Low-spin particle/hole-core excitations in 41,47,49 Ca isotopes studied by cold-neutron capture reactions
S. Bottoni, N. Cieplicka-Oryńczak, S. Leoni, B. Fornal, G. Colò, P.F. Bortignon, G. Bocchi, D. Bazzacco, G. Benzoni, A. Blanc, A. Bracco, S. Ceruti, F.C.L. Crespi, G. de France, E.R. Gamba, Ł.W. Iskra, M. Jentschel, U. Köster, C.Michelagnoli, B.Million, D.Mengoni, P.Mutti, Y.Niu, C.Porzio, G. Simpson, T. Soldner, B. Szpak, A. Türler, C.A. Ur, W. Urban
LLow-spin particle/hole-core excitations in , , Ca isotopesstudied by cold-neutron capture reactions
S. Bottoni , , N. Cieplicka-Ory´nczak , S.Leoni , , B. Fornal , G.Col`o , , P.F. Bortignon , , G. Bocchi , ,D. Bazzacco , G. Benzoni , A. Blanc , A. Bracco , , S. Ceruti , , F .C .L. Crespi , , G. de France ,E. R. Gamba , , (cid:32)L.W. Iskra , , M. Jentschel , U. K¨oster , C. Michelagnoli , B. Million , D. Mengoni , ,P. Mutti , Y. Niu , C. Porzio , , G. Simpson , T. Soldner , B. Szpak , A. T¨urler , C.A. Ur , W. Urban Dipartimento di Fisica, Universit`a degli Studi di Milano, 20133 Milano, Italy INFN Sezione di Milano, 20133, Milano, Italy Institute of Nuclear Physics, PAN, 31-342 Krak´ow, Poland INFN Sezione di Padova, 35131 Padova, Italy Institut Laue-Langevin, 38042 Grenoble CEDEX 9, France GANIL, BP 55027, 14076 Caen CEDEX 5, France Museo Storico della Fisica e Centro di Studi e Ricerche Enrico Fermi, 00184 Roma, Italy Dipartimento di Fisica e Astronomia, Universita degli Studi di Padova, 35131 Padova, Italy School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China Universit¨at Bern and Paul Scherrer Institut, Villigen, Switzerland ELI-NP, Magurele-Bucharest, Romania and Faculty of Physics, Warsaw University, 00-681 Warsaw, Poland
We present recent results on the structure of the one-valence-particle Ca and Ca, and one-valence-hole Ca, nuclei. The isotopes of interest were populated via the cold-neutron capturereactions Ca(n, γ ), Ca(n, γ ) and Ca(n, γ ), respectively. The experiments were performed atthe Institut Laue-Langevin, within the EXILL campaign, which employed a large array of HPGedetectors. The γ decay and level schemes of these nuclei were investigated by γ -ray coincidencerelationships, leading to the identification of 41, 10, and 6 new transitions in Ca, Ca, and Ca,respectively. Branching ratios and intensities were extracted for the γ decay from each state, and γ -ray angular correlations were performed to establish a number of transition multipolarities andmixing ratios, thus helping in the spin assignment of the states. The experimental findings arediscussed along with microscopic, self-consistent beyond-mean-field calculations performed with theHybrid Configuration Mixing model, based on a Skyrme SkX Hamiltonian. The latter suggeststhat a fraction of the low-spin states of the Ca, Ca, and Ca nuclei is characterized by thecoexistence of either 2p-1h and 1p-2h excitations, or couplings between single-particle/hole degreesof freedom and collective vibrations (phonons) of the doubly-magic “core”.
PACS numbers: 28.20.Np, 23.20.Lv, 23.20.En, 24.10.Cn
I. INTRODUCTION
The structure of calcium isotopes between the doubly-magic Ca ( N =20) and Ca ( N =28) nuclei has beenthe subject of many experimental studies over the pastdecades [1–13]. With six stable isotopes, calcium playsa crucial role in stellar nucleosynthesis [14–16]. The for-mation of Ca isotopes involves several astrophysical pro-cesses, such as silicon- and oxygen-burning [17], as well ass- and r-processes [18, 19], which generate, for example,the heaviest, symmetric N=Z stable nucleus, i.e. Ca,and the lightest stable doubly-magic neutron-rich system,namely Ca, in the nuclide chart. Moreover, the Z =20isotopic chain contains a rare cosmogenic radioactive nu-cleus, i.e. Ca, produced by neutron-capture reactionson Ca induced by cosmic rays [20].In this context, nuclear structure studies along Ca iso-topes are crucial to understand, for instance, the evolu-tion of single-particle states and collectivity from sym-metric to neutron-rich systems, which are properties sig-nificantly affecting the reaction rates in stellar environ-ments. Moreover, new experimental results may serve as a benchmark for the most advanced theoretical mod-els, such as state-of-the-art shell model calculations [21–24] and ab initio approaches, employing chiral two- andthree-nucleon interactions [25–28].The low-lying structure of Ca is characterized by a 0 + state at 3.4 MeV, as a first excited state - a clear sig-nature of a robust double shell closure in this nucleus -and a very collective octupole, 3 − vibration at 3.7 MeV,with a B(E3) of ≈
