Shape changes in the mirror nuclei ^{70}Kr and ^{70}Se
K. Wimmer, W. Korten, P. Doornenbal, T. Arici, P. Aguilera, A. Algora, T. Ando, H. Baba, B. Blank, A. Boso, S. Chen, A. Corsi, P. Davies, G. de Angelis, G. de France, J.-P. Delaroche, D. T. Doherty, J. Gerl, R. Gernhäuser, M. Girod, D. Jenkins, S. Koyama, T. Motobayashi, S. Nagamine, M. Niikura, A. Obertelli, J. Libert, D. Lubos, T. R. Rodríguez, B. Rubio, E. Sahin, T. Y. Saito, H. Sakurai, L. Sinclair, D. Steppenbeck, R. Taniuchi, R. Wadsworth, M. Zielinska
SShape changes in the mirror nuclei Kr and Se K. Wimmer,
1, 2, 3, ∗ W. Korten, P. Doornenbal, T. Arici,
5, 6
P. Aguilera, A. Algora,
8, 9
T. Ando, H. Baba, B. Blank, A. Boso, S. Chen, A. Corsi, P. Davies, G. de Angelis, G. de France, J.-P. Delaroche, D. T. Doherty, J. Gerl, R. Gernh¨auser, M. Girod, D. Jenkins, S. Koyama, T. Motobayashi, S. Nagamine, M. Niikura, A. Obertelli, † J. Libert, D. Lubos, T. R. Rodr´ıguez, B. Rubio, E. Sahin, T. Y. Saito, H. Sakurai,
2, 3
L. Sinclair, D. Steppenbeck, R. Taniuchi, R. Wadsworth, and M. Zielinska Instituto de Estructura de la Materia, CSIC, E-28006 Madrid, Spain Department of Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France GSI Helmholtzzentrum f¨ur Schwerionenforschung, D-64291 Darmstadt, Germany Justus-Liebig-Universit¨at Giessen, D-35392 Giessen, Germany Comisi´on Chilena de Energ´ıa Nuclear, Casilla 188-D, Santiago, Chile Instituto de Fisica Corpuscular, CSIC-Universidad de Valencia, E-46071 Valencia, Spain Institute of Nuclear Research of the Hungarian Academy of Sciences, Debrecen H-4026, Hungary CENBG, CNRS/IN2P3, Universit´e de Bordeaux, F-33175 Gradignan, France Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy Department of Physics, University of York, YO10 5DD York, United Kingdom Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Legnaro, I-35020 Legnaro, Italy GANIL, CEA/DSM-CNRS/IN2P3, F-14076 Caen Cedex 05, France CEA, DAM, DIF, F-91297 Arpajon, France Physik Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany Departamento de F´ısica Te´orica and Centro de Investigaci´on Avanzada en F´ısica Fundamental,Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain Department of Physics, University of Oslo, PO Box 1048 Blindern, N-0316 Oslo, Norway (Dated: January 8, 2021)We studied the proton-rich T z = − Kr through inelastic scattering at intermediateenergies in order to extract the reduced transition probability, B ( E
2; 0 + → + ). Comparison withthe other members of the A = 70 isospin triplet, Br and Se, studied in the same experiment,shows a 3 σ deviation from the expected linearity of the electromagnetic matrix elements as a functionof T z . At present, no established nuclear structure theory can describe this observed deviationquantitatively. This is the first violation of isospin symmetry at this level observed in the transitionmatrix elements. A heuristic approach may explain the anomaly by a shape change between themirror nuclei Kr and Se contrary to the model predictions.
