Lifetime Measurements in the Even-Even ^{102-108}Cd Isotopes
M. Siciliano, J.J. Valiente-Dobón, A. Goasduff, T.R. Rodríguez, D. Bazzacco, G. Benzoni, T. Braunroth, N. Cieplicka-Ory?czak, E. Clément, F.C.L. Crespi, G. de France, M. Doncel, S. Ertürk, C. Fransen, A. Gadea, G. Georgiev, A. Goldkuhle, U. Jakobsson, G. Jaworski, P.R. John, I. Kuti, A. Lemasson, H. Li, A. Lopez-Martens, T. Marchi, D. Mengoni, C. Michelagnoli, T. Mijatovi?, C. Müller-Gatermann, D.R. Napoli, J. Nyberg, M. Palacz, R.M. Pérez-Vidal, B. Say?i, D. Sohler, S. Szilner, D. Testov
LLifetime Measurements in the Even-Even − Cd Isotopes
M. Siciliano,
1, 2, 3
J.J. Valiente-Dob´on, A. Goasduff,
2, 3, 4
T.R. Rodr´ıguez, D. Bazzacco, G. Benzoni, T. Braunroth, N. Cieplicka-Ory´nczak,
6, 8
E. Cl´ement, F.C.L. Crespi,
6, 10
G. de France, M. Doncel, S. Ert¨urk, C. Fransen, A. Gadea, G. Georgiev, A. Goldkuhle, U. Jakobsson, G. Jaworski,
2, 16
P.R. John,
3, 4, 17
I. Kuti, A. Lemasson, H. Li, A. Lopez-Martens, T. Marchi, D. Mengoni,
3, 4
C. Michelagnoli,
9, 19
T. Mijatovi´c, C. M¨uller-Gatermann,
7, 21
D.R. Napoli, J. Nyberg, M. Palacz, R.M. P´erez-Vidal,
13, 2
B. Say˘gi,
2, 23, 24
D. Sohler, S. Szilner, and D. Testov
3, 4, 25 Irfu/CEA, Universit´e Paris-Saclay, Gif-sur-Yvette, France. INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy. Dipartimento di Fisica e Astronomia, Universit`a di Padova, Padua, Italy. INFN, Sezione di Padova, Padua, Italy. Universidad Aut´onoma de Madrid, Madrid, Spain. INFN, Sezione di Milano, Milan, Italy. Institut f¨ur Kernphysik, Universit¨at zu K¨oln, Cologne, Germany. Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow, Poland Grand Acc´el´erateur National d’Ions Lourds, Irfu/CEA/DRF and CNRS/IN2P3, Caen, France. Dipartimento di Fisica, Universit`a di Milano, Milan, Italy. Universidad de Salamanca, Salamanca, Spain. ¨Omer Halisdemir ¨Universitesi, Ni˘gde, Turkey. Instituto de F´ısica Corpuscular, CSIC-Universidad de Valencia, Valencia, Spain. IJCLab, Universit´e Paris-Saclay, Orsay, France. Department of Physics, Royal Institute of Technology, Stockholm, Sweden. Heavy Ion Laboratory, University of Warsaw, Warsaw, Poland. Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, Darmstadt, Germany. Institute of Nuclear Research ATOMKI, Debrecen, Hungary. Institut Laue-Langevin, Grenoble, France. Ruder Boˇskovi´c Institute and University of Zagreb, Zagreb, Croatia. Argonne National Laboratory, Lemont (IL), United States. Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden. Ege ¨Universitesi, ˙Izmir, Turkey. Department of Physics, Sakarya University, Sakarya, Turkey. Joint Institute for Nuclear Research, Dubna, Russia.
Background:
The heaviest T z = 0 doubly-magic nucleus, Sn, and the neighboring nuclei of-fer unique opportunities to investigate the properties of nuclear interaction in extreme conditions.Studies of light Sn nuclei are hindered by their relatively high mass, proton-rich character and thepresence of low-lying isomers. Having only two proton holes in the Z = 50 shell, the Cd isotopesare expected to present features 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 nucleiin the vicinity of
Sn.
Methods:
The neutron-deficient nuclei in the N ≈ Z ≈
50 region were populated using a multi-nucleon transfer reaction with a
Cd beam and a Mo target. The complete identification of thebeam-like reaction products was possible thanks to the VAMOS++ spectrometer, while the γ rayswere detected using the AGATA array. Lifetimes of excited states were determined using the RecoilDistance Doppler-Shift method, employing the Cologne differential plunger. Results:
Lifetimes of low-lying states were measured in the even-mass − Cd isotopes. Inparticular, multiple states with excitation energy up to ≈ Cd via inelastic scattering. The transition strengths corresponding to the measuredlifetimes were compared with those resulting from state-of-the-art beyond-mean-field calculationsusing 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 rotationalcharacter of the Cd nuclei, which have prolate-deformed ground states with β ≈ .