30 W.u. [29]. Moreover, in the spinrange 2¯ h -8¯ h , deformed and superdeformed bands havebeen observed and associated to 4p-4h and 8p-8h exci-tations, respectively [30]. These features are graduallylost in mid-shell Ca nuclei, where deformed structurestake over spherical ones already at low energies, owing toneutron p-h excitations across the pfg energy gap. Thisscenario changes again in Ca, where the presence ofa low-lying 0 + state at 4.3 MeV and a 3 − phonon withB(E3) ≈ Ca.In this framework, Ca nuclei one-particle or one-holeaway from double shell closures are of particular interest.These isotopes are ideal to investigate the interplay be-tween fermionic and bosonic degrees of freedom, as it oc- a r X i v : . [ nu c l - e x ] D ec curs in the coexistence and competition between pure p-hexcitations and the so-called particle/hole-vibration cou-pling [32]. As a matter of fact, the low-lying structure ofone-valence-particle/hole nuclei is strongly influenced bythe collective phonons of the underlying “core”. On theother hand, core excitations are perturbed and dampedby the single particle/hole motion and non-collective p-hexcitations [33, 34]. Therefore, a comprehensive inves-tigation of these mechanisms, moving along the Ca iso-topic chain, may significantly advance our understand-ing of the emergence of complex phenomena, such as thequenching of spectroscopic factors and the anarmonicityof vibrational spectra in this mass region.In this paper, we present new experimental results in Ca, Ca, and Ca, populated via cold neutron-capture reactions and studied by γ -ray spectroscopy.Neutron-capture reactions induced by cold and thermalneutrons populate the corresponding N+1 systems at theneutron separation energy S n . The spin of the capturelevel depends on the ground-state spin J of the targetnucleus and can only be J ± + neutron-capture state. The γ -ray decay is typicallydominated by high-energy, E1 primary transitions, whichpreferentially populate 1/2 − and 3/2 − states (based on γ -decay selection rules), followed by secondary electro-magnetic radiation of different character and multipolar-ity.In this context, it is clear that the combined use ofneutron-capture reactions and detectors with high en-ergy resolution (e.g. HPGe crystals) enables to performan almost-complete γ -ray spectroscopy from the neutronbinding energy to the ground state, providing an exhaus-tive picture of the low-spin structure of the nuclei of in-terest. The present experimental results on the Ca, Ca, and Ca nuclei will be discussed in the frame-work of the Hybrid Configuration Mixing Model [35, 36],with particular attention to the interplay between single-particle/hole states and couplings with core excitations.The paper is organized as follows: in Sec. II the experi-mental details will be presented along with the differentreactions performed; in Sec. III the analysis of the datawill be discussed, while in Sec.IV the experimental resultswill be outlined in connection with theoretical interpre-tations.
II. THE EXPERIMENT
The experiments were performed at Institut Lau-Langevin (ILL) in Grenoble, within the EXILL experi-mental campaign [37]. Neutron-capture reactions werestudied at the High Flux Reactor of ILL [38], which de-livers the most intense, continuous neutron beams world-wide for scientific research.In the present measurement, a high-efficiency, high-resolution composite HPGe array was installed at the [r e l . un it s ] E [keV]
100 1000 10000 Cl(n, fit EXILL
FIG. 1. (Color online) Relative γ -ray efficiency of the HPGearray used in the Ca(n, γ ) experiment. Experimental dataobtained from the Cl(n, γ ) reaction are displayed along withthe fit function (red). The efficiency of the full EXILL setupis also reported [37] (see text for details). PF1B cold-neutron beam line [39], where the neutronflux was about 10 neutrons cm − s − , after collimation.The array comprised 8 clover detectors from the EX-OGAM setup [40], 6 coaxial GASP detectors [41] and2 ILL clover detectors, providing a total photopeak effi-ciency of ≈