The strong interaction is independent of the electriccharge of a particle, its Hamiltonian commutes withthe isobaric spin operator T . Within this isobaric spinsymmetry, the proton ( T z = − /
2) and the neutron( T z = +1 /
2) are two representations of a particle, thenucleon [1]. Electromagnetic effects violate isobaric spinsymmetry and the light quark mass difference ( m u (cid:54) = m d )results in a larger neutron mass than the mass of a pro-ton [2] making the neutron unstable. The relative massdifference between neutrons and protons is only 0.0013,suggesting that the symmetry breaking related to thestrong interaction is rather small and the observable ef-fects are dominated by the electromagnetic interaction.Precise measurements of the nn , pp , and np scatteringlength, for example, and careful correction of all electro-magnetic effects nevertheless demonstrated that proton-proton ( T z = − T z = 0), or neutron-neutron ( T z = +1) interactions are different [2].In atomic nuclei, the charge independence of the nu-clear interaction implies: (i) exactly degenerate energiesof isobaric multiplets [3], (ii) pure isospin quantum num- bers and no isospin mixing in nuclear states, and (iii)identical wave functions for the members of an isobaricmultiplet. For charge dependent two-body interactionsthe masses of isobaric nuclei depend on the isospin pro-jection T z . The isobaric multiplet mass equation (IMME)relates the mass-excess of isobars as a quadratic functionof T z . Deviations from the IMME indicate isospin mix-ing, isospin symmetry breaking or the presence of three-body forces.Isospin symmetry is typically studied through MirrorEnergy Differences (MED) to test the charge symmetryof the nuclear interaction and Triplet Energy Differences(TED) to test charge independence of the nuclear interac-tion. Very recently, isospin symmetry breaking has beenobserved in the A = 73 mirror pair, where the groundstate spins of Sr and Br differ [4]. The excitation en-ergy difference of the first two states in Br is, however,only 27 keV, so that the absolute scale of this violation isvery small and comparable to other cases [5]. The anal-ysis of mirror energy differences of excited states showsthat isospin non-conserving interactions are required in a r X i v : . [ nu c l - e x ] J a n addition to the Coulomb force to reproduce the observa-tion with shell model calculations [6]. The origin of theadditional, phenomenological terms in the interaction isnot yet understood. Excitation energies alone, however,do not reveal the isospin purity of states and do not probethe mirror symmetry of the wave functions.An alternative and more rigorous way to test isospinsymmetry are electromagnetic matrix elements. In con-trast to the excitation energies, the matrix elements alsoprobe the properties (ii) and (iii) of the charge inde-pendence of the nuclear interaction. In such a case theisospin dependence of the proton matrix element for a T = 1 triplet is given by a simple linear relation [7] M p ( T z ) = 12 ( M − M T z ) (1)with the isoscalar ( M ) and isovector ( M ) matrix el-ements. Experimentally, this linearity can be testedthrough measurements of the reduced electromagnetictransition probability B ( Eλ ; J i → J f ) = | M p ( T z ) | J i + 1 (2)of the decay of the first T, J π = 1 , + to 1 , + states inthe triplet. This has been studied for T = 1 triplets with22 ≤ A ≤
50 [8–10] and no deviation from the expectedlinear trend was detected within the experimental uncer-tainties. Isospin mixing of T = 0 and T = 1 states inthe odd-odd system could potentially disturb the linear-ity, but different systematic uncertainties from differentexperiments make it difficult to draw conclusions [8].In this letter, we present the first case where the elec-tromagnetic matrix elements significantly deviate fromthe linear trend of Eq. 1. The A = 70 nuclei have beenchosen for this because previous experimental investiga-tions of the Coulomb energy differences of Se and Brfound an anomalous behavior [11], which could be in-terpreted as a shape change between the isobars [12].Spectroscopy of Kr at T z = − Se and Kr [15]. Electromagnetic transitionmatrix elements, on the other hand, also allow for thedetermination of the shape or deformation of a nucleus.In a rotational model, the B ( E