2. At thisdeformation Z = 48 becomes a closed-shell configuration, which is favored with respect to thespherical one. PACS numbers: 21.10.Tg, 23.20.Lv, 25.70.Hi, 27.60.+j, 29.30.Aj, 29.30.Kv, 29.40.Gx
In recent years, the interest in studies of nuclear structure around Z = 50 has significantly increased. a r X i v : . [ nu c l - e x ] J a n This region presents unique conditions to investigate ob-servables, such as excitation energies, quadrupole mo-ments and reduced transition probabilities, starting fromneutron-deficient nuclei close to the proton drip line, upto neutron-rich isotopes towards and beyond the N = 82neutron shell closure. Consequently, the longest isotopicchains between two experimentally accessible shell clo-sures – i.e. tellurium ( Z = 52), tin ( Z = 50) and cad-mium ( Z = 48) isotopes – are being extensively studiedin order to probe the evolution of nuclear properties inboth stable and exotic nuclei.Due to the rather constant excitation energies of the2 +1 and 4 +1 states and the presence of low-lying isomersin the even-mass nuclei, the Z = 50 semi-magic Sn iso-topes have been considered for decades to be excellentexamples of pairing dominance, showing the typical fea-tures of seniority schemes [1–3]. On the other hand, the B ( E
2; 2 +1 → g.s. ) reduced transition probabilities re-main almost constant for the 106 ≤ A ≤
114 Sn nuclei,instead of following the parabolic trend expected for thepairing domination. This observation casts doubts onthe validity of the generalized seniority interpretation.In particular, recent works [4, 5] highlighted the key roleof the 4 +1 → +1 transition strengths in revealing the del-icate balance between pairing and quadrupole correla-tions in the light Sn isotopes. Furthermore, thanks tothe precise determination of the B ( E
2; 4 +1 → +1 ) valuein Sn [4], Zuker [5] proved how the sole information onthe 2 +1 states is not sufficient for an in-depth descriptionof the nuclei in this mass region: any “sufficiently good”interaction is capable of reproducing the electromagneticproperties of the 2 +1 states. B ( E ; + → + ) [ e f m ] Neutrons
CdSn
FIG. 1: (Color online) Systematics of the experimental B ( E
2; 2 +1 → + g.s. ) reduced transition probabilities for theeven-mass Cd (red squares) and Sn (blue circles) isotopicchains. Results are taken from Ref. [6]. The Cd isotopes, which have only two proton holes inthe Z = 50 shell, are expected to present features similarto those found for the Sn nuclei. For instance, Figure 1 shows that the B ( E
2; 2 +1 → + g.s. ) values display similartrends in the Sn and Cd isotopic chains, except for thelarger collectivity of the latter. Moreover, the excitationenergies of the 2 +1 and 4 +1 states in even-mass Cd nucleiare rather constant, similarly to the Sn isotopes. There-fore, one can expect that the experimental informationon the Z = 48 nuclei may not only be important in itself,but it may also provide an insight into the structure ofthe corresponding Z = 50 isotones.Based on the excitation energies of their low-lyingstates, the cadmium isotopes have been considered atextbook example of harmonic quadrupole-vibrationalnuclei [7–11] with a two-phonon triplet and a three-phonon quintuplet of levels at approximately twice andthree times the energy of the 2 +1 state, respectively.On the other hand, the electromagnetic properties ofthe Cd isotopes, i.e. quadrupole moments and transi-tion strengths, put their vibrational character in doubt.In fact, recent experimental results from multi-stepCoulomb excitation and lifetime measurements havedemonstrated a substantial disagreement with a vibra-tional structure and revealed a systematic trend of the B ( E
2) values in the even-even − Cd isotopes [12–20]. However, the lack of precise experimental informa-tion makes it difficult to assess whether the vibrationalpicture still holds for the neutron-deficient Cd nuclei.The experiment described in this work aimed at thedetermination of the 2 +1 → + g.s. and 4 +1 → +1 transitionstrengths in neutron-deficient Z ≤
50 nuclei by measur-ing lifetimes of the 2 +1 and 4 +1 states. The results concern-ing the light Sn isotopes were discussed in Ref. [4], whilethe present manuscript focuses on the lifetimes of low-lying states in even-mass − Cd. The results are com-pared with the predictions of new beyond-mean-field cal-culations using the symmetry-conserving configuration-mixing approach. The general features and the evolutionof the ground-state structure are discussed for the wholeCd isotopic chain, with a particular focus on the varietyof excited bands in
Cd.
I. EXPERIMENT
Multi-nucleon transfer (MNT) is a reaction mecha-nism widely adopted to study neutron-rich nuclei [21–23]. However, in the present experiment this reactionmechanism was applied, somehow unusually, to popu-late neutron-deficient nuclei in the vicinity of
Sn. Forthis purpose, a
Cd beam at 770 MeV energy, providedby the separated-sector cyclotron of the GANIL facil-ity, impinged on a 0.8 mg/cm
Mo target. The life-time measurement was performed with the Recoil Dis-tance Doppler-Shift (RDDS) method [24–26]. The tar-get was mounted on the differential Cologne plungerwith a 1.6 mg/cm thick nat Mg degrader placed down-stream. Eight different target-degrader distances in the31-521 µ m range were used to measure the lifetimes ofinterest. The complete identification of the beam-likereaction products was performed with the VAMOS++magnetic spectrometer [27–29], placed at the grazing an-gle θ lab =25 ◦ . The emitted γ rays were detected by the γ -ray tracking detector array AGATA [30, 31], consistingof 8 triple-cluster detectors placed in a compact config-uration (18.5 cm from the target) at backward angleswith respect to the beam direction. The combinationof the pulse-shape analysis [32] and the Orsay Forward-Tracking (OFT) algorithm [33] allowed reconstruction ofthe trajectories of the γ rays emitted by the reactionproducts. More details can be found in Refs. [4, 34–36]. II. LIFETIME ANALYSIS