6% at 1.3 MeV. Apart from the ILL clovers,all the other HPGe detectors were equipped with BGOanti-Compton shields for background suppression. In thecase of the Ca(n, γ ) experiment, the GASP and ILLdetectors were replaced by 16 LaBr :Ce fast scintillatorsfrom the FATIMA collaboration [42] for lifetime measure-ments by using fast-timing techniques [43]. As a conse-quence, the γ -ray efficiency of this HPGe detector con-figuration, comprising the EXOGAM clovers only, differsfrom the one of the full EXILL setup. In Fig.1, the rel-ative efficiency of the HPGe array used in the Ca(n, γ )experiment is reported, arbitrary normalized at ≈ γ -ray transitions inthe Cl nucleus, populated in the Cl(n, γ ) reaction. Athigh energy, the deviation of the present efficiency fromthe EXILL curve is consistent with the reduced numberof HPGe detectors in the setup.The compact geometry of the 8 EXOGAM clovers,mounted in a symmetric, ring configuration around thescattering chamber, was used to study γ -ray angular cor-relations, with the aim of determining the multipolar-ity of the detected radiation, thus constraining the spinand parity of the observed states. All the possible angu-lar combinations between crystals (11 angles from 0 ◦ to90 ◦ ) were grouped into 3 angles only, i.e. 0 ◦ , 45 ◦ , and90 ◦ , corresponding to the angles between clover detectorswith respect to the target position. This enabled to in-crease the statistics for γ - γ coincidences, allowing us toperform angular-correlation studies also in cases of weaktransitions. Experimental data were fitted by using theanalytic function [44] W ( θ ) = 1 + a q P (cos θ ) + a q P (cos θ ) , (1)where a ii are the multipole expansion coefficients,P i (cos θ ) the Legendre Polynomials, and q i the attenu-ation parameters which take into account the finite sizeof the detectors. The latter were determined by study-ing γ -ray angular correlations of known transitions of the Eu γ -ray source, and were found to be q =0.87 andq =0.6.The Ca and Ca nuclei were populated by (n, γ ) reac-tions on an enriched target. For this purpose, a 40.6-mgCa(NO ) compound, enriched to 31.7% in Ca, wasprepared at Paul Scherrer Institute, Switzerland. Thenitrate solution was directly dried in a 25 µ m thin FEP(fluorinated ethylene propylene) bag, which has negli-gible neutron-capture cross section. It is important tonote that Ca is the isotope with the second-lowestnatural relative isotopic abundance (only 0.004%), after He. A large fraction of the target (60.5%) was com-posed by Ca, with the latter being the most abundantZ=20 isotope. This allowed us to perform Ca(n, γ ) and Ca(n, γ ) reactions at the same time. On the other hand,the Ca nucleus was populated by neutron-capture re-actions on a 350-mg CaCO compound target, enrichedto 60.5% in Ca. Also in this case, traces of other Caisotopes were present in the sample.The composition of the targets used in the current ex-periments, along with the (n, γ ) cross sections for ther-mal neutrons [45], are reported in Tab. I. The isotopes ofinterest for this work and the corresponding targets arehighlighted in bold. III. DATA ANALYSIS
Data were acquired using fast, digital electronics intriggerless mode and the analysis was performed by con-sidering coincident events, built within a 200-ns, prompttime window. The good energy resolution and efficiencyof the HPGe array turned out to be essential to observe,with high accuracy, very weak γ -ray decay paths, whereasthe BGO Compton shields significantly suppressed γ -raycoincidences with Compton-scattered radiation.The level schemes and the γ -ray decays of the Ca, Ca,and Ca nuclei were studied by using γ - γ and triple- γ coincidence techniques. At first, very selective gateson primary, high-energy γ transitions were used to iden-tify secondary γ -ray cascades and to locate new low-lyingstates. The energies of the latter were determined by cor-recting the measured γ energies by the recoil energy ofthe nucleus. This is particularly crucial for high-energytransitions, considering the relatively light mass of theisotopes studied in this work. As a second step, gates TABLE I. Isotopic composition of the Ca(NO ) and CaCO compounds used in the current experiment and correspond-ing (n, γ ) cross sections [45]. The (n, γ ) columns show thepercentage of capture reactions for a given isotope. The nu-clei of interest for the the present work are marked in bold.Details of the FEP bag are also reported (see text for details). σ (n, γ ) Ca(NO ) CaCO [barn] [atoms %] (n, γ ) [atoms %] (n, γ )TARGET Ca 0.41 60.5 34% Ca 0.68 0.63 1% 0.3 0% Ca 6.2 0.15 1% 0.1 1% Ca 0.88 5.35 6% 2.5 2% Ca 0.74 31.7 32% Ca 1.09
C 3.84 · −
0% 100 0%N 7.47 · −
200 20% 0%O 2.24 · −
600 0% 300 0%FEPC 3.84 · −
89 0% 6 0%F 9.51 · −