2) value is related to themagnitude of the (intrinsic) quadrupole moment Q , andthe absolute value of the deformation β [16].The experiment was performed at the Radioactive Iso-tope Beam Factory, operated by RIKEN Nishina Centerand CNS, University of Tokyo. Nuclei along the N = Z line were produced by projectile fragmentation of an in-tense Kr primary beam at an energy of 345 A MeV.The reaction products were selected and identified inthe BigRIPS separator [17] using the Bρ − ∆ E − T OF method. The average intensity of the Kr beam was 15 pps with a total fraction of 0.9% in the secondarybeam. At the final focus of the BigRIPS separatorthe beam impinged on 926(2) mg/cm thick Au and703(7) mg/cm thick Be targets. The targets were sur-rounded by the DALI2 detector array [18], consisting of186 individual NaI(Tl) crystals. Reaction products wereidentified in the ZeroDegree spectrometer [17] using thesame technique as for BigRIPS. Further details of theexperiment can be found in [14, 19].The Doppler-corrected γ -ray energy spectra for Krimpinging on the Au and Be targets are shown in Fig. 1.In both spectra the decay of the 885 keV excited 2 + state
600 800 1000 1200 1400 1600 1800 E γ (keV) C o un t s /10 k e V Au( Kr, Kr+ γ ) backgroundsimulationtotal600 800 1000 1200 1400 1600 1800 E γ (keV) C o un t s /16 k e V Be( Kr, Kr+ γ ) + → + ( + ) → + ( + ) → + unp l a ce d ( − ) → + FIG. 1. Doppler-corrected γ -ray energy spectrum for the in-elastic scattering of Kr on a
Au (top) and Be (bottom)targets. The Doppler correction assumes γ -ray emission at thevelocity behind the target. The data are fitted with simulatedresponse functions for the transition at 884 keV and a contin-uous background (red). For the Au target data, only forwardDALI2 crystals ( θ lab < ◦ ) are shown to reduce backgroundfrom atomic processes. For the Be target data see Ref. [14]for further details. to the ground state is observed. The γ -ray yield has beendetermined by fitting a simulated response function anda continuous double exponential background to the data.The simulated response function assumes a level lifetimeof 2.8 ps, consistent with the results extracted from theCoulomb excitation cross section (see below). For thescattering with the Au target, only the data from theforward DALI2 crystals have been taken into accountto reduce the low-energy background from atomic pro-cesses. The angular distribution for the E +1 state,indirect feeding from higher-lying states has to be sub-tracted. For the inelastic scattering off the Be target,the yield for states above the 2 +1 state was subtracted tocorrect for indirect feeding as described in Ref. [14]. Forthe data taken with the Au target, the de-excitation fromthe 2 +2 state at 1478 keV to the 2 +1 state is not observed.The cross section for the excitation of the 2 +1 state in Kramounts to 349(36) mb (see Table I). Adding a transi-tion at 594 keV to the fit of the spectrum shown in thetop panel of Fig. 1 results in an upper limit for the crosssection σ (2 +2 ) of 15 mb. States above the 2 +2 state areexpected to contribute even less. In order to accountfor them and the uncertainty related to the feeding ofthe 2 +2 state an additional 15 mb has been added to thesystematic uncertainty. The corrections and previouslyquoted systematic uncertainty for the cross section aretaken into account when the B ( E
2; 0 + → + ) values aredetermined from the measured cross sections.Within the same spectrometer setting, also the isobars Br ( E (2 +1 ) = 934 keV) and Se ( E (2 +1 ) = 954 keV)were transmitted. For the former, an isomeric 9 + stateat 2292 keV [20] allows, in principle, to only extract alower limit for the excitation cross sections of the 2 +1 state. However, no transition besides the 2 +1 → +1 de-cay has been observed in the scattering off the Au target,indicating a small isomeric ratio in the beam or a small B ( E
2) value for the states above the isomer. For Se,statistics are limited because the acceptance of BigRIPSwas optimized for the more exotic
