Axial and triaxial degrees of freedom in 72Zn

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Axial and triaxial degrees of freedom in 72Zn © 2023 Published by Elsevier B.V. Funded by SCOAP3. Published version Hellgartner, S.; Mücher, D.; Wimmer, K.; Bildstein, V.; Egido, J. L.; Gernhäuser, R.; Krücken, R.; Nowak, A. K.; Zielińska, M.; Bauer, C.; Benito, M. L. L.; Bottoni, S.; De Witte, H.; Elseviers, J.; Fedorov, D.; Flavigny, F.; Illana, A.; Klintefjord, M.; Kröll, T.; Lutter, R.; Marsh, B.; Orlandi, R.; Pakarinen, J.; Raabe, R.; Rapisarda, E.; Reichert, S.; Reiter, P.; Scheck, M.; Seidlitz, M.; Siebeck, B.; Siesling, E.; Steinbach, T.; Stora, T.; Vermeulen, M.; Voulot, D.; Warr, N.; Wenander, F. J. C. Hellgartner, S., Mücher, D., Wimmer, K., Bildstein, V., Egido, J. L., Gernhäuser, R., Krücken, R., Nowak, A. K., Zielińska, M., Bauer, C., Benito, M. L.L., Bottoni, S., De Witte, H., Elseviers, J., Fedorov, D., Flavigny, F., Illana, A., Klintefjord, M., Kröll, T., . . . Wenander, F. J. C. (2023). Axial and triaxial degrees of freedom in 72Zn. Physics Letters B, 841, Article 137933. https://doi.org/10.1016/j.physletb.2023.137933 2023 Physics Letters B 841 (2023) 137933 Contents lists available at ScienceDirect Physics Letters B journal homepage: www.elsevier.com/locate/physletb Axial and triaxial degrees of freedom in 72Zn S. Hellgartnera, D. Mücherb,c,d,∗, K. Wimmere,f,g,∗, V. Bildsteinc, J.L. Egidoi, R. Gernhäusera, R. Krückenh, A.K. Nowaka, M. Zieli´ nskaj, C. Bauerk, M.L.L. Benitol, S. Bottonim,n, H. De Witten, J. Elseviersn, D. Fedorovl, F. Flavignyo, A. Illanaf,r, M. Klintefjordp, T. Kröll k, R. Lutterq, B. Marshl, R. Orlandif, J. Pakarinenr, R. Raaben, E. Rapisardan,l,s, S. Reicherta, P. Reiter b, M. Scheckt, M. Seidlitzb, B. Siebeckb, E. Sieslingl, T. Steinbach b, T. Stora l, M. Vermeulenu, D. Voulotl, N. Warrb, F.J.C. Wenanderl aPhysik-Department, Technische Universität München, 85748 Garching, Germany bInstitut für Kernphysik, Universität zu Köln, 50937 Köln, Germany cCollege of Physics & Engineering Science, University of Guelph, 50 Stone Road East Guelph, Ontario N1G 2W1, Canada dPhysical Sciences Division, TRIUMF, Vancouver, British Columbia, V6T 2A3, Canada eGSI Helmholtzzentrum für Schwerionenforschung, D-64291 Darmstadt, Germany fInstituto de Estructura de la Materia, CSIC, E-28006 Madrid, Spain gDepartment of Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan hNuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA iDepartamento de Física Teórica, Universidad Autónoma de Madrid, E-28049 Madrid, Spain jIRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France kInstitut für Kernphysik, Technische Universität Darmstadt, Germany lISOLDE, CERN, CH-1211, Geneva, Switzerland mDipartimento di Fisica, Università degli Studi di Milano and INFN Sez. Milano, Milano I-20133, Italy nKU Leuven, Instituut voor Kern- en Stralingsfysica, 3001 Leuven, Belgium oNormandie Univ, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France pDepartment of Physics, University of Oslo, N-0316 Oslo, Norway qDepartment of Physics, Ludwig Maximilian Universität München, 85748 Garching, Germany rDepartment of Physics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland sPaul Scherrer Institut, Villigen, Switzerland tSchool of Computing, Engineering, and Physical Sciences, University of the West of Scotland, Paisley PA1 2BE, UK uDepartment of Physics, University of York, YO10 5DD, United Kingdom a r t i c l e i n f o a b s t r a c t Article history: Received 22 June 2021 Received in revised form 24 March 2023 Accepted 21 April 2023 Available online 26 April 2023 Editor: B. Blank Keywords: Multiple Coulomb excitation 72Zn N=40 sub-shell closure Triaxiality The unstable N=42 nucleus 72Zn has been studied using multiple safe Coulomb excitation in inverse kinematics. The experiment was performed at the REX-ISOLDE facility at CERN making first use of the silicon detector array C-REX in combination with the γ-ray spectrometer Miniball. The high angular coverage of C-REX allowed to determine the reduced transition strengths for the decay of the yrast 0+ 1, 2+ 1 and 4+ 1as well as of the 0+ 2and 2+ 2states in 72Zn. The