Low-energy Coulomb excitation of ${}^{96,98}\mathrm{Sr}$ beams

The structure of neutron-rich ${}^{96,98}\mathrm{Sr}$ nuclei was investigated by low-energy safe Coulomb excitation of radioactive beams at the REX-ISOLDE facility, CERN, with the MINIBALL spectrometer. A rich set of transitional and diagonal $E2$ matrix elements, including those for non-yrast structures, has been extracted from the differential Coulomb-excitation cross sections. The results support the scenario of a shape transition at $N=60$, giving rise to the coexistence of a highly deformed prolate and a spherical configuration in ${}^{98}\mathrm{Sr}$, and are compared to predictions from several theoretical calculations. The experimental data suggest a significant contribution of the triaxal degree of freedom in the ground state of both isotopes. In addition, experimental information on low-lying states in ${}^{98}\mathrm{Rb}$ has been obtained.

Figure 1

(a) Total $\gamma$-ray spectrum from MINIBALL, following Coulomb excitation of the ${}^{96}\mathrm{Sr}$ beam impinging on the ${}^{109}\mathrm{Ag}$ target, Doppler corrected for the projectile. (b) Same spectrum for the excitation of ${}^{96}\mathrm{Sr}$ on the ${}^{120}\mathrm{Sn}$ target. The inset shows a zoom on the weak $0_2^{+}\to 2_1^{+}$ transition.

Figure 2

Partial level schemes of ${}^{96}\mathrm{Sr}$ (left) and ${}^{98}\mathrm{Sr}$ (right), showing the transitions observed in the present measurement. Transition and level energies are given in keV. Arrow widths reflect the total measured intensities. The intensity of the transition at 870 keV, marked with a dashed arrow, was too low to be included in the Coulomb excitation analysis (see Sec. 4).

Figure 3

(a) Total $\gamma$-ray spectrum from MINIBALL, following Coulomb excitation of the ${}^{98}\mathrm{Sr}$ beam impinging on a ${}^{60}\mathrm{Ni}$ target, Doppler corrected for the projectile. (b) Same spectrum for the excitation of ${}^{98}\mathrm{Sr}$ on the ${}^{208}\mathrm{Pb}$ target. The origin of the unknown transitions marked with asterisks is discussed in the text. Reproduced from Ref. [28].

Figure 4

Doppler corrected spectra for ${}^{98}\mathrm{Sr}$ on ${}^{208}\mathrm{Pb}$, for three ranges of scattering angles (blue: ${15}^{\circ }–{22}^{\circ }$; red: ${23}^{\circ }–{30}^{\circ }$; green: ${31}^{\circ }–{39}^{\circ }$, in the laboratory frame) zoomed in on the $2_1^{+}\to 0_1^{+}$ decay in ${}^{98}\mathrm{Sr}$. In the inset the RDDS analysis for the $2_1^{+}$ state is presented. Top: Standard decay function as a function of the target-to-DSSSD distance. $I_s$ denotes the intensity of the shifted component, while $I_{us}$ that of the unshifted one. Bottom: Deduced $2_1^{+}$ lifetime for each distance and the global fit.

Figure 5

Proposed level scheme of ${}^{98}\mathrm{Rb}$. Arrow widths reflect total intensities of transitions observed in the present experiment. Transition and level energies are given in keV.

Figure 6

Measured $\gamma$-ray energies, after Doppler correction for the velocity of ${}^{98}\mathrm{Rb}/{}^{98}\mathrm{Sr}$ nuclei, as a function of the cosine of the angle between the velocity vector of the scattered projectile and the direction of the $\gamma$-ray emission.

Figure 7

A two-dimensional ${\chi }^2$ surface with respect to $\langle 2_1^{+}\parallel E2\parallel 0_1^{+}\rangle$ and $\langle 2_1^{+}\parallel E2\parallel 2_1^{+}\rangle$ in ${}^{96}\mathrm{Sr}$, obtained for Coulomb excitation of ${}^{96}\mathrm{Sr}$ on ${}^{120}\mathrm{Sn}$. A $1\sigma$ cut is applied with the condition that ${\chi }^2<{\chi }_{\min }^2+1$. The vertical green lines correspond to the previously reported value of the $2_1^{+}$ lifetime (solid) and its uncertainty (dashed). The black dotted lines illustrate how the uncertainties of matrix elements were determined. The red open square and the circle represent the results of theoretical calculations using the complex excited VAMPIR approach [40] and beyond-mean-field calculations with the Gogny D1S interaction, respectively (see Sec. 5).

Figure 8

Quadrupole deformation parameters obtained with the sum-rule method for the $0_{1,2}^{+}$ states in ${}^{98}\mathrm{Sr}$ and $0_1^{+}$ in ${}^{96}\mathrm{Sr}$.

Figure 9

Transitional quadrupole moments $Q_0^t$ calculated from the $B(E2;J\to J-2)$ values in the ground-state band for Mo, Zr, Sr, and Kr $N=58$ and $N=60$ isotones. Some of the values are slightly displaced on the $x$ axis for clarity.

Figure 10

Spectroscopic quadrupole moments $Q_s$ calculated from diagonal $E2$ matrix elements measured for Mo, Sr, and Kr isotopes. Some of the values are slightly displaced on the $x$ axis for clarity.

Figure 11

Diagonal matrix element $\langle 2_1^{+}\parallel E2\parallel 2_1^{+}\rangle$ in ${}^{98}\mathrm{Sr}$, normalized to the $\langle 2_1^{+}\parallel E2\parallel 0_1^{+}\rangle$, as a function of the $\gamma$ deformation parameter. The black dashed line represents the results of the Davydov-Filippov model, the blue shaded area corresponds to the values of matrix elements determined in the present study (with $1\sigma$ uncertainty), and the yellow shaded area to the result of the quadrupole sum rules approach for the $0_1^{+}$ state in ${}^{98}\mathrm{Sr}$.

Figure 12

Comparison between the theoretical (black) and experimental (red) systematics of excitation energies of the $0_2^{+}$ (top) and $2_1^{+}$ (middle) states and the $B(E2,2_1^{+}\to 0_1^{+})$ values (bottom). The theoretical values for ${}^{90,92}\mathrm{Sr}$ are taken from [72]. For most experimental points uncertainties are smaller than the marker size.

Figure 13

Assignment of calculated levels in ${}^{96}\mathrm{Sr}$ to specific bands. All $J\to J-2$ transitions with $B\left(E2\right)$ values higher than 10 $e^2{\mathrm{fm}}^4$ are marked with lines. Dotted lines correspond to $B(E2$) values lower than 100 $e^2{\mathrm{fm}}^4$, while for those higher than 100 $e^2{\mathrm{fm}}^4$ solid lines are used, and their width is proportional to $B(E2;J\to J-2)$. Energies of $0_2^{+}$ and $2_3^{+}$ levels are offset by 40 keV for clarity. For the same reason spectroscopic moments of $0^{+}$ states are presented as being equal to those of the corresponding $2^{+}$ states. Bands are labeled as in Refs. [22, 23].

Figure 14

Same as Fig. 13 but for ${}^{98}\mathrm{Sr}$. Energies of $2_2^{+}$ and $4_3^{+}$ levels are offset by 40 keV for clarity.

Figure 15

Same as Fig. 13 but for calculated $E0$ transition strengths in ${}^{96}\mathrm{Sr}$. All transitions with ${\rho }^2\left(E0\right)$ values higher than ${10}^{-2}$ are marked with lines. Dotted lines correspond to ${\rho }^2\left(E0\right)$ values lower than $4\times {10}^{-2}$, while for those higher than $4\times {10}^{-2}$ solid lines are used, and their width is proportional to ${\rho }^2\left(E0\right)$. Bands are labeled as in Refs. [22, 23] and Fig. 13. The $0_3^{+}$ state, not attributed to any of these bands, is marked with a red cross at $Q_{sp}=0$ and offset by 30 keV for clarity.

Figure 16

Same as Fig. 15 but for ${}^{98}\mathrm{Sr}$. Energy of the $0_3^{+}$ level is offset by 60 keV for clarity.