Evolution of collectivity in 126 , 128 Xe studied in Coulomb excitation measurements

. The characteristics of 126 , 128 Xe were investigated in Coulomb excitation measurements performed at the National Superconducting Cyclotron Laboratory (NSCL) Re-accelerator facility, ReA3, Michigan State University (MSU). The Xe nuclei were accelerated to sub-barrier energies and were impinged on 196 Pt and 208 Pb targets in separate experimental runs. The scattered nuclei and the de-excitation γ -rays were detected using the JANUS setup. Electromagnetic matrix elements were extracted from the experimental data with the help of the GOSIA / GOSIA2 codes. The results were compared to schematic Davydov-Filippov γ -rigid rotor theoretical calculations and large-scale calculations within a newly-established microscopic shell model (called PMMU model). The experimental results agree well with the theoretical predictions, except for the quadrupole moments of the second 2 + states in both nuclei, therefore challenging the interpretation of the γ -bands structure.


Introduction
The experimental properties of many nuclei in the stable even-A Xe mass region suggest a transitional behavior and a certain degree of triaxiality. The exhibit of such characteristics seems to be straight forward given the spherical features of the Sn nuclei and the expected changes when moving towards the mid-shell region.
Non-axially symmetric shapes can be related both to two extreme phenomenological models: (1) the γ-rigid rotor model of Davydov and Filippov (DF) [1] and (2) the γ-soft model of Wilets and Jean (WJ) [2] (which is also closely related to the O(6) dynamical symmetry of the Interacting Boson Model).
Recent works on the even-A Xe isotopes consider 128 Xe as a possible E(5) critical point in the transition from spherical to O(6)-structure γ-soft nuclei [3], while other studies [4] suggest that 130 Xe is the most likely candidate for E(5) among the Xe isotopes. Deviations from the O (6) predictions for certain transition strengths were also observed [4,5].
The energy gaps in the ground state bands in the even -A Xe nuclei indicate a possible O(6)-like structure. How-ever, a real probe to test the relation to the O(6) dynamical symmetry are the electromagnetic properties of the nuclei. In particular, one can emphasize the importance of the electric quadrupole moments which vanish within the γ-soft framework.
Previous studies provide experimental information about transition strengths in the even-A Xe nuclei but the information about quadrupole moments is sparse [4][5][6][7][8]. The present work reports results from measurements of the electromagnetic properties of 126,128 Xe, with focus on the quadrupole moments of several excited states.

Experimental Details and Data Analysis
A sub-barrier Coulomb excitation experiment was performed at the National Superconducting Cyclotron Laboratory (NSCL) ReA3 facility (stand-alone mode), Michigan State University (MSU). 126,128 Xe nuclei were accelerated to energies of 3.74 MeV/nucleon and 3.81 MeV/nucleon, respectively, and were impinged on a 208 Pb target with a thickness of 0.92 mg/cm 2 and a 196 Pt target with a thickness of 1.59 mg/cm 2 in separate experimental runs. The beam intensity was on average ∼3×10 5 pps for 128 Xe and ∼2×10 5 pps for 126 Xe. The scattered nuclei and the γ-rays following the Coulomb excitation were detected using the Joint Array for NUclear Structure (JANUS) [9] setup. The configuration consisted of a pair of segmented Micron S3-type double-sided Si detectors (Bambino2) and the Segmented Ge Array (SeGA) [10] (comprising of sixteen 32-fold segmented HPGe detectors with cylindrical crystals).
The experimental data were sorted and processed using the GRUTinizer software [11]. The setup arrangement allowed to split by angle the Si detectors data in 4 datasets for each of the Xe nuclei and construct Doppler-corrected SeGA γ-ray distributions. Spectra from such datasets with a 208 Pb target are presented in Fig. 1. Partial level schemes of 126,128 Xe are shown in Fig. 2. The transitions observed in the present work are highlighted in red.
The efficiency corrected experimental γ-ray yields were compared to calculations performed with the semiclassical Coulomb excitation code, GOSIA [12,13], and electromagnetic properties of the nuclei were extracted. The <2 + 1 E2 0 + 1 > matrix elements in both nuclei were first determined using the 196 Pt target datasets and a GOSIA/GOSIA2 iterative analysis procedure [14,15], as described in Ref. [16]. Full sets of transition and diagonal matrix elements were futher obtained in a standard GOSIA χ 2 minimization with both target datasets for each nucleus.
Known literature data for branching ratios, mixing ratios, and lifetimes were used in addition to the present experimental γ-ray yields during the minimization procedures (Table 1).