quadrupole moments of the 2+ 1, 4+ 1and 2+ 2states were extracted. Using model independent quadrupole invariants, the ground state of 72Zn was found to have an average deformation in the γdegree of freedom close to maximum triaxiality. In comparison to experimental data in zinc isotopes with N<40, the collectivity of the 4+ 1state in neutron-rich 72Zn is significantly larger, indicating a collective yrast band based on the ground state of 72Zn. In contrast, a low experimental B(E2; 0+ 2→2+ 1)strength was determined, indicating a different structure for the 0+ 2 state. Shell-model calculations propose a 0+ 2state featuring a larger fraction of the (spherical) N=40 closed-shell configuration in its wave function than for the 0+ 1ground state. The results were also compared with beyond mean field calculations which corroborate the large deformation in the γdegree of freedom, while pointing to a more deformed 0+ 2state. These experimental and theoretical findings establish the importance of the γdegree of freedom in the ground state of 72Zn, *Corresponding authors. E-mail addresses: [email protected]oeln.de (D. Mücher), [email protected] (K. Wimmer). https://doi.org/10.1016/j.physletb.2023.137933 0370-2693/©2023 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). Funded by SCOAP3. S. Hellgartner, D. Mücher, K. Wimmer et al. Physics Letters B 841 (2023) 137933 located between the 68,70Ni nuclei that have spherical ground states, and 76Ge, which has a rigid triaxial shape. ©2023 Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/). Funded by SCOAP3. One of the fundamental properties of the atomic nucleus is its shape. Nuclei with a closed-shell configuration are spherical, while deformation can arise from quadrupole correlations in open-shell nuclei. The collective, or Bohr Hamiltonian [1] describes the dynamics of nuclei in terms of the deformation parameters βand γ. The quantity βmeasures the axial-symmetric deformation of an ellipsoid, while γrelates to the deviation from axial symmetry. Two approximations of the Bohr Hamiltonian are often used when discussing the γdegree of deformation: the triaxial γ-rigid rotor of the Davydov-Filippov model [2], and the γindependent (or γunstable) Wilets-Jean model [3]. A transitional region is observed close to the harmonic oscillator shell gap N=40. The magic Ni isotopic chain at Z=28 exhibits spherical ground states across N=40. Adding four protons leads to more deformed Ge isotopes at Z=32; Ge isotopes were found to undergo a transition from γ-soft in 72Ge [4]to γrigid in 76Ge. 76Ge is one of the few cases where rigid triaxiality in low-lying states has been observed through the staggering of states in the γband [5] and from electromagnetic matrix elements [6]. Indication for changes in deformation can already be found for the Z=30 chain of Zn isotopes. Coulomb excitation and lifetime measurements [7–11]find increased B(E2; 2+ 1→0+ gs)values in 72,74Zn compared to the Zn isotopes below N=40 indicating the onset of deformation, but these earlier experiments were not sensitive to the γdegree of deformation. For 72Zn, the g-factor of the 2+ 1state, which is close to the hydro-dynamical limit, indicates deformation [12] and triaxiality was suggested to be present in 73Zn based on the observation of a deformed 5/2+isomeric state [13]. This onset of deformation and triaxiality in Zn is supported by beyond mean field calculations. Calculations employing the symmetry conserving configuration mixing approach [14]presented in Ref. [12], as well as the five-dimensional collective quadrupole Hamiltonian (5DCH) treatment [15]predict significant triaxial deformation of the ground states of 70−74Zn. Large-scale Monte-Carlo shell-model calculations predict triaxiality for 72−74Zn, but do not find triaxial shapes for the ground states of 71,75Zn [13]. A recent experimental study of the 66Zn nucleus also highlights the importance of the triaxial degree of freedom before N=40 and suggests large fluctuations of the wave functions around γ≈30◦[16]. Nuclei in this region around neutron number N=40 also show interesting occurrences of shape coexistence [17]. In 70Ni, triple shape coexistence