178 2% 13 0% on γ transitions depopulating low-lying states enabledto determine new decay paths from the neutron-capturelevel and to measure precisely the value of the S n neu-tron separation energy for all three nuclei. The latterwere obtained by considering all the possible combina-tions of γ rays decaying directly from the neutron-capturelevel. The S n values obtained in this work are presentedin Tab II, along with those ones reported in the litera-ture [46–51]. TABLE II. Neutron separation energies (S n ) for the Ca, Ca, and Ca nuclei obtained in this work, compared withvalues reported in the literature [46–51].Isotope S n [keV] S n [keV](this work) (literature) Ca Ca Ca The γ -ray intensities and branching ratios for each levelwere evaluated using γ - γ matrices, constructed consider-ing all HPGe detectors in the array. Gates were set ontransitions feeding the level of interest and relative inten-sities of deexciting transitions, with respect to a given γ ray in the level scheme, were extracted, taking into ac-count efficiency corrections. Branching ratios were deter-mined by taking the ratio of the intensity of a γ transi-tion to the summed intensity of all transitions deexcitinga given state. Concerning primary γ rays, all possible de-cay paths for each transition were considered, based onthe analysis of the γ - γ coincidence matrix, and the in-tensity balance was used to extract γ -ray intensities and (cid:1) (cid:2)(cid:3)(cid:4) (cid:5) (cid:6) (cid:7)(cid:8)(cid:9)(cid:7)(cid:9)(cid:8)(cid:10)(cid:7)(cid:10)(cid:8) (cid:1) (cid:11) (cid:12) (cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18) (cid:7) (cid:9)(cid:7)(cid:7)(cid:7) (cid:10)(cid:7)(cid:7)(cid:7) (cid:19)(cid:7)(cid:7)(cid:7) (cid:20)(cid:7)(cid:7)(cid:7) (cid:8)(cid:7)(cid:7)(cid:7) (cid:1) (cid:21)(cid:22)(cid:5)(cid:16)(cid:13)(cid:10)(cid:7)(cid:9)(cid:7)(cid:23)(cid:9)(cid:19)(cid:24)(cid:7) (cid:20)(cid:24)(cid:25)(cid:19) (cid:9) (cid:6)(cid:5) (cid:9)(cid:25)(cid:26)(cid:10)(cid:9)(cid:19)(cid:26)(cid:27) (cid:28)(cid:29) (cid:8)(cid:9)(cid:9) (cid:19)(cid:25)(cid:9)(cid:7)(cid:19)(cid:8)(cid:27)(cid:8)(cid:8)(cid:20)(cid:20) (cid:1) (cid:2)(cid:3)(cid:4) (cid:5) (cid:6) (cid:7)(cid:8)(cid:7)(cid:7)(cid:7)(cid:9)(cid:7)(cid:7)(cid:7)(cid:7)(cid:9)(cid:8)(cid:7)(cid:7)(cid:7) (cid:1) (cid:10) (cid:11) (cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17) (cid:7) (cid:8)(cid:7)(cid:7) (cid:9)(cid:7)(cid:7)(cid:7) (cid:9)(cid:8)(cid:7)(cid:7) (cid:18)(cid:7)(cid:7)(cid:7) (cid:18)(cid:8)(cid:7)(cid:7) (cid:1) (cid:19)(cid:20)(cid:5)(cid:15)(cid:12)(cid:21)(cid:21)(cid:9)(cid:22) (cid:18)(cid:7)(cid:9)(cid:7) (cid:9) (cid:6)(cid:5) (cid:9)(cid:21)(cid:22)(cid:18)(cid:23)(cid:18)(cid:23) (cid:9)(cid:18)(cid:23)(cid:21)(cid:8)(cid:21)(cid:21) (cid:20)(cid:24) (cid:25)(cid:25)(cid:7) (cid:9) (cid:6)(cid:5) (cid:18) (cid:4)(cid:26) (cid:8)(cid:9)(cid:9) (cid:18)(cid:7)(cid:7)(cid:9)(cid:9)(cid:27)(cid:21)(cid:28) (cid:18) (cid:4)(cid:26) (cid:8)(cid:18)(cid:7) (cid:9)(cid:7)(cid:29) (cid:1) (cid:2)(cid:3)(cid:4) (cid:5) (cid:6) (cid:7)(cid:8)(cid:9)(cid:9)(cid:7)(cid:10)(cid:9)(cid:11)(cid:7)(cid:7)(cid:11)(cid:8)(cid:9)(cid:11)(cid:9)(cid:7) (cid:1) (cid:12) (cid:13) (cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19) (cid:7) (cid:11)(cid:7)(cid:7)(cid:7) (cid:8)(cid:7)(cid:7)(cid:7) (cid:20)(cid:7)(cid:7)(cid:7) (cid:21)(cid:7)(cid:7)(cid:7) (cid:9)(cid:7)(cid:7)(cid:7) (cid:22)(cid:7)(cid:7)(cid:7) (cid:1) (cid:23)(cid:24)(cid:5)(cid:17)(cid:14)(cid:11)(cid:25)(cid:21)(cid:20)(cid:26)(cid:9)(cid:8)(cid:7) (cid:8)(cid:8)(cid:25)(cid:7) (cid:27)(cid:28) (cid:9)(cid:11)(cid:11) (cid:11)(cid:8)(cid:22)(cid:29)(cid:11)(cid:11)(cid:9)(cid:11) (cid:11)(cid:21)(cid:29)(cid:8) (cid:9)(cid:25)(cid:7)(cid:7)(cid:20)(cid:22)(cid:11)(cid:7) (cid:11) (cid:6)(cid:5) (cid:8) (cid:4)(cid:27) (cid:1) (cid:2)(cid:3)(cid:4) (cid:5) (cid:6) (cid:7)(cid:8)(cid:9)(cid:7)(cid:9)(cid:8)(cid:10)(cid:7)(cid:10)(cid:8)(cid:11)(cid:7)(cid:11)(cid:8) (cid:1) (cid:12) (cid:13) (cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19) (cid:7) (cid:8)(cid:7)(cid:7) (cid:9)(cid:7)(cid:7)(cid:7) (cid:9)(cid:8)(cid:7)(cid:7) (cid:10)(cid:7)(cid:7)(cid:7) (cid:10)(cid:8)(cid:7)(cid:7) (cid:1) (cid:20)(cid:21)(cid:5)(cid:17)(cid:14)(cid:9)(cid:22)(cid:23)(cid:11)(cid:24)(cid:25)(cid:10)(cid:25) (cid:9)(cid:22)(cid:11)(cid:11)(cid:9)(cid:10)(cid:25)(cid:23) (cid:26)(cid:27) (cid:8)(cid:9)(cid:9) (cid:22)(cid:23)(cid:11)(cid:28)(cid:8)(cid:29)(cid:11)(cid:25)(cid:22) (cid:10)(cid:23)(cid:7)(cid:10) FIG. 2. (Color online) Projection of the γ - γ coincidence matrix measured in the Ca(n, γ ) reaction a), gated on the 4418-keVtransition of the Ca nucleus. Projections of the triple- γ coincidence matrix, gated on the 2010- and 1390-keV b), 1943- and727-keV c), and 1943- and 520-keV d) transitions. New γ rays, observed for the first time, are marked in red, while thosealready known in the literature [46–49, 52] are marked in black. Transitions associated to (n, γ ) reactions on contaminantspresent in the target are labelled by circles. First and second escape peaks for high-energy transitions are marked by 1 st and2 nd , respectively. branching ratios from the neutron-capture state. Sys-tematic errors associated to the partial angular coverageof the detectors (i.e., ≈ γ -ray intensities. Uncertainties originatingfrom efficiency correction (see Fig. 1) were also consid-ered. Possible uncertainties coming from self-absorptionof the targets and summing effects were not included, be-ing negligible when compared to other sources of error.In particular, the former can be excluded due to the smallsamples used in the current experiment (see Sec. II), andthe latter was estimated to be lower than 10 − . A. Ca The level scheme of the Ca nucleus was built by us-ing both γ - γ and γ - γ - γ ray coincidence relationships, set-ting gates on the most intense γ rays. This was possiblethanks to the high statistics collected and the rather highlevel density between the neutron-capture level and the ground state, which results in the emission of γ rays withmultiplicity greater than two. Fig. 2 presents examplesof γ -ray spectra measured in the current experiment, forthe Ca(n, γ ) reaction. The γ rays observed here, for thefirst time, are marked in red, while all other correspond totransitions reported in the literature [46–49, 52]. Peakscoming from transitions associated to (n, γ ) reactions oncontaminants present in the target are labelled by circles.Panel a) presents the projection of the γ - γ coincidencematrix, obtained by setting a gate on the known 4418-keV, primary γ -ray transition, populating the 1/2 − stateat 3944 keV. Two new γ rays with energies 544 keV and1274 keV, depopulating the 3944-keV, 1/2 − state canbe seen. Panels b), c), and d) show the projections ofthe γ - γ - γ coincidence matrix, gated on the 2010-1390-,1943-727-, and 1943-520-keV combinations of γ rays. Inthese cases, all the spectra are almost background-free,enabling the identification of very weak γ rays. In partic-ular, it is worth noting the 1672-keV b) and the 2402-keVc) lines ( ≈ − stateat 5072 keV.The level and γ -ray decay scheme of the Ca nucleus ispresented in Fig 3, where 41 new transitions and 2 new �� �� � / � - �� / � - ����� / � + ����� / � - ����� / � - ����� / � + ����� / � + ����� / � + ����� / � + ����� / � + ���� ( � / � ± �� / � ± ) ����� / � - ����� / � - ����� / � + ����� / � - ����� / � + ����� / � - ����� / � + ����� / � - ����� / � + ����� / � + ����� / � - ����� / � - ����� / � - ����� / � - ����� / � - ����� / � - ���� ( � / � ± �� / � ± ) ����� / � - ����� / � + ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� ���� FIG. 3. (Color online) Level scheme of Ca, as measured in the current experiment. Newly-observed γ -ray transitions andlevels are reported in red. Levels with tentative spin assignment are marked by dashed lines. Dashed arrows indicate γ rayswith no firm placement in the level scheme (i.e., the 4799- and 1622-keV cascade) or very weakly observed (see text for details). �� �� � / � - �� / � - ����� / � + ����� / � - ����� / � - ����� / � + ����� / � + ����� / � + ����� / � + ����� / � + ���� ( � / � ± �� / � ± ) ����� / � - ����� / � - ����� / � + ����� / � - ����� / � + ����� / � - ����� / � + ����� / � - ����� / � + ����� / � + ����� / � - ����� / � - ����� / � - ����� / � - ����� / � - ����� / � - ���� ( � / � ± �� / � ± ) ����� / � - ����� / � + ���� �������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� FIG. 3. - continued a)520 - 19433/2 - →3/2 - →7/2 - b)727 - 19431/2 + →3/2 - →7/2 - c)1671 - 1943 1/2 - →3/2 - ,3/2 - →7/2 - d)1151 - 19431/2 - →3/2 - →7/2 - e)1151 - 5201/2 - →3/2 - →3/2 - f)1482 - 5201/2 - →3/2 - →3/2 - + →3/2 - →3/2 - g) h)5900 - 5201/2 + →3/2 - →3/2 - θ γγ [deg] W ( θ ) / W ( ) - FIG. 4. (Color online) Panels a)-d): γ -ray angular correlation between the 3/2 − → − , 1943-keV transition and the 520-, 720-,1671-, and 1151-keV γ rays, depopulating the 3/2 − , 1/2 + , 1/2 − states at 2462, 2670, and 3613 keV, respectively. Panels e)-h):angular correlations between the 3/2 − → − , 520-keV transition and the 1151-, 1482-, 2290-, 5900-keV lines depopulatingthe 1/2 − , 1/2 − , 1/2 − , and 1/2 + states at 3613, 3944, 4753, and 8362 keV, respectively. Experimental fits are shown as solidred lines, while theoretical predictions are displayed as dashed blue lines (see Sec. III