Experimental Results and Discussion
Experimental results for transition matrix elements and B(E2) values determined in the current work are presented in Table 2, along with a comparison to theoretical values Table 1. Experimental branching ratios (BR), E2/M1 mixing ratios (δ), and lifetimes (τ) in 126,128 Xe known prior to the present work [17][18][19][20]. These values were used during the GOSIA χ 2 minimizations.
0.268 (5)  3.5 (2) and previous measurements. The uncertainties of the experimental values were obtained by taking into account all correlations between the matrix elements. The experimental diagonal matrix elements, and the respective quadrupole moments, determined in the GOSIA χ 2 minimizations are presented in Table 3, along with a comparison to theoretical calculations.
The new experimental results for the transition strengths in 126,128 Xe are in general in a good agreement with literature values. Although the experiemntal B(E2; 2 + 1 →0 + 1 ) = 56 (5) W.u. for the decay of the first 2 + state in 126 Xe differs from the evaluated data in Ref. [19,21], it agrees well with recently performed measurements [22][23][24]. The newly determined quadrupole moments suggest deviations from the O(6)-like structure in the ground state bands. The low magnitude of the quadrupole moments of the 2 + 2 states in both nuclei are notable. These values are consistent in a systematic way with results from measurements in 130 Xe [8].
Schematic claculations within the asymmetric rotor approach [1] were performed for 126,128 Xe. The triaxiality parameters γ were determined from the experimental features of the second 2 + states. Transition strengths were calculated based on the model predictions for the respective values of γ, with a normalization to the experimental <2 + 1 E2 0 + 1 > matrix elements (Table 2 and 3). A more detailed theoretical approach was applied using microscopic shell model calculations. The model employs a realistic Hamiltonian (called PMMU) which combines the pairing plus multipole terms with the monopole interaction obtained by the monopole based universal force [25]. It has been proven to describe well energy spectra and electromagnetic transitioins in a wide range of nuclei [26,27]. The present PMMU calcula-   21 29 (5) 29.6 (29) a From Ref. [5] b From Ref. [6] c From Ref. [4]  tions adopt a large model space including 1g 9/2 , 1g 7/2 , 2d 5/2 , 2d 3/2 , 3s 1/2 , 1h 11/2 , 2 f 7/2 (referred to as gdsh f ). Conventional shell-model techniques cannot be used with such a large model space. Thus, the recently developed Hartree-Fock-Bogoliubov plus generator coordinate method (HFB+gcm) [28] was applied. It uses the HFB method to generalize a suitable mean field and applies the gcm calculation to obtain the wave functions through superimposing many configurations in the quadrupoledeformation plane.
The present PMMU shell-model calculations for 126,128 Xe were performed using the same HFB+gcm code as in Ref. [29], with the same parameters as in Ref. [30]. A comparison between the PMMU level energies and experimental data is shown in Fig. 3. Transition probabilities and quadrupole moments were calculated (with effective charges taken as e p = 1.0e and e n = 0.4e) and are presented in Tables 2 and 3  quadrupole moments of the 2 + 2 states. It is remarkable that the model calculations reproduce well the small experimental B(E2; 2 + 2 → 0 + 1 ) in 126 Xe and 128 Xe. The γ-soft model predicts a small B(E2) value for this transition, consistent with the experimental data.  Fig. 4 shows the potential energy surfaces for 126,128 Xe as a function of the quadrupole moments Q 0 and Q 2 . The plots suggest very soft energy surfaces for a wide area in the parameter plane. Perturbations can easily drive the nucleus moving in the Q 0 -Q 2 plane. Thus, the nuclei can have a variety of possibilities to develop different shapes (including unusual ones) once they get excited.

Summary
The electromagnetic properties of 126,128 Xe were studied in sub-barrier Coulomb excitation measurements performed at NSCL, MSU. Transition and diagonal matrix elements were determined and compared to theoretical calculations within the schematic γ-rigid asymmetric rotor model and the microscopic PMMU shell model. The experimental results are in a reasonable agreement with the model predictions. The theoretical approaches do not reproduce well the low-magnitude experimental quadrupole moment of the 2 + 2 states, resulting in difficulties to interpret the structure of the nuclei in the γ-bands.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.