of a spherical ground state with prolate and oblate deformed excited 0+states is predicted by Monte-Carlo shell-model calculations [18]. Experimentally, a candidate for an excited 0+state has been observed at 1567 keV [19]. In 72Ge, the first excited state is a 0+state and has been classified as an intruder state of spherical nature [4], while the ground state is deformed. In this letter, we study the unstable Z=30 isotope 72Zn at N=42 via multi-step safe Coulomb excitation. This method is sensitive to the reduced transition probabilities and the spectroscopic quadrupole moments. Furthermore, the data set also allows the extraction of approximate shape invariants, which give access to the shape of the nucleus in a model independent way. This way, we test triaxiality in the direct vicinity of the “doubly magic” 68Ni nucleus and gain new insight into the coexisting shapes in this region. The experiment was performed at the REX-ISOLDE facility at CERN [20,21]. The radioactive 72Zn beam was produced by the 1.4 GeV proton beam of the PS booster impinging on a UCx ISOL production target. To select the 72Zn atoms from other reaction products from the primary target, they were laser ionized at the Resonant Ionization Laser Ion Source [22], accelerated to 30 keV, and mass separated in the High Resolution Separator. The singly-charged 72Zn ions were bunched in the penning trap REX-TRAP and bred to a higher charge state, Q=20, in the REXEBIS. Finally, the 72Zn ions were accelerated to beam energies of Ebeam =2.85 MeV/nucleon in the normal-conducting linear accelerator REX. The average beam intensity of 72Zn was 3.5(3) ·107 ions/s. A small fraction of surface-ionized 72Ga was transmitted as well. The isobaric contamination, 6.9(6)% of the total beam, was determined using a modified laser on/off method. The postaccelerated beam impinged on a 1.17 mg/cm2thick 109Ag target located in the center of the C-REX array [23] and surrounded by 8 six-fold segmented high-purity Ge triple cluster detectors of the Miniball array [24]used for high resolution spectroscopy of the γradiation emitted by the Coulomb excited nuclei. The silicon detector array C-REX was designed and first used for this experiment, allowing for the selection of projectile and target-like reaction products with a large angular coverage. The design of C-REX is based on the transfer reaction setup T-REX [25]featuring the same scattering chamber and type of detectors, but it is optimized for Coulomb-excitation experiments. In particular, it offers a good coverage of large center-of-mass scattering angles for normalkinematics experiments, increasing the experimental sensitivity to multi-step processes in Coulomb excitation. C-REX features two annular double-sided silicon strip detectors (DSSSD) covering laboratory angles θlab =[21.0◦−60.2◦]and [153◦−172◦]. Each detector is divided into four quadrants with 16 annular rings (r= 2 mm) and 24 radial strips (φ=3.4◦), each. In addition to the DSSSDs, C-REX is equipped with four squared single sided silicon strip detectors (θlab =[102◦−153◦]) which are arranged in a box. Their 16 resistive strips feature a pitch of d =3.125 mm each and are orientated perpendicular to the beam axis. The electronics of C-REX is identical to the T-REX one and allows for high particle count rates and particle-particle coincidences. For this, in contrast to T-REX, the trigger signals were generated for each of the quadrants independently. More details can be found in Ref. [23]. The high beam intensity, in combination with the large angular coverage of the C-REX array, allows the study of multi-step Coulomb excitation of 72Zn with high precision. Since the γrays originating from the de-excitation of the ejectile and recoil nuclei are emitted in flight, a good Doppler correction is essential. Fig. 1 shows a γ-ray energy spectrum coincident with 72Zn particles detected in the forward part of C-REX. The main peaks in Fig. 1are the yrast 2+ 1→0+ gs (Eγ=653 keV) and 4+ 1→2+ 1(Eγ=847 keV) transitions as well as the decay of the 2+ 2state to the ground (Eγ=1658 keV) and 2+ 1states (Eγ=1004 keV). The level scheme with the observed transitions is shown in