Lifetime measurements in the even-even ${}^{102–108}\mathrm{Cd}$ isotopes

Background: The heaviest $T_z=0$ doubly-magic nucleus, ${}^{100}\mathrm{Sn}$, and the neighboring nuclei offer unique opportunities to investigate the properties of nuclear interaction. For instance, the structure of light-Sn nuclei has been shown to be affected by the delicate balance between nuclear-interaction components, such as pairing and quadrupole correlations. From Cd to Te, many common features and phenomena have been observed experimentally along the isotopic chains, leading to theoretical studies devoted to a more general and comprehensive study of the region. In this context, having only two proton holes in the $Z=50$ shell, the Cd isotopes are expected to present properties similar to those found in the Sn isotopic chain.

Purpose: The aim of this work was to measure lifetimes of excited states in neutron-deficient nuclei in the vicinity of ${}^{100}\mathrm{Sn}$.

Methods: The neutron-deficient nuclei in the $N\approx Z\approx 50$ region were populated using a multinucleon transfer reaction with a ${}^{106}\mathrm{Cd}$ beam and a ${}^{92}\mathrm{Mo}$ target. The beamlike products were identified by the VAMOS$++$ spectrometer, while the $\gamma$ rays were detected using the AGATA array. Lifetimes of excited states were determined using the recoil distance Doppler-shift method, employing the Cologne differential plunger.

Results: Lifetimes of low-lying states were measured in the even-mass ${}^{102–108}\mathrm{Cd}$ isotopes. In particular, multiple states with excitation energy up to $\approx 3$ MeV, belonging to various bands, were populated in ${}^{106}\mathrm{Cd}$ via inelastic scattering. The transition strengths corresponding to the measured lifetimes were compared with those resulting from state-of-the-art beyond-mean-field calculations using the symmetry-conserving configuration-mixing approach.

Conclusions: Despite the similarities in the electromagnetic properties of the low-lying states, there is a fundamental structural difference between the ground-state bands in the $Z=48$ and $Z=50$ isotopes. The comparison between experimental and theoretical results revealed a rotational character of the Cd nuclei, which have prolate-deformed ground states with ${\beta }_2\approx 0.2$. At this deformation $Z=48$ becomes a closed-shell configuration, which is favored with respect to the spherical one.

Figure 1

Systematics of the experimental $B(E2;2_1^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+})$ reduced transition probabilities for the even-mass Cd (red squares) and Sn (blue circles) isotopic chains. Results are taken from Ref. [17].

Figure 2

Partial level scheme of the even-mass ${}^{102–108}\mathrm{Cd}$ presenting the transitions observed in the current measurement. The arrow widths represent the efficiency-corrected transition yields normalized over the $2_1^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+}$ one: intensities below $1\%$ are shown with dashed arrows, while transitions with yield lower than $0.1\%$ and not useful for the analysis have been omitted. The excitation energy of the states is highlighted in red. Spin, parity, and excitation energy of the states are assigned according to the NNDC On-Line Data Service from the ENSDF database [52] (file revised as of August 2009 for ${}^{102}\mathrm{Cd}$, September 2009 for ${}^{104}\mathrm{Cd}$, June 2008 for ${}^{106}\mathrm{Cd}$, and October 2008 for ${}^{108}\mathrm{Cd}$).

Figure 3

Doppler-corrected $\gamma$-ray energy spectrum of ${}^{102}\mathrm{Cd}$ before (black) and after (green) the gate on the total kinetic-energy loss (TKEL), obtained by summing the statistics of all the target-degrader distances. The $2^{+}\to 0_{\mathrm{g}.\mathrm{s}.}$ (red), $4^{+}\to 2^{+}$ (blue), and $6^{+}\to 4^{+}$ (orange) transitions are marked, indicating the unshifted and shifted centroids with solid and dashed lines, respectively.

Figure 4

DDCM analysis for the lifetime measurement of the $2_1^{+}$ excited state of ${}^{102}\mathrm{Cd}$. Top: Area of the shifted (red diamonds) and feeding-corrected unshifted (blue triangles) components, normalized to the number of ions detected in VAMOS$++$. The dashed line represents a fit to the shifted-component points. Bottom: Corresponding lifetimes obtained for individual distances. The solid line denotes the weighted average of the lifetimes, while the filled area corresponds to $1\sigma$ statistical uncertainty.

Figure 5

Decay curve as a function of the lifetime of the $2_1^{+}$ excited state in ${}^{106}\mathrm{Cd}$, obtained with the $R_{\mathrm{sum}}$ approach. The black line represents the experimental value obtained by summing the statistics of all target-degrader distances and gating on the shifted component of the $4_1^{+}\to 2_1^{+}$ transition. The red curve is the expected value calculated with Eq. (2). The interception between the experimental and expected values (green line) represents the lifetime of the state. All the dashed curves denote the $1\sigma$ uncertainty.

Figure 6

Ratio of the component intensities as a function of the target-degrader distance for the $6_1^{+}\to 4_1^{+}$ transition in ${}^{106}\mathrm{Cd}$. The solid red line represents the fitted decay curve obtained with second-order Bateman equations, whose components are shown with dashed and dotted blue lines for the feeder and the direct population, respectively.

Figure 7

PNVAP potential energy surfaces as a function of the (${\beta }_2, \gamma$) deformation parameters for the even-mass ${}^{100–130}\mathrm{Cd}$ isotopes. The results are obtained with the Gogny-D1S interaction within the SCCM approach.

Figure 8

Reduced transition probabilities (a) $B(E2;2_1^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+})$ and (b) $B(E2;4_1^{+}\to 2_1^{+})$, and (c) $2_1^{+}$ and $4_1^{+}$ excitation energy systematics for even-mass Cd isotopes. The experimental results [55, 56, 58, 60, 61, 63, 64, 65, 66, 67, 75, 76, 77, 78, 79, 80, 81, 82] are compared with the recent large-scale shell-model (LSSM) calculations of Ref. [16] (blue open pentagons) and the present SCCM predictions (red open circles and squares). The results of this work (black open squares) are obtained by adopting the weighted average of the DCM and DDCM lifetimes, reported in Table 1.

Figure 9

Collective wave functions (CWFs) as a function of the (${\beta }_2, \gamma$) deformation parameters for the $2_1^{+}$ states in the even-mass ${}^{100–130}\mathrm{Cd}$ isotopes. The results are obtained with the Gogny-D1S interaction within the SCCM approach.

Figure 10

Collective wave functions (CWF) as a function of the (${\beta }_2, \gamma$) deformation parameters for the $4_1^{+}$ states in the even-mass ${}^{100–130}\mathrm{Cd}$ isotopes. The results are obtained with the Gogny-D1S interaction within the SCCM approach.

Figure 11

Top: Level scheme of ${}^{106}\mathrm{Cd}$ predicted by the present SCCM calculations. The numbers on the arrows represent the $B\left(E2\right)$ reduced transition probabilities in Weisskopf units. Bottom: The CWFs are shown for all the investigated states and grouped in bands with different $({\beta }_2,\gamma )$: prolate-deformed ground-state band (black), predominantly $K=2$ pseudogamma band (red), predominantly $K=4$ band (green), oblate-deformed shape-mixed band (magenta), prolate-deformed shape-mixed band (orange), and the continuation of the latter at larger angular momenta (blue).