N < Z nuclei. Basedon the systematics of the less exotic Kr isotopes, a low-lying excited 0 + state might be present in the beam asan isomeric contamination. In the mirror nucleus Seno such state is known. The 0 +2 state candidate is lo-cated at 2010 keV [21] and is thus short lived. Theoret-ical calculations predict the 0 +2 in Kr at considerablyhigher energy than the 2 +1 state [22, 23] as well, so thatits lifetime would be much shorter than the flight timeto the BigRIPS focal plane. In the analysis of nucleonremoval reactions from the same experiment no evidencefor a low-lying 0 + state was found in either of the twonuclei [14]. Due to the low beam intensity, it was notpossible to search for an isomeric state in Kr as it wasdone for Kr [19], where an isomeric ratio of 4(1)% wasfound. In the following extraction of the B ( E
2) values itwas assumed that the A = 70 beam particles are in theirrespective ground state, when reaching the secondary re-action target.The cross sections measured with both targets for allthree beams are listed in Table I. Besides the statisticaluncertainty resulting from the fitting of the γ -ray spec-trum and the subtraction of feeding in case of the Be tar-get data, a number of systematic uncertainties contribute TABLE I. Measured cross sections and deduced nuclear de-formation length δ N , proton matrix elements M ( E B ( E
2; 0 + → + ) values for the A = 70 , T = 1 triplet. Thetotal uncertainties are listed together with the individual con-tributions of statistical, systematic, and theoretical uncertain-ties. Kr Br Se E (2 + ) (keV) 885 934 954 σ (2 +1 ) Be (mb) 14.5(46)(10) 14.6(5)(8) 15.1(53)(28) σ (2 +1 ) Au (mb) 349(36)(27) 201(11)(20) 234(70)(43) δ N (fm) 1.10(19) 1.09(3) 1.11(22)∆ stat δ N (fm) 0.19 0.02 0.20∆ syst δ N (fm) 0.04 0.03 0.10 M ( E
2) (efm ) 52.2(43) 38.1(31) 40.7(82)∆ stat M ( E
2) (efm ) 2.8 1.2 6.8∆ syst M ( E
2) (efm ) 2.1 2.2 4.1∆ theo M ( E
2) (efm ) 2.5 1.8 2.0 B ( E
2) (e fm ) 2726(451) 1454(233) 1659(659)∆ stat B ( E
2) (e fm ) 294 91 543∆ syst B ( E
2) (e fm ) 224 165 336∆ theo B ( E
2) (e fm ) 258 137 164 to the total uncertainty for the cross section. These in-clude the full-energy peak detection efficiency of DALI2(5%), target thickness (1%), ZeroDegree efficiency andtransmission (3% for Kr and Br, 10% for Se), trig-ger efficiency (2%), effects of the γ -ray angular distribu-tion (2%), and the unobserved indirect feeding discussedabove.The excitation of the 2 + states of interest is causedby both the electro-magnetic and the nuclear interactionbetween target and projectile. These two contributionsinterfere and can not be disentangled. The extractionof the nuclear deformation length δ N and the reducedtransition probability B ( E
2) from the measured crosssections requires therefore a consistent reaction modelanalysis. The procedure is described in [19] in detail.Reaction model calculations were performed with a mod-ified version of the distorted wave coupled channels codeFRESCO [24] using optical model potentials calculatedusing the method described in [25]. Both the input nu-clear deformation length and the B ( E
2) value for theprojectile nucleus were adjusted to reproduce simultane-ously the measured cross sections for the Be and Au tar-get. The resulting nuclear deformation lengths and E δ N which propagate to the determi-nation of the B ( E
2) values. A detailed discussion of theexperimental and theoretical uncertainties is presentedin [19].In order to validate the analysis procedure, the resultsfor the present experiment are compared to previous mea-surements of neighboring nuclei using both Coulomb ex-citation and lifetime measurements. The B ( E
2) valuesfor the N = Z nuclei Kr and Se as well as the A = 70isobars are shown in Fig. 2. In all four cases the presentresults agree with the previous measurements. Kr Se Kr Br Se B ( E ; + → + ) this workprev. data FIG. 2. Summary of the results for the B ( E
2; 0 +1 → +1 ) val-ues extracted in the present work. The results for Kr werealready presented in [19]. The error bars indicate statisticaluncertainties, while the additional caps show the total un-certainties including statistical, systematical, and uncertain-ties arising from the reaction theory calculations. For Krand Se the statistics uncertainty is smaller than the sym-bol size. Previous experimental results are taken from [26–30]and shown by the open symbols.