Fig. 4. The decays of excited states in the 109Ag target nucleus are observed as broad peaks when the Doppler correction assumes the Zn trajectory. A small peak at 166 keV is associated with the 72Ga beam contamination. When gating on 72Zn ions that are scattered to laboratory backward angles, the 858-keV 0+ 2→2+ 1transition is clearly identified (see inset of Fig. 1), indicating that this state is mostly populated by multi-step excitation with growing differential cross sections for larger θc.m. angles. With the available beam intensities, the 0+ 2state could therefore only be studied with the newly de- 2 S. Hellgartner, D. Mücher, K. Wimmer et al. Physics Letters B 841 (2023) 137933 Fig. 1. Doppler corrected and background subtracted γ-ray energy spectrum measured in coincidence with 72Zn ions detected in the forward C-REX detectors. The Doppler correction has been performed assuming the γrays are emitted from the 72Zn. Known transitions in 72Zn are indicated. Indicated in gray are contributions from the strongest 109Ag γ-ray transitions as well as from the isobaric beam contaminant 72Ga. The inset shows a comparison of the Doppler corrected γ-ray spectrum of the forward (gray, filled) and backward (blue) C-REX detectors. In backward direction, additionally the 0+ 2→2+ 1transition of 72Zn at Eγ=858 keV is present. veloped C-REX array covering large scattering angles and not with the previous setup at REX-ISOLDE [24]. In Coulomb excitation [26]the excitation cross section for final states Jfdepends not only on the transitional E2matrix elements for the direct excitation, 0+ gs||E2|| Jf, and second order effects from multi-step excitations through intermediate states, 0+ gs||E2|| JiJi||E2|| Jf, but also on the diagonal matrix elements (quadrupole moments Q) and their signs. In contrast to previous lifetime measurements and Coulomb excitation at intermediate beam energies the excellent statistics of the present experiment and the high angular coverage of the new C-REX detector allow to analyze the angular distributions of the Coulomb-excitation cross sections and determine the matrix elements. The matrix elements and their respective signs were obtained by fitting the detected γ-ray yields with a multi-step Coulomb-excitation calculation obtained with the CLX [27,28] and GOSIA [29] codes. To avoid systematic uncertainties introduced by an evaluation of absolute luminosity and detection efficiencies, a relative measurement is performed, i.e. the yields are normalized using a γ-ray transition with a known (partial) lifetime. In the present work, a set of 26 electric and magnetic matrix elements of 109Ag was used for the normalization [23]. For this, the data for the γ-ray yields for 109Ag were divided into 14 angular bins to obtain in total 110 γ-ray yield data points used in the fit. The data for the γ-ray yields for 72Zn were divided into the same angular bins. Since the yield could not be determined in every bin for each of the five observed transitions, bins have been combined and a total of 45 γ-ray yield data points were used in the global minimization procedure for 72Zn. In addition, upper limits, for example for the observation of the 6+→4+ transition, have been introduced. The matrix elements for 72Zn have then been obtained following the GOSIA-GOSIA2 procedure described in Ref. [30]. The strong sensitivity of the data to the spectroscopic quadrupole moments (diagonal matrix elements) is shown for the 2+ 1,2and 4+ 1states in Fig. 2and the results of the minimization are listed in Table 1. As a cross check, additionally the measured lifetime of the 2+ 1state of 72Zn [9–11] has been used for normalization, which results in a consistent set of matrix elements. The results for the B(E2)values of the 2+ 1→0+ 1and 4+ 1→ 2+ 1transitions in 72Zn are shown in Fig. 3and compared with the neighboring Zn isotopes and previous experimental results. The B(E2; 2+ 1→0+ 1)value agrees very well with previous Coulomb-excitation and lifetime measurements [7,9–11]. However, the measured B(E2; 4+ 1→2+ 1)values indicate larger val- Fig. 2. Differential cross section for the excitation of the 2+ 1(top), 4+ 1(middle) and 2+ 2(bottom) states. The data, divided into the 14 angular ranges are shown in black and for the horizontal error bars it has been assumed that the counts are uniformly distributed in that angular bin. The green (solid) curve shows the calculated angular distribution using the best fit values for the transitional and diagonal matrix elements. For comparison, also calculations using QS=0(blue, dashed) or a positive quadrupole moment (red, dashed-dotted) are shown. In these cases the transitional matrix element has been adjusted to the data point at θc.m.