Combining the precise determination of the ion veloc-ity vector given by VAMOS++ and the identification ofthe first interaction point of each γ ray inside AGATA,Doppler correction was applied on an event-by-event ba-sis. The magnetic spectrometer directly measured the ionvelocity after the degrader ( β after ≈ γ -ray energy. Thevelocity of the ions before the degrader ( β before ≈ γ -ray before the Mg foil ( shifted compo-nent ) and after it ( unshifted component ). The relativeintensities of the unshifted ( I u ) and shifted ( I s ) com-ponents depend on the ratio between the lifetime of theinvestigated state and the target-degrader time of flight,which depends on the β before velocity and the plungerdistance [25]. Specifically, the ratio R ( x ) ≡ I u I u + I s , called decay curve , is described by the Bateman equations.The lifetimes of the excited states were extracted usingthe NAPATAU software [37], applying the DifferentialDecay Curve Method (DDCM) [25] by fitting the areaof both the shifted and the unshifted components witha polynomial piecewise function. These intensities werescaled according to an external normalization, given bythe number of ions identified in VAMOS++ [38]. Thisnormalization is not only proportional to the beam in-tensity and duration of the measurement, but it alsoprovides a measure of possible degradation of the tar-get during the experiment. For each i -th target-stopperdistance the lifetime τ i is obtained as τ i = I ui − Σ j ( Br α I ui ) jddt I si , (1)where the summation is extended over j feeding transi-tions, each with a certain branching ratio ( Br ) and pa-rameter α , which includes the efficiency correction andthe angular correlation between the transition of inter-est and the feeding one. The α parameters were ex-tracted from a γ -ray energy spectrum obtained by sum-ming up the statistics collected for all target-degrader distances [25, 37]. In the case of the γ - γ coincidence pro-cedure with a gate placed on the feeding transition, thecontributions from feeding transitions are eliminated andthis term is null. The final result is given by a weightedaverage of the lifetimes within the sensitive region of thetechnique, i.e. where the derivative of the fitting functionis largest.For the less intense channels, the Decay-Curve Method(DCM) was adopted. Since it relies on well-defined fittingfunctions, whose parameters can be deduced experimen-tally, this technique permits to measure lifetimes even ifthe number of experimental points is limited. A particu-lar application of the method is the R sum approach [39].In this approach, if the statistics is not sufficient to de-termine the area of the γ -ray transition components foreach target-degrader distance, the spectra obtained forthe different target-degrader distances are summed up.The lifetime is then calculated from the solution of theweighted average of the decay curves R j ( x j , τ ) R sum ≡ Σ j I uj Σ j ( I uj + I sj ) = Σ j n j R ( x j , τ ) (2)where x j denotes the plunger distance and n j is the nor-malization factor for each distance. The normalizationfactors n j were given by the total number of γ rays de-tected by AGATA in time coincidence with the shiftedcomponent. III. RESULTS
The coupling of the AGATA and VAMOS++ spec-trometers represents a powerful tool for high precisionspectroscopy. By requiring a time coincidence betweenthe identified reaction products and the detected γ rays,it is possible to clearly select the channels of interest.Additionally, the combination of a MNT reaction,which is a binary mechanism (secondary processes, suchas particle evaporation, are negligible), with the com-plete recoil identification in VAMOS++ allowed us toreconstruct the Total Kinetic-Energy Loss (TKEL) onan event-by-event basis [40]. This quantity is propor-tional to the total excitation energy of the investigatednucleus [26, 41]. While, in the Sn case, a TKEL gate wasapplied to control the direct feeding of the states [4, 36],in the present analysis it was used to reduce the possiblepresence of the inelastically scattered Cd beam, whichcould contaminate other channels despite the extraordi-nary performance of the magnetic spectrometer.In the following, the lifetime measurements in the even-mass − Cd isotopes are presented, with each casediscussed in detail. Figure 2 shows the partial levelschemes of the investigated nuclei with the γ -ray tran-sitions observed in the current measurement. Table Isummarizes the measured lifetimes obtained with the dif-ferent techniques. FIG. 2: Partial level scheme of the even-mass − Cd presenting the transitions observed in the current measurement. Spinand parity of the states are assigned according to the NNDC On-Line Data Service from the ENSDF database [42] (file revisedas of August 2009 for
Cd, September 2009 for
Cd, June 2008 for
Cd, and October 2008 for
Cd). A. Cd In order to reduce the contamination caused by the
Cd beam, a TKEL >
32 MeV condition was imposed.Such threshold permitted to limit as much as possiblethe presence of γ -ray peaks related to the inelasticallyscattered beam, without decreasing the statistics in thetransitions of interest. In these conditions, the 2 +1 → + g.s. , 4 +1 → +1 and 6 +1 → +1 transitions in Cd wereclearly identified in the γ -ray energy spectrum obtainedby summing up the statistics from all the distances, asshown in Figure 3.Since the statistics of the 4 +1 → +1 shifted componentwere not sufficient for a coincidence measurement, thelifetime of the 2 +1 state was obtained via DDCM by sub-tracting the contribution of the unshifted component ofthe 4 +1 → +1 transition. Figure 4 presents the DDCManalysis, resulting in a lifetime τ (2 +1 ) = 5 . +1 → +1 tran-sition it was not possible to determine the area of the shifted and unshifted components for the individual dis-tances. The lifetime was, therefore, measured via DDCMadopting the so-called “gate from below” approach [45],resulting in τ (4 +1 ) = 3 . B. Cd Figure 2 presents the partial level scheme of