The matrix elements for the A = 70 triplet are shownin Fig. 3. It can be seen that the value for Kr clearlydeviates from the negative trend indicated by the pre-viously known M p values for Br and Se. A lin-ear fit for these latter two nuclei with Eq. 1 results in M = 76(4) efm and M = − . In order togauge the deviation from this trend, the confidence in-terval has been determined. The weighted average ofthe previously and the presently determined value for Br [29] as well as the weighted average of two previousmeasurements for Se [29, 30] were fitted using linearregression according to Eq. 1 and then extrapolated to T z = −
1. The result of the extrapolation, shown by thegreen Gaussian curve in the top panel Fig. 3 amountsto M ( E , T z = −
1) = 35 . . The experimen-tal value for Kr ( M ( E
2) = 52 . ) deviates bymore than 3 σ from this extrapolation.In many cases, especially for medium heavy nuclei,the B ( E
2) value for the proton-rich T z − M and M were extractedfrom a fit of Eq. 1 only to the T z = 0 and +1 mem-bers of the triplet [31]. The isovector matrix element M was found to be very small. For the cases with A ≥ M are all slightly negative, Kr Br Se25303540455055 M p ( E )( e f m ) weighted averagebest fit T z = 0 , +1-1 0 +1 T z M p ( E )( e f m ) HFB-5DCH HFB-SCCM FRDM12GXPF1AJUN45 VAMPIRthis workprev. data
FIG. 3. E T z for the A = 70 nuclei. (Top panel) Weighted averagesof the present and previous data are shown by the black datapoints. The linear fit of the data points at T z = +1 and 0is shown by the solid green line. The red dashed, dotted,and dash-dotted lines show the 1, 2, and 3 σ confidence in-tervals of the linear fit. The extrapolation of the probabilitydistribution to T z = − T z axis forvisualization purposes only. Squares show the results of thetheoretical calculations shown in Table II. albeit compatible, within errors, with zero [31]. The workwas extended to include the newest available data for A = 78 [32, 33] and a negative isoscalar matrix elementwas found again. The negative trend for the T z = 0 and+1 members observed for the present A = 70 case is thusnot unique.If the M p data for all three values of T z are fitted witha curve using simple linear regression, both the matrixelements for Br and Kr deviate by about 2 σ from thecurve. Fitting the matrix elements shown in Fig. 3 bya quadratic curve ( M p = a + bT z + cT z ) such as sug-gested in Ref. [9] to test isospin symmetry results in a c = 8 . coefficient, or, to compare with Fig. 5of Ref. [9], c/ a = c/M = 0 . M p may be explainedby isospin mixing of T = 0 and T = 1 states in theodd-odd system. However, isospin mixing alone cannotexplain the observed change in collectivity in Kr. Thedramatic change in the magnitude of the B ( E
2) valuebetween Se and Kr suggests a change in deformationwith larger deformation for Kr than for its mirror nu-cleus.The theoretical predictions for the A = 70 triplet aresummarized in Table II and also shown in the bottompanel of Fig. 3. Few calculations have been performed TABLE II. Selected theoretical predictions for the B ( E
2; 0 +1 → +1 ) values of the A = 70 triplet. For theshell model calculations effective charges e n = 0 . e and e p = 1 . e were used.Method B ( E
2; 0 + → + ) (e fm ) Reference Kr Br SeExperiment 2726(451) a b c HFB-5DCH 3289 2767 this workHFB-SCCM 4725 3450 this workFRDM12 4725 4465 3715 [34]GXPF1A [35] 1910 1990 2075JUN45 [36] 2325 2085 1885VAMPIR 2945 2630 2365 [37] a present work b weighted average of present work and Ref. [29] c weighted average of present work and Refs. [29, 30] for all three A = 70 isotopes within the same theo-retical framework. The Hartree-Fock-Bogoliubov-based(HFB) models [22, 23] were only applied to the even-even nuclei. They predict shape coexistence between anoblate and a triaxial shape with very little difference inboth Se and Kr, but generally too large B ( E
2) val-ues. The increase in deformation towards Kr is morepronounced in the symmetry conserving configuration-mixing (SCCM) method [23] than in the five-dimensionalcollective quadrupole Hamiltonian (5DCH) treatment toaccount for the configuration mixing. The Finite-RangeDroplet-Model (FRDM12) [34] predicts only the ground-state deformation parameter β on the mean-field level.The B ( E