=50◦. ues than deduced from lifetime measurements. For N=42 and 44, Coulomb-excitation experiments, including the present study, yield higher B(E2)values compared to lifetime measurements of Refs. [10,11,37]. Indirect feeding through transitions from higherlying states can result in systematically too large lifetimes extracted in those experiments. This effect was investigated in Ref. [10]by gating on the excitation energy in the reaction residue. The present experimental data also allowed to determine the diagonal matrix elements. Both the 2+ 1and the 4+ 1have negative spectroscopic quadrupole moments, while the value obtained for the 2+ 2state is positive (see Fig. 2). The data also allowed to deduce the E2/M1 mixing ratio for the 2+ 2→2+ 1transition with a negligible M1 contribution [23]. The influence of the 6+ 1||E2||4+ 1 matrix element on the results has been investigated. Using the conservative upper limit for the observation of the 6+ 1→4+ 1tran- sition from the present data or the lifetime measured in Ref. [11] results in negligible changes of the deduced 4+ 1||E2||2+ 1matrix element. 3 S. Hellgartner, D. Mücher, K. Wimmer et al. Physics Letters B 841 (2023) 137933 Table 1 Transition energies, matrix elements, reduced transitions strengths B(πλ), and quadrupole moments for states and transitions in 72Zn determined from the GOSIA-GOSIA2 analysis. The uncertainties are listed separately for statistical and systematic contributions. The statistical uncertainties include the uncertainties of the normalization as well as the statistical errors of the count rates in the individual peaks. In addition, a systematic uncertainty of 5% is added to the matrix elements to account for the approximations used in the GOSIA code [29,30]. Theoretical results based on shell-model calculations with the jj44c and JUN45 effective interactions and mean-field generator coordinate method (GCM) calculations are also presented. Experiment SM jj44c SM JUN45 Triaxial GCM Transition EγJi||E2|| JfB(E2)EγB(E2)EγB(E2)EγB(E2) (keV) (eb) (e2fm4)(keV)(e 2fm4)(keV)(e 2fm4)(keV)(e 2fm4) 2+ 1→0+ 1653 0.424+0.002 −0.002 ±0.021 360+3 −3±36 818 384.4 1007 315.0 789 547.8 2+ 2→0+ 11658 0.074+0.005 −0.004 ±0.004 11.0+1.4 −1.1±1.1 1929 3.9 1906 2.9 2043 37.0 2+ 2→2+ 11004 0.32+0.01 −0.01 ±0.02 205+12 −17 ±21 1111 326.3 899 421.2 1254 385.0 4+ 1→2+ 1847 0.68+0.01 −0.01 ±0.03 514+9 −9±52 861 508.6 954 327.6 1182 817.2 0+ 2→2+ 1858 0.14+0.01 −0.03 ±0.01 196+33 −73 ±20 1009 93.8 769 126.4 1591 260.0 Transition EγJi||M1|| JfB(M1)EγB(M1)EγB(M1)EγB(M1) (keV) (μN)(10 −4μ2 N)(keV)(10 −4μ2 N)(keV)(10 −4μ2 N)(keV)(10 −4μ2 N) 2+ 2→2+ 11004 −0.06+0.07 −0.03 ±0.001 7.2+16.8 −7.2±0.1 1111 372 899 2202 1254 3.73 State EJi||E2|| JiQSEQ SEQ SEQ S (keV) (eb) (efm2)(keV)(efm 2)(keV)(efm 2)(keV)(efm 2) 2+ 1653 −0.31+0.04 −0.04 ±0.01 −24+3 −3±1 818 −27.5 1007 −4.7 789 -38.5 4+ 11500 −0.36+0.06 −0.10 ±0.02 −27+5 −7±1 1679 −45.3 1961 −41.2 1971 -49.9 0+ 21511 1828 1776 2380 2+ 21658 +0.52+0.05 −0.03 ±0.03 +39+4 −3±2 1929 +17.7 1906 +3.8 2043 +38.4 Compared to the less neutron-rich isotopes, a significant increase is observed in the B(E2; 2+ 1→0+ 1)and B(E2; 4+ 1→2+ 1) values for N=42 compared to 64−70Zn. For the B(E2; 4+ 1→2+ 1) value the data for the nucleus 70Zn at N=40 remain conflicting. Such an increase in deformation is in agreement with the reduction of the excitation energies of the 2+ 1and 4+ 1states by adding four neutrons to 68Zn. The increase in collectivity at and beyond N=42 is in also agreement with earlier observations for the Zn nuclei [9] and the evolution along the Ni isotopic chain. The present results are compared