Cd,populated via two-neutrons stripping, showing the tran-sitions observed in the singles γ -ray energy spectrum andin the γ − γ matrix obtained by summing the statistics ofthe different target-degrader distances. Due to the com-plexity of the decay pattern and the presence of γ -raytransitions with a similar energy, coincidence techniqueswere necessary to extract the lifetimes of several excitedstates.The lifetime of the 2 +1 state, fed by a single transition,was measured via DDCM by subtracting the contribu-tion from the unshifted component of the 4 +1 → +1 tran- C oun t s / ke V C oun t s / ke V Energy [keV]
FIG. 3: (Color online) Doppler-corrected γ -ray energy spec-trum of Cd before (black) and after (green) the gate onthe Total Kinetic-Energy Loss (TKEL), obtained by sum-ming up the statistics of all the target-degrader distances.The 2 + → g.s. (red), 4 + → + (blue) and 6 + → + (orange)transitions are marked, indicating the unshifted and shiftedcentroids with a solid and a dashed line, respectively. N o r m a li z e d a r ea L i f e t i m e [ p s ] Distance [ µ m] FIG. 4: (Color online) DDCM analysis for the lifetime mea-surement of the 2 +1 excited state of Cd. (top) Area of theshifted (red diamonds) and feeding-corrected unshifted (bluetriangles) components, normalized to the number of ions de-tected in VAMOS++. The dashed line represents a fit to theshifted-component points. (bottom) Corresponding lifetimesobtained for individual distances. The solid line denotes theweighted average of the lifetimes, while the filled area corre-sponds to 1 σ statistical uncertainty. sition, yielding τ (2 +1 ) = 9 . R sum approach, resultingin τ (2 +1 ) = 10 . +0 . − . ps. These results are in a perfectagreement with values reported in literature [43, 44, 46].Due to the presence of multiple feeding transitions,the lifetime of the 4 +1 state was measured via DDCM by gating on the unshifted component of the 2 +1 → + g.s. transition, and via the R sum approach by gating on theshifted component of the 878-keV 6 +1 → +1 transition.The two techniques yielded τ (4 +1 ) = 1 . τ (4 +1 ) = 1 . +0 . − . ps, respectively. Both results are com-patible with the most accurate and recent measurement,reported in Ref. [43].Due to the limited statistics and the presence of var-ious feeding transitions, the 6 +1 excited state was stud-ied only via the R sum method. Unfortunately, due tothe close proximity of the unshifted component of the4 +1 → +1 transition, a very narrow gate had to be set onthe shifted component of the 841-keV 8 +2 → +1 transi-tion. The consequent limited statistics was not sufficientto determine the lifetime of the 6 +1 state, but an upperlimit τ (6 +1 ) < +1 , 4 +1 and 6 +1 states are presentedin Table I. C. Cd The lifetime of the 2 +1 excited state in Cd, equal to10.4 (2) ps, was obtained via DDCM from the presentdata [35]. The same result was obtained via DCM, seeTable I, and the excellent agreement between the resultsof the two approaches validated the calibration of theplunger device. In the R sum approach, since it is basedon the Bateman equations, the knowledge of the absolutetarget-degrader distances is crucial to properly measurelifetimes [35]. In the spectrum obtained by summing upthe statistics of all the distances and then gating on theshifted component of the 4 +1 → +1 transition, the inten-sity ratio resulted 0.56 (1). Considering this experimen-tal value and exponential functions as R j ( x j , τ ) decaycurves, Figure 5 presents the R sum analysis for the 2 +1 state, yielding a lifetime of τ (2 +1 ) = 10 . − excited state was obtained viathe R sum approach by gating on the shifted componentof the 691-keV transition de-exciting the 6 − state. Theresulting lifetime is τ (5 − ) = 8 . γ -ray transitions were observed feedingthe 4 +1 state and the statistics in their shifted compo-nents were not sufficient for a coincidence measurement,the lifetime of the 4 +1 state was extracted via DDCMby gating on the unshifted component of the 2 +1 → + g.s. transition. The analysis yielded τ (4 +1 ) = 1 . +2 ( E x = 1795 . +2 ( E x = 1716 . +3 ( E x = 2304 . + ( E x =2347 . +2 ( E x = 2503 . + , + , + ( E x =2485 . γ -ray spectra and γ − γ matrices. Therefore, theirlifetimes were determined via DCM using an exponentialfunction and, for the most intense channels, via DDCMas well.The 4 +2 excited state was investigated via DCM us-ing second-order Bateman equations. This state was ob-served, in both singles γ -ray energy spectra and γ - γ ma-trices, to be fed only by the 5 + → +2 and 5 − → +2 transitions. The direct population of these states, whichis a parameter of Bateman equations, was extracted fromthe γ -ray spectrum obtained by summing up the statis-tics of all target-degrader distances [25]. The directpopulations resulting from the efficiency and branching-ratio corrected intensities of the observed transitions were59(4)%, 28(3)% and 14(3)% for the 4 +2 , 5 + and 5 − states,respectively. Taking into account the known lifetime τ (5 + ) = 0 . τ (5 − ) measured in thepresent work, the lifetime of the 4 +2 level was determinedto be τ (4 +2 ) = 4 . +1 excited state, no feeding transitions wereobserved in singles γ -ray spectra and γ − γ matrices. Onthe other hand, the decay curve of the 6 +1 → +1 transi-tion (see Figure 6) suggested feeding from a state witha lifetime longer than that of the 6 +1 state. Assuming atwo-step decay cascade, the lifetime of the 6 +1 state wasdetermined to be τ (6 +1 ) = 1 . +1 lifetime was determined by fitting the decaycurve with an exponential function.An upper limit of 0.3 ps can be set for the lifetimesof the 2 +4 , 3 − and 5 − excited states, since in the γ -rayspectra only the shifted components of the 2 +4 → +1 ,3 − → +1 and 5 − → +1 transitions were observed for theshortest plunger distances. The same limit can be de-termined for a state de-exciting via a ≈ + R s u m Lifetime [ps]