2) values shown in Table II have been calculatedassuming a simple rotor model. These calculations againover-estimate the absolute deformation, and show an in-crease of deformation towards Kr similar to the SCCM.Shell-model calculations with the GXPF1A [35, 38] andJUN45 [36] effective interactions were previously per-formed for Se and Br and reproduce the observed B ( E
2) values quite well [29]. We have extended thesecalculations to include Kr and find a decreasing lin-ear trend as expected from Eq. 1 for the GXPF1A effec-tive interaction, in contrast to our experimental findings.The results obtained with the JUN45 effective interactionshow a positive trend. Both shell model calculations areable to reproduce the absolute magnitude of the B ( E Kr compared to Br. It should benoted that the inclusion of isospin non-conserving termsinto the interaction, that are commonly added to explainmirror energy differences [5, 39], have a negligible effecton the calculated B ( E
2) values. Finally, several calcu-lations have been published using the complex excitedVAMPIR model [15, 37, 40, 41]. The published valuesvary considerably over time, demonstrating the difficultyto correctly describe these shape-changing nuclei. Thelatest results show shape coexistence between oblate andprolate shapes with a moderate, continuous increase ofthe B ( E
2) values toward Kr [37], again in contradic-tion to the experimental result. However, it is interest-ing to note that this model is the only one to predict ashape change along the isobars since the wave functionsof low-lying yrast states in Kr and Br are dominatedby prolate components, while the oblate component be-comes more important in Se. The latter is also consis-tent with the conclusions of [30].In conclusion, while several calculations show a slightincrease of the matrix element (and hence the deforma-tion) from Se to Kr no calculation is able to describethe absolute B ( E
2) values and the strong increase be-tween the mirror nuclei Se and Kr observed experi-mentally.In summary, we have determined the B ( E
2; 0 +1 → +1 )value for the T z = − Kr for the first time. Inaddition, previously known B ( E
2; 0 +1 → +1 ) value val-ues for the isobars Br and Se were confirmed. The A = 70 triplet is the heaviest one where the B ( E
2) val-ues for all three members are experimentally determined.The B ( E
2) value of Kr is significantly larger than in theother members of the T = 1 triplet Br and Se. Protonmatrix elements for the triplet have been extracted fromthe B ( E
2) values, and they should exhibit a simple linearrelation as a function of isospin T z . The large value de-termined for Kr deviates by 3 σ from the extrapolationbased on the other two nuclei. This suggests that a sub-stantial shape change occurs between the oblate Se [30]and (presumably prolate) Kr [37]. None of the currentnuclear structure models is able to explain the increaseof the B ( E
2) value determined in this work.We would like to thank the RIKEN accelerator andBigRIPS teams for providing the high intensity beams.This work has been supported by UK STFC undergrant numbers ST/L005727/1 and ST/P003885/1, theSpanish Ministerio de Econom´ıa y Competitividad un-der grants FPA2011-24553 and FPA2014-52823-C2-1-P,the Program Severo Ochoa (SEV-2014-0398), and theSpanish MICINN under PGC2018-094583-B-I00. K. W.acknowledges the support from the Spanish Ministeriode Econom´ıa y Competitividad RYC-2017-22007. A. O.acknowledges the support from the European ResearchCouncil through the ERC Grant No. MINOS-258567. ∗ Corresponding author: [email protected] † Present address: Institut f¨ur Kernphysik Technische Uni-versit¨at Darmstadt Germany[1] W. Heisenberg, Z. Phys. , 1 (1932).[2] G. A. Miller, A. K. Opper, and E. J. Stephenson, Ann.Rev. Nucl. Part. Sci. , 253 (2006).[3] E. Wigner, Phys. Rev. , 106 (1937).[4] D. E. M. Hoff, A. M. Rogers, S. M. Wang, P. C. Bender,K. Brandenburg, K. Childers, J. A. Clark, A. C. Dom-bos, E. R. Doucet, S. Jin, R. Lewis, S. N. Liddick, C. J.Lister, Z. Meisel, C. Morse, W. Nazarewicz, H. Schatz,K. Schmidt, D. Soltesz, S. K. Subedi, and S. Wani-ganeththi, Nature , 52 (2020).