to shell-model calculations in the jj44 model space (1 f5/2, 2p3/2, 2p1/2and 1g9/2for both protons and neutrons) using the jj44c [34,35] and JUN45 [36] residual interactions. The calculations have been performed with the KSHELL code [38]. In all shell-model calculations, effective charges determined for this model space, (ep, en) =(1.5, 1.1), and g-factors, geff s=0.7gfree s[36,39], were used when calculating transition probabilities. For the harmonic oscillator potential employed to calculate the transition rates, we used ¯ hω=41A−1/3. The results are presented in Table 1and Figs. 3and 4. The calculations all reproduce the excitation energies as well as the magnitude and trend of the B(E2)values well. The rather steep increase in B(E2; 2+ 1→0+ 1)values from 68Zn to 72Zn is better described using the jj44c interaction. Experimentally, we observe a similar increase for the B(E2; 4+ 1→2+ 1)strength which is not fully reflected in any of our calculations, but a kink is observed at N=38 using the jj44c interaction. Overall, the two interactions produce rather similar results. Looking now into the wave function composition of states, for 72Zn the majority of neutron configurations for the 0+ 1, 2+ 1, and 4+ 1states have two neutrons excited from the ν0f5/2, 1p1/2, or 1p3/2orbitals to the 0g9/2orbital above N=40. This scattering of neutron pairs above the N=40 harmonic oscillator gap can be understood as an effect arising from polarization of the Z=28 core [40,41]. Note that all interactions used here reflect the core polarization only indirectly through their fitted effective matrix elements and effective charges. Our calculations are consistent with the assumption that core polarization and increased ν0g9/2occu- pation play a vital role in the increased B(E2)values and lowering of excitation energies in the ground state band beyond N=38. The experimental results clearly indicate an enhanced B(E2; 4+ 1→2+ 1) strength beyond N=40. These results will serve future more sophisticated calculations as bench mark. It is now interesting to study the shape and the nature of the deformation of 72Zn. As shown in Fig. 4the 4+ 1, 2+ 2, and 0+ 2states lie close in energy as expected in a vibrational model, where the two-phonon excitations are located at twice the energy of the one-phonon 2+ 1state. However, for Ji=4+ 1, 2+ 2, 0+ 2a constant ratio B(E2; Ji→2+ 1)/B(E2; 2+ 1→0+ 1) =2would be expected in the vibrational model, while this is clearly not experimentally observed. On the other hand, the 72Zn nucleus can also not be described assuming a rigid axial deformation, in which case the quadrupole moment is related to the B(E2)value |QS(2+ 1)|=2 716πB(E2;2+ 1→0+ 1). (1) For the present case, this yields |QS(2+ 1)| =38.4(19)efm2, significantly larger than determined from the present Coulomb-excitation measurement, suggesting that triaxiality plays a major role in 72Zn. It is therefore intriguing to compare 72Zn to the geometric triaxial Davydov-Filippov model [2]. The ratios of the excitation energies of the 2+ 2and 4+ 1states to the one of the first excited 2+ 1state as well as the B(E2; 2+ 2→0+ 1)/B(E2; 2+ 1→0+ 1)and B(E2; 2+ 2→2+ 1)/B(E2; 2+ 1→0+ 1)ratios are well reproduced by assuming a static triaxial deformation with γ=22 −25◦. It is, however, impossible to experimentally distinguish between γ-soft and γ-rigid deformation based on these arguments. Previous experimental and theoretical studies [9]of 72Zn suggested γ-softness for 72Zn based on the energy ratio R22 =E(2+ γ)/E(2+ 1) =2.54 close to the value 2.5 expected in the γ-soft Wilets-Jean model [3]. Although a tentative assignment for the 3+and 4+states belonging to the γband in 72Zn [9]points towards γ-softness rather than γ-rigid deformation, the γband in 72Zn is experimentally not established. 