FIG. 5: (Color online) Decay curve as a function of the life-time of the 2 +1 excited state in Cd, obtained with the R sum approach. The black line represents the experimental valueobtained by summing up the statistics of all target-degraderdistances and gating on the shifted component of the 4 +1 → +1 transition. The red curve is the expected value calculatedwith Equation 2. The interception between the experimentaland expected values (green line) represents the lifetime of thestate. All the dashed curves denote the 1 σ uncertainty. I u / ( I u + I s ) Distance [ µ m] FIG. 6: (Color online) Ratio of the component intensities asa function of the target-degrader distance for the 6 +1 → +1 transition in Cd. The solid red line represents the fitteddecay curve obtained with second-order Bateman equations,whose components are shown with dashed and dotted bluelines for the feeder and the direct population, respectively. of the levels at the excitation energy of 2717.9 keV and2720.6 keV, so it is not possible to unambiguously at-tribute this upper limit.The results obtained via both DCM and DDCM aresummarized in Table I. D. Cd In both singles γ -ray energy spectra and γ - γ matricesno contamination from the Cd beam was observed.This result is important for the study of
Cd, since thetwo isotopes present similar structures with γ -ray transi-tions very close in energy. Figure 2 shows the partial levelscheme of Cd, indicating the observed transitions.The lifetime of the 2 +1 excited state was measured withthe R sum method by gating on the shifted component ofthe 4 +1 → +1 transition, resulting in τ (2 +1 ) = 10 +3 − ps.The large uncertainty of the lifetime is mostly due to thelimited statistics resulting from the use of γ − γ coinci-dences, even though the statistics of all target-degraderdistances were summed together. Therefore a DDCManalysis for this lifetime was performed by subtractingthe intensity of the unshifted component of the 4 +1 → +1 transition, yielding τ (2 +1 ) = 10 . +1 state wasobtained via DDCM by gating on the unshifted compo-nent of the 2 +1 → + g.s. transition. This approach yielded τ (4 +1 ) = 1 . +1 and4 +1 states. TABLE I: Measured lifetimes of the excited states I πi in the even-mass − Cd isotopes. The lifetimes are extracted fromthe I πi → I πf transitions and the results are compared with literature values. I πi I πf E γ [keV] τ [ps]DDCM DCM Literature Cd 2 +1 + g.s.
777 5.6 (6) - 5.9 (5) [43]5.2 (7) [44]4 +1 +1
861 3.6 (12) - < . Cd 2 +1 + g.s.
658 9.6(3) 10 . +0 . − . +1 +1
834 1.6 (5) 1 . +0 . − . < +1 +1
878 - < < Cd 0 +2 +1 +1 + g.s.
633 10.4 (2) [35] 10.7 (4) [35] 10.5 (1) [51]10.1 (3) 9.4 (4) [49]10.1 (8) [52]10.1 (6) [53]7.0 (3) [54]2 +2 + g.s. +1 +4 +1 < . + +1 + ,3 + ,4 + +1 − +1 < . +1 +1
861 1.4 (2) - 1.26 (16) [49, 53]2.5 (2) [54]4 +2 +1 ≤ . +3 +1
811 - 1.1 (1) 1.1 (1) [54]5 − +2
525 - 8.2 (4) 7 +6 − [47]5 − +1 < . +1 +1
998 - < +2 +1 Cd 2 +1 + g.s.
633 10.8 (9) 10 +3 − +1 +1
876 1.4 (5) - 1.28 (16) [49]
IV. DISCUSSION
In view of the measured lifetimes, the even-mass Cdnuclei were studied within a self-consistent beyond-mean-field framework [55, 56], i.e. the symmetry-conservingconfiguration mixing (SCCM) [57, 58] method, with theGogny-D1S [59, 60] interaction and a model space con-sisting of N h.o. = 9 harmonic-oscillator orbitals. Thecalculations are based on the mixing of a set of intrin-sic states with different quadrupole (axial and non axial)deformations. These states are Hartree-Fock-Bogoliubov(HFB) like wave functions obtained self-consistentlythrough the particle-number variation-after-projection (PNVAP) method [61]. Since the HFB states break therotational invariance of the system, this symmetry is con-sequently restored by projecting onto good angular mo-mentum (particle-number and angular-momentum pro-jection, PNAMP). The final spectrum and the nuclearwave functions are obtained by mixing such PNAMPstates within the generator coordinate method.A first estimation of the structure of the Cd isotopescan be obtained by analysing the calculated potential-energy surfaces (PES) as a function of deformation pa-rameters. Figure 7 presents the PNVAP energies as afunction of the ( β , γ ) deformation parameters for theeven-mass − Cd. For all studied isotopes a well de- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β 𝛾 ⤴ (deg) C d C d β 𝛾 ⤴ (deg) C d β 𝛾 ⤴ (deg) C d β ⤴ β 𝛾 ⤴ (deg) C d C d β 𝛾 ⤴ (deg) C d β 𝛾 ⤴ (deg) C d β ⤴ β 𝛾 ⤴ (deg) C d C d β 𝛾 ⤴ (deg) C d β 𝛾 ⤴ (deg) C d β ⤴ β 𝛾 ⤴ (deg) C d C d β 𝛾 ⤴ (deg) C d β 𝛾 ⤴ (deg) C d β ⤴ PNVAP (MeV)(MeV)(MeV)(MeV)
