Coulomb excitation of ${}^{80,82}\mathrm{Kr}$ and a change in structure approaching $N=Z=40$

Background: Nuclei approaching $N=Z=40$ are known to exhibit strongly deformed structures and are thought to be candidates for shape coexistence. In the krypton isotopes, ${}^{80,82}\mathrm{Kr}$ are poorly characterized, preventing an understanding of evolving deformation approaching $N=40$.

Purpose: The present work aims to determine electric quadrupole transition strengths and quadrupole moments of ${}^{80,82}\mathrm{Kr}$ in order to better characterize their deformation.

Methods: Sub-barrier Coulomb excitation was employed, impinging the isotopes of krypton on ${}^{196}\mathrm{Pt}$ and ${}^{208}\mathrm{Pb}$ targets. Utilizing a semiclassical description of the safe Coulomb-excitation process $E2$ matrix elements could then be determined.

Results: Eleven new or improved matrix elements are determined in ${}^{80}\mathrm{Kr}$ and seven in ${}^{82}\mathrm{Kr}$. The new $B(E2;0_1^{+}\to 2_1^{+})$ value in ${}^{82}\mathrm{Kr}$ disagrees with the evaluated value by $3\sigma$, which can be explained in terms of deficiencies in a previous Coulomb-excitation analysis.

Conclusions: Comparison of measured $Q_s\left(2_1^{+}\right)$ and $B(E2;0_1^{+}\to 2_1^{+})$ values indicates that neutron-deficient ($N\le 42$) isotopes of krypton are closer to axial deformation than other isotopic chains in the mass region. A continuation of this trend to higher $Z$ may result in Sr and Zr isotopes exhibiting near-axial prolate deformation.

Figure 1

Doppler corrected TIGRESS-S3 coincidence spectra in the beam (${}^{80}\mathrm{Kr}$, black) and target (${}^{196}\mathrm{Pt}$, red) frame for: (a) Beam-like nuclei scattered and detected in the downstream S3 detector. (b) Target-like nuclei scattered and detected in the downstream S3 detector. (c) Beam-like nuclei scattered and detected in the upstream S3 detector. Transitions relevant to the present work are indicated. Note that the resolution for beam-like $\gamma$ rays identified in coincidence with target-like scattered ions is worsened due to the slowing of the beam-like recoil within the target.

Figure 2

Doppler corrected TIGRESS-S3 coincidence spectra in the beam (${}^{82}\mathrm{Kr}$, black) and target (${}^{196}\mathrm{Pt}$, red) frame for: (a) Beam-like nuclei scattered and detected in the downstream S3 detector. (b) Target-like nuclei scattered and detected in the downstream S3 detector. (c) Beam-like nuclei scattered and detected in the upstream S3 detector. Transitions relevant to the present work are indicated.

Figure 3

Levels and transitions relevant to the present work for (a) ${}^{80}\mathrm{Kr}$ and (b) ${}^{82}\mathrm{Kr}$.

Figure 4

Confidence intervals for the $\langle 0_1^{+}|E2|2_1^{+}\rangle$ and $\langle 2_1^{+}|E2|2_1^{+}\rangle$ matrix elements in ${}^{80}\mathrm{Kr}$, calculated using the fitting technique described in the text. Filled points correspond to the ${\chi }^2+1$ distribution calculated from a two-dimensional scan [14]. The dashed red ellipse (“limited”) is the corresponding $1\sigma$ confidence interval using the minuit method described here, where only the $\langle 0_1^{+}|E2|2_1^{+}\rangle$ and $\langle 2_1^{+}|E2|2_1^{+}\rangle$ matrix elements were permitted to vary. The solid ellipses correspond to confidence intervals from a minimization in which all relevant matrix elements were allowed to vary. The two solid ellipses correspond to minimisations using only the ${}^{196}\mathrm{Pt}$ data (red) and the complete data set, incorporating both ${}^{196}\mathrm{Pt}$ and ${}^{208}\mathrm{Pb}$ data (black). The points correspond to the central values obtained from the minuit method.

Figure 5

Confidence intervals for the $\langle 0_1^{+}|E2|2_1^{+}\rangle$ and $\langle 2_1^{+}|E2|2_1^{+}\rangle$ matrix elements in ${}^{82}\mathrm{Kr}$, calculated using the fitting techniques described in the text. As near-identical confidence intervals are obtained from the limited and full analysis when compared to Fig. 4 only the confidence limit from the full minimization is shown. This results from the fact that other matrix elements included in the full minimization are not strongly correlated with $\langle 2_1^{+}|E2|2_1^{+}\rangle$. The point corresponds to the central value from the minuit minimization.

Figure 6

$B(E2;0_1^{+}\to 2_1^{+})$ and $Q_s\left(2_1^{+}\right)$ values in isotopes of krypton, with the present results indicated. Also shown are the $Q_s\left(2_1^{+}\right)$ values expected for an axially symmetric rotor, as described in Eq. (1). Literature data taken from Ref. [15].

Figure 7

(a) $cos{\left(3\delta \right)}_{2_1^{+}}$ as defined in Eq. (2) for isotopes of Zn, Ge, Se, and Kr. A value of 1 corresponds to an axially symmetric prolate central deformation, $-1$ to an axially symmetric oblate central deformation and 0 to a central maximally triaxial configuration. It can be seen that isotopes of krypton are approaching the axially symmetric prolate limit. (b) $\beta$ values, as calculated using Eq. (3) for the same isotopes, as well as for isotopes of strontium. The $\beta$ values are seen to systematically increase towards a maximum in the strontium and krypton isotopes. ${}^{72}\mathrm{Zn}$ values taken from Ref. [28] and ${}^{76,78}\mathrm{Sr}$ values from the weighted average presented in Ref. [2], all other literature values from Ref. [15].

Figure 8

(a) $\langle 0_1^{+}\left|\widehat{Q^2}\right|0_1^{+}\rangle$ plotted against $cos\left(3\delta \right)$ in ${}^{80}\mathrm{Kr}$, calculated with samples drawn from the multivariate normal distribution defined by the mean values from the fitted matrix elements in Table 3 and the covariance matrix extracted from the fit (Table 6). See text for details of the calculation of the $Q^2$ and $cos\left(3\delta \right)$ values. (b) Contributions to $cos\left(3\delta \right)$ from matrix element products.