4 S. Hellgartner, D. Mücher, K. Wimmer et al. Physics Letters B 841 (2023) 137933 Fig. 3. Comparison of the B(E2)values for the 2+ 1→0+ 1(a) and 4+ 1→2+ 1(b) transitions with previous measurements. Adopted values [31,32]are shown as black circles. Previous Coulomb-excitation studies at REX-ISOLDE [8]and GANIL [7]are labeled with black squares and crosses, respectively. Gray triangles and circles represent the results from lifetime measurements [9–11]. The results of the present study are highlighted as red stars. Note that for the 4+ 1→2+ 1transition of 70Zn, the adopted value is the weighted average of the lifetime measurements of Refs. [10,11], while an older value is much higher, but potentially the transition is contaminated by the 3− 1→2+ 1transition of the same energy [32,33]. The solid and dotted-dashed lines show the results of shell-model calculations using the jj44c and JUN45 effective interactions [34–36]. A model independent measure of the nuclear shape can be obtained from rotationally invariant zero-coupled products [42,43]. The deformation is expressed in terms of the two parameters Q and δ. For a certain state s, expanding all intermediate states i, Q2=5 2Is+1 i s||E2||ii||E2||s220 IsIsIi(2) yields the quadrupole invariant Q2, which is related to the deformation βby Q2=3 4πZR3 02 β2(3) with R0=r0A1/3. The asymmetry, related to the parameter γin the Bohr Hamiltonian, is described by cos3δ, and can be obtained by summation over all combinations of intermediate states iand j Q3cos3δ=− 35 2 1 2Is+1×  i,j s||E2||ii||E2|| j j||E2||s222 IsIjIi.(4) The angle δcan then be obtained by assuming Q3cos3δ ≈ Q23/2cos3δ. Summing over the experimentally observed states and the extracted matrix elements these quantities amount to Q2 =0.185(18)e2b2and cos3δ =0.34(10)for the 0+ 1ground state. Using Eq. (3) and associating δwith the Bohr parameter γ, these yield β=0.241(12)and γ=23.3(21)◦. This suggests that 72Zn is moderately deformed and shows a significant deformation in the γdegree of freedom. Obviously, the sums in Eqs. (2) and (4)are truncated and include only experimentally measured matrix elements. The results should therefore be regarded as an approximation [44]. It would be interesting to determine the variance of Q2and cos3δto determine the rigidity in the deformation and triaxiality directions and gauge if 72Zn is γ-soft or rigid in nature. However, the statistics of the present study is not sufficient for this analysis. The method of extracting the quadrupole invariants can also be applied to the shell-model calculations. Including up to 200 states in the calculation, these yield Q(0+ 1)2 =0.203(69)and 0.166(70) for the jj44c and JUN45 effective interactions, respectively, while the triaxiality parameters amount to cos3δ =0.45(45)and 0.30(50), where the values in parentheses give the variance of the deformation parameters. This suggests a larger degree of triaxiality in the JUN45 calculations, and points to the fact that the small calculated quadrupole moment for the 2+ 1state shown in Table 1 is resulting from the superposition of oblate and prolate configurations in a γ-soft nucleus. The calculated values β=0.252(87)and β=0.228(99)for jj44c and JUN45 are in good agreement with experiment, as are the values γ=21(22)◦and γ=24(22)◦, respectively. The 5DCH calculations of Ref. [15]give β=0.239(80) and γ=26(13)◦. The obtained set of E2matrix elements was not sufficient to obtain quadrupole invariants for the 0+ 2state. It is, however, clear that the 0+ 2state is of different nature than the ground state, as indicated by the much weaker transition to the 2+ 1state, compared to the 0+ 1→2+ 1one. The structural difference can be further explored by looking at the calculated wave function composition of the 0+states. In all our calculations, the ground state has dominant wave function contributions where two neutrons from the 0f5/2, 1p1/2, or 1p3/2orbitals are excited above the N=40 subshell gap to the 0g9/2level. The excited 0+ 2state on the other hand is dominated by closed-shell configurations with only two neutrons in the 0g9/2level. In order to get more insights, we have also performed Generator Coordinate Method (GCM) calculations with the deformation parameters (β, γ)as coordinates and exact particle number and angular momentum projection (PNAMP) [14]. In the calculations, the Gogny force with the D1S parametrization was used. In Fig. 5 a), we display the potential energy surface in the PNAMP approach, i.e., without mixing of the different (β, γ)values, for J=0 ¯ h. A broad triaxial