delta$) of the transitions in ${}^{82}\mathrm{Kr}$. E2 gates set on the ${140}^{\circ }$ and ${90}^{\circ }$ detectors to extract DCO ratios whereas ${90}^{\circ }$ detectors used for the linear polarization measurements.

Figure 3

Experimental spectra created by the gate on top of the transition along with the fitted Doppler shapes (total in red and deconvoluted in blue), contaminant (stopped) peaks (in green), and background (as a black solid line) for the (a) 1044.2-, (b) 1317.1-, and (c) 1092.3-keV transitions in ${}^{82}\mathrm{Kr}$. The vertical dotted lines represent the stopped peak position of the transition.

Figure 4

Level scheme of the E(5) symmetry ($\tau =0–4$ levels of $\xi =1$ family and the $\tau =0–2$ levels with $\xi =2$) for (a) infinite-$N_B$ and (b) $N_B=6$ in the IBA calculation compared with (c) the level scheme of ${}^{82}\mathrm{Kr}$ that was obtained experimentally. Width of the transitions represents B(E2) values in the Weisskopf units (W.u.s) and normalized with the measured value of $B(\mathrm{E}2:2_{1,1}^{+}\to 0_{1,0}^{+})=21.3\mathrm{W}.\mathrm{u}.$ In (c) present measurements are colored by blue whereas green-colored one adopted from Ref. [27]. Branching ratio of the $0_{1,3}^{+}$ level is normalized with $B(\mathrm{E}2\text{:}{0}_{1,3}^{+}\to 2_{1,2}^{+})=33\mathrm{W}.\mathrm{u}.$ (IBA calculation for $N_B=6$), and the depopulating transitions from this state are colored by red. The level energies both for the experimental and calculated scheme are rounded off to the nearest keV (represented by the magenta-colored numbers).

Figure 5

Variation (represented by red solid curves) of (a) $B_{412\to 211}$, (b) $B_{613\to 412}{B}_{613\to 412}$, (c) $B_{413\to 212}=\frac{B(E2:4_{1,3}^{+}\to 2_{1,2}^{+})}{B(E2:2_{1,1}^{+}\to 0_{1,0}^{+})}$, (d) $B_{514\to 313}=\frac{B(E2:5_{1,4}^{+}\to 3_{1,3}^{+})}{B(E2:2_{1,1}^{+}\to 0_{1,0}^{+})}$, (e) $B_{514\to 413}=\frac{B(E2:5_{1,4}^{+}\to 4_{1,3}^{+})}{B(E2:2_{1,1}^{+}\to 0_{1,0}^{+})}$, (f) $B_{020\to 211}$, (g) $\frac{E\left(4_{2,2}^{+}\right)}{E\left(2_{1,1}^{+}\right)}$, (h) $B_{221\to 020}$, and (i) $B_{221\to 212}$ from U(5), through E(5) (denoted by vertical solid lines), to O(6) against the ratio $\epsilon /A.\epsilon /A$ values for U(5), E(5), and O(6) are $\infty$, 10, and 0, respectively. Experimental values and uncertainty ranges are indicated by the horizontal dashed lines and shaded rectangles, respectively.