[5] M. A. Bentley and S. M. Lenzi, Prog. Part. Nucl. Phys. , 497 (2007).[6] A. P. Zuker, S. M. Lenzi, G. Mart´ınez-Pinedo, andA. Poves, Phys. Rev. Lett. , 142502 (2002).[7] A. M. Bernstein, V. R. Brown, and V. A. Madsen, Phys.Rev. Lett. , 425 (1979).[8] F. M. Prados Est´evez, A. M. Bruce, M. J. Taylor,H. Amro, C. W. Beausang, R. F. Casten, J. J. Ressler,C. J. Barton, C. Chandler, and G. Hammond, Phys.Rev. C , 014309 (2007).[9] A. Boso, S. Milne, M. Bentley, F. Recchia, S. Lenzi,D. Rudolph, M. Labiche, X. Pereira-Lopez, S. Afara,F. Ameil, T. Arici, S. Aydin, M. Axiotis, D. Barrien-tos, G. Benzoni, B. Birkenbach, A. Boston, H. Boston,P. Boutachkov, A. Bracco, A. Bruce, B. Bruyneel,B. Cederwall, E. Clement, M. Cortes, D. Cullen,P. D´esesquelles, Z. Dombr´adi, C. Domingo-Pardo,J. Eberth, C. Fahlander, M. Gelain, V. Gonz´alez,P. John, J. Gerl, P. Golubev, M. G´orska, A. Got-tardo, T. Grahn, L. Grassi, T. Habermann, L. Harkness-Brennan, T. Henry, H. Hess, I. Kojouharov, W. Ko-rten, N. Lalovi´c, M. Lettmann, C. Lizarazo, C. Louchart-Henning, R. Menegazzo, D. Mengoni, E. Merchan,C. Michelagnoli, B. Million, V. Modamio, T. Moeller,D. Napoli, J. Nyberg, B. N. Singh], H. Pai, N. Pietralla,S. Pietri, Z. Podolyak, R. P. Vidal], A. Pullia, D. Ralet,G. Rainovski, M. Reese, P. Reiter, M. Salsac, E. San-chis, L. Sarmiento, H. Schaffner, L. Scruton, P. Singh,C. Stahl, S. Uthayakumaar, J. Valiente-Dob´on, andO. Wieland, Phys. Lett. B , 134835 (2019).[10] M. M. Giles, B. S. Nara Singh, L. Barber, D. M. Cullen,M. J. Mallaburn, M. Beckers, A. Blazhev, T. Braun-roth, A. Dewald, C. Fransen, A. Goldkuhle, J. Jolie,F. Mammes, C. M¨uller-Gatermann, D. W¨olk, K. O. Zell,S. M. Lenzi, and A. Poves, Phys. Rev. C , 044317(2019).[11] G. de Angelis, T. Martinez, A. Gadea, N. Marginean,E. Farnea, E. Maglione, S. Lenzi, W. Gelletly, C. Ur,D. Napoli, T. Kroell, S. Lunardi, B. Rubio, M. Axio-tis, D. Bazzacco, A. B. Sona, P. Bizzeti, P. Bednarczyk,A. Bracco, F. Brandolini, F. Camera, D. Curien, M. D.Poli, O. Dorvaux, J. Eberth, H. Grawe, R. Menegazzo,G. Nardelli, J. Nyberg, P. Pavan, B. Quintana, C. R. Al-varez, P. Spolaore, T. Steinhart, I. Stefanescu, O. Thelen,and R. Venturelli, Eur. Phys. Jour. A , 51 (2001).[12] B. S. Nara Singh, A. N. Steer, D. G. Jenkins,R. Wadsworth, M. A. Bentley, P. J. Davies, R. Glover,N. S. Pattabiraman, C. J. Lister, T. Grahn, P. T. Greenlees, P. Jones, R. Julin, S. Juutinen, M. Leino,M. Nyman, J. Pakarinen, P. Rahkila, J. Sar´en, C. Sc-holey, J. Sorri, J. Uusitalo, P. A. Butler, M. Dimmock,D. T. Joss, J. Thomson, B. Cederwall, B. Hadinia, andM. Sandzelius, Phys. Rev. C , 061301 (2007).[13] D. M. Debenham, M. A. Bentley, P. J. Davies, T. Haylett,D. G. Jenkins, P. Joshi, L. F. Sinclair, R. Wadsworth,P. Ruotsalainen, J. Henderson, K. Kaneko, K. Aura-nen, H. Badran, T. Grahn, P. Greenlees, A. Herza´aˇn,U. Jakobsson, J. Konki, R. Julin, S. Juutinen, M. Leino,J. Sorri, J. Pakarinen, P. Papadakis, P. Peura, J. Par-tanen, P. Rahkila, M. Sandzelius, J. Sar´en, C. Scholey,S. Stolze, J. Uusitalo, H. M. David, G. de Angelis, W. Ko-rten, G. Lotay, M. Mallaburn, and E. Sahin, Phys. Rev.C , 054311 (2016).[14] K. Wimmer, W. Korten, T. Arici, P. Doornenbal,P. Aguilera, A. Algora, T. Ando, H. Baba, B. Blank,A. Boso, S. Chen, A. Corsi, P. Davies, G. de Ange-lis, G. de France, D. Doherty, J. Gerl, R. Gernh¨auser,D. Jenkins, S. Koyama, T. Motobayashi, S. Nagamine,M. Niikura, A. Obertelli, D. Lubos, B. Rubio, E. Sahin,T. Saito, H. Sakurai, L. Sinclair, D. Steppenbeck,R. Taniuchi, R. Wadsworth, and M. Zielinska, Phys.Lett. B , 441 (2018).[15] A. Petrovici, Phys. Rev. C , 014302 (2015).[16] A. Bohr and B. R. Mottelson, Nuclear Structure , Vol. 2(Benjamin, Reading, Massachusetts, 1975).[17] T. Kubo, D. Kameda, H. Suzuki, N. Fukuda,H. Takeda, Y. Yanagisawa, M. Ohtake, K. Kusaka,K. Yoshida, N. Inabe, T. Ohnishi, A. Yoshida,K. Tanaka, and Y. Mizoi, Prog. Theo.Exp. Phys. (2012), 10.1093/ptep/pts064,03C003, http://oup.prod.sis.lan/ptep/article-pdf/2012/1/03C003/11595011/pts064.pdf.[18] S. Takeuchi, T. Motobayashi, Y. Togano, M. Matsushita,N. Aoi, K. Demichi, H. Hasegawa, and H. Murakami,Nucl. Instr. Meth. A , 596 (2014).