A and Tab. III for details). levels obtained in this work are displayed in red (tenta-tive levels and γ rays are marked as dashed lines). Ofparticular note are the two new levels at 3564- and 6374-keV of excitation energy. In the decay scheme, the orderof γ rays was assigned on the basis of previously-knownlevels, as well as on the observation of parallel cascades.In the case of the 4799-1622-keV decay chain, involvingthe newly-found 3564-keV level, the high energy transi-tion was tentatively assumed to depopulate directly theneutron-capture state. However, since it is not possibleto firmly constrain their right order, the two transitionsare displayed as dashed lines.The scarce statistics collected for the γ decays involvingthe new levels did not allow for the study of γ -ray an-gular correlations, therefore their (1/2 ± ,3/2 ± ) spin andparity is tentatively assigned on the basis of the mostprobable γ -ray multipolarities. On the contrary, angularcorrelations could be performed for a number of levels ofknown spin and parity, as presented in Fig. 4. The toppanels a)-d) show angular correlations between the pureE2 1943-keV ground-state decay (3/2 − → − ), and the520-, 727-, 1671-, and 1151-keV transitions, depopulating the 3/2 − , 1/2 + , 1/2 − states at 2462 keV, 2670 keV, and3613 keV, respectively. The solid red curve correspondsto the experimental fit, which enabled to determine the δ mixing ratios between the two most probable multipolar-ities, by using a χ minimization procedure. Theoreticalpredictions are also shown as dashed blue lines. In partic-ular, a M1+E2 character, with δ = 0.13 (19), was foundfor the 520-keV, 3/2 − → − transition. This value is inagreement, within the error, with the one reported in theliterature ( δ = 0.03 (12) [49]) and was used to extractangular correlations for γ rays in coincidence with the520-keV line, as displayed in the bottom panels e)-h) ofFig. 4. These are the 1151-, 1482-, 2290-keV transitionsdepopulating the 1/2 − , 1/2 − , and 3/2 + states at 3613,3944 and 4753 keV, respectively, and the 5900-keV, 1/2 + → − primary transition, for which a E1(+M2) char-acter was found ( δ = 0.00(1)), confirming the expecteddipole nature of this high-energy, primary γ ray.The energies of levels and γ rays, along with the γ -raymultipolarities, mixing ratios, branching ratios, and γ -ray intensities are presented in Tab. III. New results ob-tained in this work are marked by stars. TABLE III: Initial and final states, γ -ray energies, multipolarities, mix-ing ratios, branching ratios, and γ -ray intensities normalized to the1942.5-keV transition (100 units) of Ca, as observed in this work. Newfindings are marked by a star. Multipolarities and mixing ratios notmeasured in this work are also reported [46–49].E i [keV] J πi E f [keV] J πf E γ [keV] Multipolarity δ BR γ I γ − − + − TABLE III —Continued E i [keV] J πi E f [keV] J πf E γ [keV] Multipolarity δ BR γ I γ − + − − − + − + + − + + + + − + + + + + + − ± ,3/2 ± ) 1942.5(1) 3/2 − − + − +11 − * 0.349(21) 1.138(78)1942.5(1) 3/2 − +12 − * 0.533(22) 1.736(122)3730.5(1) 3/2 − − − − − + + − − + + − − + + + − − + + − + − + + − + + − + − + + + + + + − TABLE III —Continued E i [keV] J πi E f [keV] J πf E γ [keV] Multipolarity δ BR γ I γ + + + − + + − − − − − − − − − − + − − ± ,3/2 ± ) 3730.5(1) 3/2 − − − − − + − ± ,3/2 ± ) 1989.5(4)* 0.0018(1) 0.198(10)5888.8(1) 1/2 − − − − − − + + − + − + − + − − ± ,3/2 ± ) 4798.5(2)* 0.0007(1) 0.075(7)3525.3(1) 3/2 + + + + − + − B. Ca Projections of the γ - γ coincidence matrix for the Canucleus are presented in Fig. 5, where new γ rays areshown in red, while transitions reported in the literatureare marked in black [47, 48, 50, 52]. The spectrum inpanel a) is obtained by gating on the 565-keV transition.Of particular interest are the 4697-, 4676-, and 2825-keV γ rays, which depopulate directly the neutron-capturelevel, feeding previously-known low-lying states at 2578, 2599 and 4450 keV, respectively. Panel b) shows a spec-trum obtained by gating on the 862-keV line in whichnew γ rays at 1182 keV and 1933 keV, populating thestate at 2876 keV, can be seen. Finally, panel c) presentsthe spectrum measured in coincidence with the 3218-keVtransition. In the picture, besides the 1182 keV transi-tion discussed above, new γ rays with energies 1458 keVand 1479 keV are visible. These feed the 1/2 + and 3/2 + states at 2599 keV and 2578 keV, respectively, from thestate at 4057 keV.0The level and γ -decay scheme of the Ca nucleus, ob-tained in this work, is presented in Fig.6 (left), with new γ rays