FIG. 7: (Color online) PNVAP potential energy surfaces as a function of the ( β , γ ) deformation parameters for the even-mass − Cd isotopes. The results are obtained with the Gogny-D1S interaction within the SCCM approach. fined prolate minimum with β = 0 . − . − Cd, which exhibit practicallyspherical minima, due to the vicinity of the N = 82 shellclosure. Moreover, for the − Cd isotopes a shallowsecond triaxial-prolate minimum with ( β , γ )=(0.3, 20 ◦ )is obtained.The final theoretical spectra, obtained by mixing in-trinsic HFB-like states with different quadrupole defor-mations, are presented in Figure 8(c), which shows thesystematics of the excitation energies for the 2 +1 and 4 +1 states. The theoretical predictions correctly reproducethe trends observed experimentally for the 2 +1 and 4 +1 states, although they overestimate the absolute values,especially for the 2 +1 energies above N = 64 and for the4 +1 energies in the whole range of neutron numbers. Thisis a well-known effect in the present form of the SCCMmethod where only static intrinsic shapes are consideredin the mixing. Thus, the ground state is variationallyfavored with respect to the excited states and, as a re-sult, a stretched spectrum is obtained. A better approachwould be an SCCM method that includes intrinsically ro-tating (cranking) states. Within such a framework it ispossible to explore on an equal footing collective groundand excited states and the variational approach does notproduce such a stretching. However, this improvement isvery demanding from the computational point of view,especially for nuclei in this medium-mass region. Never-theless, the inclusion of the triaxial degree of freedom in the SCCM calculations improves significantly the agree-ment with the experimental data with respect to previousaxial calculations [69]. Notably, the intriguing loweringof the 2 +1 energy from Cd to
Cd is still reproducedby the present calculations. In the vicinity of a shell clo-sure, the excitation energy of the 2 +1 state is expectedto increase and display a parabolic trend as a functionof nucleon number. Not only such a parabolic increasehas not been observed experimentally, but the excitationenergy of the 2 +1 states slightly decreases. This patternwas reproduced, for the first time, by the previous axialcalculations [69] and a better agreement is found withthe present ones.In Figure 8 the experimental B ( E
2; 2 +1 → + g.s. ) and B ( E
2; 4 +1 → +1 ) strengths are compared to the theo-retical results of SCCM, together with the predictionsof Ref. [5] for the neutron-deficient isotopes. An un-usual behaviour is found for Cd, where prolate andtriaxial-prolate configurations cross for J π = 4 + , leadingto a decrease of the 4 +1 excitation energy and a conse-quent increase of the B ( E
2; 4 +1 → +1 ) reduced transitionprobability. Except for this single case, the calculatedstrengths of the 2 +1 → + g.s. and 4 +1 → +1 transitions wellreproduce the trend of the experimental results, slightlyoverestimating the β deformation. This slight overesti-mation is a plausible explanation for the almost perfectreproduction of the 2 +1 excitation energies in − Cd.The theoretical values, indeed, should be larger than the (b) B ( E ; + → + ) [ e f m ] (a) B ( E ; + → + ) [ e f m ] ORNLBNLANU OLLNBIIoffe Inst. INFN-LNLUni. CologneREX-ISOLDE LBNLThis work (c) E xc . E n e r g y [ M e V ] Mass Number
Exp. 2 Exp. 4 SCCM 2 SCCM 4 FIG. 8: (Color online) Reduced transition probabilities (a) B ( E
2; 2 +1 → + g.s. ) and (b) B ( E
2; 4 +1 → +1 ), and (c) 2 +1 and4 +1 excitation energy systematics for even-mass Cd isotopes.The experimental results [43, 44, 46, 49–54, 62–68] are com-pared with the recent Large-Scale Shell-Model (LSSM) cal-culations of Ref. [5] (blue open pentagons) and the presentSCCM predictions (red open circles and squares). experimental ones for a SCCM method without crankingterms.The collective wave functions (CWF), i.e. the weightsof the intrinsic quadrupole deformations in each nuclearstate, are presented for the 2 +1 and 4 +1 excited statesin Figure 9 and Figure 10, respectively. For all 0 +1 states (not shown), the SCCM calculations predict a well-defined prolate minimum with deformation β = 0 . N ≥
76 due to the proxim-ity of the neutron shell closure. A non-zero deformationof the ground states in the Cd nuclei was also deducedfrom the LSSM calculations of Zuker [5] and its originwas attributed to the pseudo-SU(3) symmetry, due tothe evident quadrupole dominance in the nuclear inter-action. Similar behavior is predicted for the 2 +1 and 4 +1 states, except for the − Cd nuclei presenting a sec-ond triaxial-prolate minimum in the PES of Figure 7. For those nuclei, the CWFs of the 2 +1 and 4 +1 states are spreadin both β and γ . This suggests that they constitute per-fect candidates for shape coexistence, as investigated for , Cd in the recent work of Garrett et al. [20].As the ground-state bands are expected to presentthe features of prolate-deformed rotors, the intrinsicquadrupole moments and, consequently, the β defor-mation parameters can be extracted from the measuredlifetimes of the 2 +1 and 4 +1 states, as discussed in detailin Ref. [70, Sec.IV]. Assuming an axially symmetric ro-tational model, the deduced average β parameters are0.14 and 0.16 for Cd and for the even-mass − Cd,respectively. These results are in agreement with the con-stant deformation predicted by the SCCM calculations,even though the theoretical predictions slightly overesti-mate its magnitude. On the other hand, the quadrupoledeformation deduced from the experimental results as-suming the axial rotor model slightly decreases with spinin the ground-state bands. However, due to the paucityof experimental information on the lifetimes of higher-spin states, this trend cannot be investigated further.Contrary to what is observed in the neutron-deficientSn isotopes [4], no unusual trends are present for thereduced transition probabilities between the low-lyingstates in the light Cd nuclei. The Z = 48 nuclei behaveinstead as prolate-deformed rotors, as suggested also inRef. [5]. Even though these two isotopic chains differby only two protons, the Sn and Cd nuclei present com-pletely different structures whose origin can be attributedto a rearrangement of the nuclear orbitals. While for theGogny-D1S interaction the spherical Z = 50 gap remainsrather constant along the Cd isotopic chain and has thesame size as for the Sn nuclei, in the Nilsson plots a gapis produced at a prolate-deformed configuration, due tothe lowering of the d / and g / , and the