[19] K. Wimmer, T. Arici, W. Korten, P. Doornenbal,J. P. Delaroche, M. Girod, J. Libert, T. R. Rodr´ıguez,P. Aguilera, A. Algora, T. Ando, H. Baba, B. Blank,A. Boso, S. Chen, A. Corsi, P. Davies, G. de Angelis,G. de France, D. T. Doherty, J. Gerl, R. Gernh¨auser,T. Goigoux, D. Jenkins, G. Kiss, S. Koyama, T. Mo-tobayashi, S. Nagamine, M. Niikura, S. Nishimura,A. Obertelli, D. Lubos, V. H. Phong, B. Rubio, E. Sahin,T. Y. Saito, H. Sakurai, L. Sinclair, D. Steppenbeck,R. Taniuchi, V. Vaquero, R. Wadsworth, J. Wu, andM. Zielinska, Eur. Phys. Jour. A , 159 (2020).[20] M. Karny, L. Batist, D. Jenkins, M. Kavatsyuk, O. Ka-vatsyuk, R. Kirchner, A. Korgul, E. Roeckl, andJ. ˙Zylicz, Phys. Rev. C , 014310 (2004).[21] R. Wadsworth, L. P. Ekstrom, G. D. Jones, F. Kearns,T. P. Morrison, P. J. Twin, and N. J. Ward, J. Phys. G, 1403 (1980).[22] J. P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hi-laire, S. P´eru, N. Pillet, and G. F. Bertsch, Phys. Rev.C , 014303 (2010).[23] T. R. Rodr´ıguez, Phys. Rev. C , 034306 (2014).[24] I. J. Thompson, Comp. Phys. Rep. , 167 (1988);A. Moro, (2018), priv. comm.[25] T. Furumoto, W. Horiuchi, M. Takashina, Y. Yamamoto,and Y. Sakuragi, Phys. Rev. C , 044607 (2012).[26] A. Gade, D. Bazin, A. Becerril, C. M. Campbell, J. M.Cook, D. J. Dean, D.-C. Dinca, T. Glasmacher, G. W. Hitt, M. E. Howard, W. F. Mueller, H. Olliver, J. R.Terry, and K. Yoneda, Phys. Rev. Lett. , 022502(2005).[27] H. Iwasaki, A. Lemasson, C. Morse, A. Dewald,T. Braunroth, V. M. Bader, T. Baugher, D. Bazin, J. S.Berryman, C. M. Campbell, A. Gade, C. Langer, I. Y.Lee, C. Loelius, E. Lunderberg, F. Recchia, D. Smalley,S. R. Stroberg, R. Wadsworth, C. Walz, D. Weisshaar,A. Westerberg, K. Whitmore, and K. Wimmer, Phys.Rev. Lett. , 142502 (2014).[28] A. Obertelli, T. Baugher, D. Bazin, J. P. Delaroche,F. Flavigny, A. Gade, M. Girod, T. Glasmacher, A. Goer-gen, G. F. Grinyer, W. Korten, J. Ljungvall, S. McDaniel,A. Ratkiewicz, B. Sulignano, and D. Weisshaar, Phys.Rev. C , 031304 (2009).[29] A. Nichols, R. Wadsworth, H. Iwasaki, K. Kaneko,A. Lemasson, G. de Angelis, V. Bader, T. Baugher,D. Bazin, M. Bentley, J. Berryman, T. Braunroth,P. Davies, A. Dewald, C. Fransen, A. Gade, M. Hack-stein, J. Henderson, D. Jenkins, D. Miller, C. Morse,I. Paterson, E. Simpson, S. Stroberg, D. Weisshaar,K. Whitmore, and K. Wimmer, Phys. Lett. B , 52(2014).[30] J. Ljungvall, A. G¨orgen, M. Girod, J.-P. Delaroche,A. Dewald, C. Dossat, E. Farnea, W. Korten, B. Melon,R. Menegazzo, A. Obertelli, R. Orlandi, P. Petkov,T. Pissulla, S. Siem, R. P. Singh, J. Srebrny, C. Theisen,C. A. Ur, J. J. Valiente-Dob´on, K. O. Zell, andM. Zieli´nska, Phys. Rev. Lett. , 102502 (2008).[31] C. Morse, H. Iwasaki, A. Lemasson, A. Dewald,T. Braunroth, V. Bader, T. Baugher, D. Bazin, J. Berry-man, C. Campbell, A. Gade, C. Langer, I. Lee, C. Loelius, E. Lunderberg, F. Recchia, D. Smalley,S. Stroberg, R. Wadsworth, C. Walz, D. Weisshaar,A. Westerberg, K. Whitmore, and K. Wimmer, Phys.Lett. B , 198 (2018).[32] A. Lemasson, H. Iwasaki, C. Morse, D. Bazin,T. Baugher, J. S. Berryman, A. Dewald, C. Fransen,A. Gade, S. McDaniel, A. Nichols, A. Ratkiewicz,S. Stroberg, P. Voss, R. Wadsworth, D. Weisshaar,K. Wimmer, and R. Winkler, Phys. Rev. C , 041303(2012).[33] R. D. O. Llewellyn, M. A. Bentley, R. Wadsworth,H. Iwasaki, J. Dobaczewski, G. de Angelis, J. Ash,D. Bazin, P. C. Bender, B. Cederwall, B. P. Crider,M. Doncel, R. Elder, B. Elman, A. Gade, M. Grinder,T. Haylett, D. G. Jenkins, I. Y. Lee, B. Longfellow,E. Lunderberg, T. Mijatovi´c, S. A. Milne, D. Muir, A. Pa-store, D. Rhodes, and D. Weisshaar, Phys. Rev. Lett. , 152501 (2020).[34] P. M¨oller, A. Sierk, T. Ichikawa, and H. Sagawa, At.Data Nucl. Data Tables , 1 (2016).[35] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki,Eur. Phys. J. A , 499 (2005).[36] M. Honma, T. Otsuka, T. Mizusaki, and M. Hjorth-Jensen, Phys. Rev. C , 064323 (2009).[37] A. Petrovici, O. Andrei, and A. Chilug, Phys. Scr. ,114001 (2018).[38] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki,Phys. Rev. C , 034335 (2004).[39] M. A. Bentley, S. M. Lenzi, S. A. Simpson, and C. A.Diget, Phys. Rev. C , 024310 (2015).[40] A. Petrovici, Phys. Scr. , 064003 (2017).[41] A. Petrovici, Phys. Rev. C97