minimum has an expectation value of β=0.32(2) with γ=22(331)◦. Again, values in parentheses show the variance of the expectation values. The surface is rather soft in γ and very steep for larger βvalues, except along the prolate axis, where it shows a rather soft behavior. In panel b), we display the collective wave function of the ground state. The wave function is very extended at a deformation that is somewhat larger than the energy minimum. This is a configuration mixing effect that drives the wave function to more deformed, symmetrical shapes, while remaining soft along the γdirection. The 2+ 1and 4+ 1states (not shown) exhibit very similar wave functions to the ground state one. These results agree with the experimental findings of a ground state with an average γclose to maximum triaxiality and the increased collectivity in the 4+yrast state. The wave function of the 2+ 2state, the head of the γ-band in our calculation, is shown in panel d). Its maximum is at γ=30◦and it has the same βvalue as the ground state and is also soft in γ. Lastly, in panel c) the wave function of the 0+ 2state is displayed. It represents a welldefined configuration peaking at β≈0.45, γ≈20◦and contains 5 S. Hellgartner, D. Mücher, K. Wimmer et al. Physics Letters B 841 (2023) 137933 Fig. 4. A comparison of the experimental 72Zn level scheme to shell-model calculations in the jj44 model space (see text for details) using the jj44c [34,35]and JUN45 effective interactions [36]. The results of the beyond mean field GCM calculations are shown on the right. The width of the arrows represent the reduced E2 transition strengths. Fig. 5. Panel a): Potential energy surface in the PNAMP approach. The energy minimum has been set to zero. The black contour lines start at 1 MeV and increase in steps of 1MeV. The white dotted contours start at 0.2 MeV and increase by 0.2 MeV up to 0.8 MeV. Panels b), c) and d): Collective wave functions of the ground, the 0+ 2, and the 2+ 2states, respectively. The latter is the predicted head of the γ-band. The eight contours start at 0.2 (white dotted line) and increase in steps of 0.2. In Ref. [12] similar calculations were performed for the ground state of 72Zn with a less dense grid of (β, γ)points. small admixtures of nearly spherical shapes. The main component of this wave function, at variance with the other states, corresponds to a configuration with six particles in the ν0g9/2orbital. In conclusion, the transitional nucleus 72Zn has been studied by multiple Coulomb excitation. The high angular coverage of C- REX allowed extraction of electromagnetic matrix elements. The B(E2)value for the 2+ 1→0+ 1transition agrees with previous measurements, while the 4+ 1→2+ 1transition is more collective than previous lifetime measurements suggested. Quadrupole invariants extracted from the data show that the ground state of 72Zn is moderately deformed and with an average γclose to maximum triaxiality, in agreement with beyond mean-field GCM and the shell-model calculations. The quadrupole moments of the first and second 2+states have different signs indicating different deformation. The shell-model calculations indicate that the structure of the excited 0+ 2state is of a more spherical nature, indicated by increased shell-model configurations with two neutrons occupying the 0g9/2orbital. This is in contrast to the GCM results where the 0+ 2state is more deformed and rigid triaxial. Overall, our results place the 72Zn nucleus between the spherical 68Ni at the N=40 sub-shell closure and the (rigid) triaxial deformed Ge isotopes. Our findings also indicate the presence of distinct configurations with different shapes in 72Zn at low excitation energies. We want to thank the ISOLDE accelerator group as well as the RILIS team providing us with an excellent pure and highly intense 72Zn beam. We would like to thank B.A. Brown for providing us with the interaction input file and for stimulating discussions. This work was supported by the German BMBF under grant numbers (05P12 WOFNF, 05P12PKFNE), by the DFG (EXC 153) and by ENSAR. D.M. acknowledges the support from NSERC. 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