rise of the g / proton orbitals. At this deformation, Z = 48 isa closed-shell configuration and is favored with respectto the spherical one [69]. Thus, the structure of theground-state band changes completely between the Cdand the Sn isotopes: the former are dominated by rota-tional structures, while the latter have seniority spectraassociated to particle-pair breaking. This fundamentalstructural change is obscured by the observed similari-ties between the Z = 48 and Z = 50 nuclei in termsof several experimental observables. The present studydemonstrates that, contrary to what one could naivelyimagine, it is not possible to infer details of the struc-ture of Z = 50 nuclei from the properties of the Z = 48ones and vice versa. This does not preclude, however,using the experimental data on one chain in order to re-fine the model description of the neighbouring one. Forinstance, Zuker [5] tuned the adopted nuclear interac-tion to the experimental information on the Cd isotopes,where the quadrupole dominance is evident, and subse-quently used this interaction to investigate the Sn nuclei.Finally, the lowering of the d / , g / orbits and the riseof the g / neutron orbitals could also favor the proton-neutron coupling that would eventually produce the 5 / + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β 𝛾 ⤴ (deg) C d C d C d C d β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) C d C d C d C d β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) C d C d C d C d β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) C d C d C d C d β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) FIG. 9: (Color online) Collective wave functions (CWF) as a function of the ( β , γ ) deformation parameters for the 2 +1 statesin the even-mass − Cd isotopes. The results are obtained with the Gogny-D1S interaction within the SCCM approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β 𝛾 ⤴ (deg) C d C d C d C d β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) C d C d C d C d β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) C d C d C d C d β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) C d C d C d C d β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) β 𝛾 ⤴ (deg) FIG. 10: (Color online) Collective wave functions (CWF) as a function of the ( β , γ ) deformation parameters for the 4 +1 statesin the even-mass − Cd isotopes. The results are obtained with the Gogny-D1S interaction within the SCCM approach. − Cd [71]. Nevertheless, a detailed SCCM calcu-lation that could confirm such ground state propertiesof the odd-even Cd isotopes is beyond the scope of thepresent work. A. Cd Contrary to the common view of the Cd nuclei asharmonic quadrupole vibrators, the theoretical calcula-tions predict the low-lying bands to be due to rotationof deformed structures (see Figure 11). This result isin agreement with the conclusions of a recent study on , Cd [20], which suggested that an interpretation ofthese nuclei as exhibiting coexistence of multiple distinctstructures is more appropriate than a vibrational picture.
FIG. 11: (Color online) Excitation energy of low-lyingstates in
Cd as a function of their angular momentum, cal-culated with the Gogny-D1S interaction within the SCCMapproach. Rotational bands with different ( β , γ ) defor-mations are predicted: prolate-deformed ground-state band(black), pseudo-gamma band (red), oblate-deformed shape-mixed band (blue), K = 4 band (green), and prolate-deformed shape-mixed band (magenta). As discussed previously, the calculations pre-dict a prolate-deformed ground-state band with( β , γ ) ≈ (0 . , ◦ ). This prediction of the quadrupole-deformation strength is slightly larger than the valuededuced by assuming the axial-rotor model and themeasured lifetimes of the 2 +1 , 4 +1 and 6 +2 states, resulting β = 0 .
167 (3). However, the lifetime of the 6 +2 stateleads to β = 0 .
12 (2), which is not compatible withthe quadrupole-deformation strengths estimated for theother two states. Such a difference may be attributedto the mixing between the 6 +1 and 6 +2 states, sincethese levels are very close in excitation energy and they both decay to the 4 +1 state. A pseudo-gamma bandbuilt on top of the 2 +2 state, having ( β , γ ) ≈ (0 . , ◦ ),and a K = 4 band are associated to the ground-stateband. Additionally, another prolate-deformed bandwith ( β , γ ) ≈ (0 . , ◦ ) develops above the 4 +4 state.Below the 4 +4 state this band splits into two branchescorresponding to strongly mixed configurations. Oneof them includes the 0 +2 (oblate shape-mixing) and 2 +3 (triaxial shape-mixing) states, coupled by a strong E2transition of 52 W.u. The second branch is formed bythe 0 +3 (prolate-deformed) and the 2 +4 (oblate-prolateshape-mixing) states.Figure 12 shows the comparison of the theoretical pre-dictions with experimental data, including excitation en-ergies and reduced transition probabilities deduced fromthe lifetimes measured in the present work. This compar-ison allowed us to identify four of these configurations. Inparticular, thanks to a good agreement between the theo-retical and experimental transition strengths it is possibleto firmly identify the pseudo-gamma band. However, theexperimental information on the structures built on topof the 0 +2 and 0 +3 states is too limited to draw conclusionsregarding the strongly mixed configurations. V. CONCLUSIONS
The structure of even-mass − Cd isotopes was in-vestigated via lifetime measurements at GANIL. Theseneutron-deficient nuclei were populated via an uncon-ventional use of a multi-nucleon transfer reaction and,thanks to the powerful capabilities of the AGATA andVAMOS++ spectrometers, an unambiguous identifica-tion of the channels of interest was possible. Moreover,the combination of the magnetic spectrometer with theadopted binary reaction mechanism permitted the recon-struction of the TKEL on an event-by-event basis, whichwas used in the present work to reduce the contaminationof the γ -ray spectra by the scattered Cd beam.Using the RDDS technique, the lifetimes of the 2 +1 and4 +1 states in even-mass − Cd were obtained. Ad-ditionally, lifetimes of other 8 states in
Cd were de-termined, providing a deep insight into the structure ofexcited bands in this nucleus.In view of these experimental results, state-of-the-art beyond-mean-field calculations were performed forthe even-mass − Cd nuclei using the symmetry-conserving configuration-mixing approach. Except forthe nuclei in proximity of the neutron shell closures, thesecalculations predict prolate-deformed ground-state bandsin the whole Cd isotopic chain. The quadrupole de-formation β is in fair agreement with the estimationobtained from the measured lifetimes by assuming anaxially-symmetric rotor model. According to the LSSMcalculations of Zuker [5], the presence of deformationalong the Cd isotopic chain can be attributed to thequadrupole dominance observed for the Z = 48 nuclei.The calculations within the SCCM approach show that,2 FIG. 12: (Color online) Partial level scheme of
Cd, reporting low-lying excited states. The numbers on the arrows representthe B ( E
2) reduced transition probabilities in Weisskopf units. The experimental results are compared with the energy spectrumpredicted by the SCCM approach, for which the band-head CWFs are shown. due to a rearrangement of the d / and g / orbitals, a de-formed closed-shell configuration is obtained for Z = 48.As discussed in details in Ref. [69], the semi-magic char-acter of Cd nuclei impacts the N = 82 shell quenchingproblem: all the observables that were attributed to apossible reduction of the N = 82 shell closure in prox-imity of Sn can simply be described by invoking thestructure of the Cd nuclei.Despite the similarities between the Z = 48 and Z = 50 nuclei, in particular with regard to the electro-magnetic properties of the 2 +1 states, the structures of thetwo isotopic chains are completely different. This resultsupports the conclusions of our previous work [4] con-cerning the limited role of the observables related to the2 +1 states in investigations of the structure of the Z ≈ +1 states allows us to shed light on the structure of the nuclei in question. Experimental in-formation on these states has been shown to be crucial todisentangle between different models and interpretations.Further experimental and theoretical studies are nec-essary to fully understand the structure of neutron-deficient Cd nuclei. Theoretical and experimental re-sults suggest that the multiple shape-coexistence inter-pretation, proposed by Garrett et al. for − Cd,can be extended to the neutron-deficient region. Inthis context, future multi-step Coulomb-excitation mea-surements, benefiting from the lifetimes measured inthe present work, will permit to directly determine the( β , γ ) deformation parameters of the individual nuclearstates. Moreover, in view of the predictions presented inthis manuscript, the investigation of the structures builton top of the two excited 0 + states is of great interest.The identification of these bands together with a precise3determination of E Acknowledgement
The authors would like to thank the AGATA and VA-MOS collaborations. Special thanks go to the GANILtechnical staff for their help in setting up the apparatusesand the good quality beam. This research was partiallysupported by the European Union’s Seventh FrameworkProgramme for Research and Technological Development(grant no. 262010). A.G. acknowledges the support ofthe Fondazione Cassa di Risparmio Padova e Rovigo un-der the project CONPHYT, starting grant in 2017. Thework of T.R.R. was supported by the Spanish MICINNunder Grant No. PGC2018-094583-B-I00. The researchwas also supported (H.L., J.N. and U.J.) by Swedish Re-search Council under the grant agreements nos. 822-2005-3332, 821-2010-6024, 821-2013-2304, 621-2014-5558 and 2017-0065, and by the Knut and Alice WallenbergFoundation grant no. 2005.0184, (B.S.) by the Scien-tific and Technological Council of Turkey (TUBITAK)under the project no. 114F473, (A.G. and R.P.) by theMinisterio de Ciencia e Innovac´ıon under the contractsnos. SEV-2014-0398, FPA2017-84756-C4 and EEBB-I-15-09671, by the Generalitat Valenciana under the grantagreement no. PROMETEO/2019/005 and by the EU-FEDER funds, (T.M. and S.S.) by the Croatian ScienceFoundation under the project no. 7194, (I.K. and D.S.)by the Hungarian National Research and Innovation Of-fice (NKFIH) under the project nos. K128947, PD124717and GINOP-2.3.3-15-2016-00034, (M.P. and G.J.) bythe Polish National Science Centre with the grants nos.2014-14-M-ST2-00738, 2016-22-M-ST2-00269 and 2017-25-B-ST2-01569, and the COPIN-IN2P3, COPIGAL andPOLITA projects, (C.M.-G.) by the U.S. Department ofEnergy, Office of Science, Office of Nuclear Physics, undercontract number DE-AC02-06CH11357. This manuscriptowes much to the collaboration with M. Zieli´nska. [1] J.J. Ressler, R.F. Casten, N.V. Zamfir, C.W. Beausang,R.B. Cakirli et al., Phys. Rev. C , 034317 (2004)[2] I.O. Morales, P. Van Isacker, I. Talmi, Phys. Lett. B ,606 (2011)[3] B. Maheshwari, H. Abu Kassim, N. Yusof, A. Ku-mar Jain, Nucl. Phys. A , 121619 (2019)[4] M. Siciliano, J. Valiente-Dob´on, A. Goasduff, F. Nowacki,A. Zuker, D. Bazzacco, A. Lopez-Martens, E. Cl´ement,G. Benzoni, T. Braunroth et al., Phys. Lett. B ,135474 (2020)[5] A. Zuker, Phys. Rev. C (2020), accepted for publication[6] A. Sonzogni et al., Nudat 2 (2016)[7] A. Arima, F. Iachello, Ann. Phys. , 253 (1976)[8] F. Iachello, A. Arima, The interacting boson model (Cambridge University Press, 1987)[9] A. Bohr, B. Mottelson,
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