shown in red. In this case, γ -ray angular corre-lations allowed to further characterize a number of tran-sitions in terms of multipolarity and mixing ratios, en-abling to firmly assign the spin and parity of the statesinvolved in the decays, as presented in Fig.6 (right). Ex-perimental fits are shown as solid red lines, while the-oretical predictions are displayed as dashed blue lines.Angular correlations are performed against the 3/2 − → − decay to the ground state, since the 2013-keV γ ray has a pure E2 character. Panel a) shows the angu-lar correlation for the 2044-keV line, depopulating thestate at 4057 keV. The results suggest a M1+E2 charac-ter for this transition, with a mixing ratio δ =0.66 +0 . − . .The shape of the angular correlation is compatible witha 3/2 − spin-parity assignment for the 4057-keV state.Panel b) presents similar results for the 2795-keV γ -ray.The experimental fit indicates a M1+E2 character, witha mixing ratio δ =0.58 +0 . − . . In this case, the angular cor-relation is well reproduced assuming a 1/2 − spin-parityassignment for the 4808-keV state. Finally, panel c)shows the angular correlation for the 4400-keV, primarytransition. A dominant E1 character is found, with aM2 mixing with δ =-0.23(6). Moreover, the shape of theangular correlation suggests a 3/2 − spin-parity assign-ment for the 2876-keV state. The energies of levels and γ rays, along with the γ -ray multipolarities, mixing ra-tios, branching ratios, and γ -ray intensities are presentedin Tab. IV, where new results are marked by stars. C. Ca The level scheme of the Ca nucleus obtained in thiswork is shown on the left side of Fig. 7. New γ raysare marked in red, with tentative transitions displayedas dashed lines. The known γ rays reported in the liter-ature are shown in black [48, 51, 52].An example of γ -ray spectrum, gated on the 1/2 − → − , 2023-keV transition, is presented on the right sideof Fig. 7. The most intense γ -ray, namely the 3123-keV,corresponds to the direct populations of the 1/2 − stateat 2023 keV, from the neutron capture level. Two new γ lines with energies 1074 keV and 2049 keV are alsopresent. The first corresponds to the primary transitionpopulating the 3/2 − state at 4072 keV, while the latteris the 3/2 − → − decay, depopulating the 4072-keVstate. Finally, the 2249-keV line depopulating the 4272-keV level and reported in [48, 51] was not observed inthe present work, suggesting a branch from this level tothe 2023-keV state ≤ Ca sample used in the current measurementwas contaminated by other nuclei with a non-negligiblecross section for neutron capture. This is the case, forexample, of the
Cd nucleus, the neutron-capture crosssection of which is about 20 kb. Therefore, even if presentin small quantity, the γ -ray decay of the Cd isotope C oun t s gate 565 st nd a) nd st
511 1182 C oun t s gate 862 st nd b) st nd
5× 10× C oun t s E γ [keV] gate 3218
862 1458 c) C oun t s FIG. 5. (Color online) Projections of the γ - γ coincidencematrix measured in the Ca(n, γ ) reaction. Gates on the565-keV a), 862-keV b), and 3218-keV c) γ rays are presented,with new transitions, observed for the first time, displayed inred. First and second escape peaks for high-energy transitionsare marked by 1 st and 2 nd , respectively. is rather strong. In Fig. 7, γ rays corresponding to (n, γ )reactions on target contaminants are marked by circles.The energies of levels and γ rays, along with branchingratios and γ -ray intensities are presented in Tab. V. Newresults obtained in this work are marked by stars, while γ -ray multipolarities are taken from [48, 51]. It is impor-tant to note that in the case of the 5146-keV transition,the procedure described in Sec. III to extract branchingratios cannot be applied, since the γ ray feeds directlythe ground state. According to the literature [48, 51],the 5146-keV transition is 3 times larger than the 3123--keV transition, therefore such a value was adopted toproperly normalize the branching ratios for the γ raysdepopulating the capture state.1 a)2044 - 20143/2 - - - b)2795 - 20141/2 - - - + - ;3/2 - - W ( W ( ) [deg] (cid:1)(cid:2) (cid:1)(cid:2) (cid:1) / (cid:2) - (cid:3)(cid:4) / (cid:2) - (cid:2)(cid:3)(cid:5)(cid:4)(cid:4) / (cid:2) + (cid:2)(cid:6)(cid:1)(cid:7)(cid:5) / (cid:2) + (cid:2)(cid:6)(cid:8)(cid:8)(cid:4) / (cid:2) - (cid:2)(cid:7)(cid:1)(cid:6)(cid:4) / (cid:2) - (cid:9)(cid:3)(cid:6)(cid:1) ( (cid:5) / (cid:2) - (cid:10) (cid:4) / (cid:2) - ) (cid:9)(cid:9)(cid:6)(cid:3)(cid:5) / (cid:2) - (cid:9)(cid:7)(cid:3)(cid:7)(cid:5) / (cid:2) + (cid:1)(cid:2)(cid:1)(cid:6) (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:3)(cid:5)(cid:6)(cid:4)(cid:3)(cid:6)(cid:3)(cid:4)(cid:4)(cid:7)(cid:7)(cid:8)(cid:2)(cid:9)(cid:10)(cid:2)(cid:10)(cid:2)(cid:1)(cid:2)(cid:4)(cid:3)(cid:6)(cid:2)(cid:6)(cid:5)(cid:1)(cid:2)(cid:2)(cid:7)(cid:5)(cid:9)(cid:5)(cid:8)(cid:1)(cid:9)(cid:10)(cid:6)(cid:2)(cid:2)(cid:7)(cid:4)(cid:4)(cid:9)(cid:4)(cid:6)(cid:5)(cid:9)(cid:4)(cid:1)(cid:10)(cid:9)(cid:9)(cid:10)(cid:2)(cid:10)(cid:3)(cid:2)(cid:2)(cid:5)(cid:6)(cid:1)(cid:10)(cid:3)(cid:1)(cid:3)(cid:1)(cid:2)(cid:7)(cid:9)(cid:8) (cid:1)(cid:2) (cid:1)(cid:2) (cid:1) / (cid:2) - (cid:3)(cid:4) / (cid:2) - (cid:2)(cid:3)(cid:5)(cid:4)(cid:4) / (cid:2) + (cid:2)(cid:6)(cid:1)(cid:7)(cid:5) / (cid:2) + (cid:2)(cid:6)(cid:8)(cid:8)(cid:4) / (cid:2) - (cid:2)(cid:7)(cid:1)(cid:6)(cid:4) / (cid:2) - (cid:9)(cid:3)(cid:6)(cid:1) ( (cid:5) / (cid:2) - (cid:10) (cid:4) / (cid:2) - ) (cid:9)(cid:9)(cid:6)(cid:3)(cid:5) / (cid:2) - (cid:9)(cid:7)(cid:3)(cid:7)(cid:5) / (cid:2) + (cid:1)(cid:2)(cid:1)(cid:6) (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:3)(cid:5)(cid:6)(cid:4)(cid:3)(cid:6)(cid:3)(cid:4)(cid:4)(cid:7)(cid:7)(cid:8)(cid:2)(cid:9)(cid:10) (cid:1)(cid:2)(cid:1)(cid:3) (cid:2)(cid:4)(cid:3)(cid:6)(cid:2)(cid:6)(cid:5)(cid:1)(cid:2)(cid:2)(cid:7)(cid:5)(cid:9)(cid:5)(cid:8)(cid:1)(cid:9)(cid:10)(cid:6)(cid:2)(cid:2)(cid:7)(cid:4)(cid:4)(cid:9)(cid:4)(cid:6)(cid:5)(cid:9)(cid:4)(cid:1)(cid:10)(cid:9)(cid:9)(cid:10)(cid:2)(cid:10)(cid:3)(cid:2)(cid:2)(cid:5)(cid:6)(cid:1)(cid:10)(cid:3)(cid:1)(cid:3)(cid:1)(cid:2)(cid:7)(cid:9)(cid:8) (cid:1)(cid:2) (cid:1)(cid:2) (cid:1) / (cid:2) - (cid:3)(cid:4) / (cid:2) - (cid:2)(cid:3)(cid:5)(cid:4)(cid:4) / (cid:2) + (cid:2)(cid:6)(cid:1)(cid:7)(cid:5) / (cid:2) + (cid:2)(cid:6)(cid:8)(cid:8)(cid:4) / (cid:2) - (cid:2)(cid:7)(cid:1)(cid:9)(cid:4) / (cid:2) - (cid:10)(cid:3)(cid:6)(cid:1) ( (cid:5) / (cid:2) - (cid:11) (cid:4) / (cid:2) - ) (cid:10)(cid:10)(cid:6)(cid:3)(cid:5) / (cid:2) - (cid:10)(cid:7)(cid:3)(cid:7)(cid:5) / (cid:2) + (cid:1)(cid:2)(cid:1)(cid:6) (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:3)(cid:5)(cid:6)(cid:4)(cid:3)(cid:6)(cid:3)(cid:4)(cid:4)(cid:7)(cid:7)(cid:8)(cid:2)(cid:9)(cid:10) (cid:1)(cid:2)(cid:1)(cid:3) (cid:2)(cid:4)(cid:3)(cid:6)(cid:2)(cid:6)(cid:5)(cid:1)(cid:2)(cid:2)(cid:7)(cid:5)(cid:9)(cid:5)(cid:8)(cid:8)(cid:9)(cid:10)(cid:6)(cid:2)(cid:2)(cid:7)(cid:4)(cid:4)(cid:9)(cid:4)(cid:6)(cid:5)(cid:9)(cid:4)(cid:1)(cid:10)(cid:9)(cid:9)(cid:10)(cid:2)(cid:10)(cid:3)(cid:2)(cid:2)(cid:5)(cid:6)(cid:1)(cid:10)(cid:3)(cid:1)(cid:3)(cid:1)(cid:2)(cid:7)(cid:9)(cid:8) (cid:1)(cid:2) (cid:1)(cid:2) (cid:1) / (cid:2) - (cid:3)(cid:4) / (cid:2) - (cid:2)(cid:3)(cid:5)(cid:4)(cid:4) / (cid:2) + (cid:2)(cid:6)(cid:1)(cid:7)(cid:5) / (cid:2) + (cid:2)(cid:6)(cid:8)(cid:8)(cid:4) / (cid:2) - (cid:2)(cid:7)(cid:1)(cid:9)(cid:4) / (cid:2) - (cid:10)(cid:3)(cid:6)(cid:1) ( (cid:5) / (cid:2) - (cid:11) (cid:4) / (cid:2) - ) (cid:10)(cid:10)(cid:6)(cid:3)(cid:5) / (cid:2) - (cid:10)(cid:7)(cid:3)(cid:7)(cid:5) / (cid:2) + (cid:1)(cid:2)(cid:1)(cid:6) (cid:1)(cid:2)(cid:3)(cid:2)(cid:4)(cid:3)(cid:5)(cid:6)(cid:4)(cid:3)(cid:6)(cid:3)(cid:4)(cid:4)(cid:7)(cid:7)(cid:8)(cid:2)(cid:9)(cid:10) (cid:1)(cid:2)(cid:1)(cid:3) (cid:2)(cid:4)(cid:3)(cid:6)(cid:2)(cid:6)(cid:5)(cid:1)(cid:2)(cid:2)(cid:7)(cid:5)(cid:9)(cid:5)(cid:8)(cid:8)(cid:2)(cid:4)(cid:8)(cid:6)(cid:9)(cid:10)(cid:6)(cid:2)(cid:2)(cid:7)(cid:4)(cid:4)(cid:9)(cid:4)(cid:6)(cid:5)(cid:9)(cid:4)(cid:1)(cid:10)(cid:9)(cid:9)(cid:10)(cid:2)(cid:10)(cid:3)(cid:2)(cid:1)(cid:10)(cid:3)(cid:1)(cid:3)(cid:1)(cid:2)(cid:7)(cid:9)(cid:8) FIG. 6. (Color online) (Left) Level scheme of Ca as measured in the current experiment. Newly-observed γ -ray transitionsare reported in red. (Right) Angular correlations in Ca which enabled to pin down the multipolarity of the 2044-keV a),2795-keV b), and 4400-keV c) transitions (see Sec. III B for discussion). Experimental data are presented as dots along withexperimental fits (solid red line), while theoretical predictions are shown as dashed blue line.TABLE IV. Initial and final states, γ -ray energies, multipolarities, mixing ratios, branching ratios, and γ -ray intensities nor-malized to the 2013.2-keV transition (100 units) of Ca, as observed in this work. New findings are marked by a star.Multipolarities and mixing ratios not measured in this work are also reported [47, 48, 50].E i [keV] J πi E f [keV] J πf E γ [keV] Multipolarity δ BR γ I γ − − + − + − − * 2013.2(1) 3/2 − − * 2875.6(3) 3/2 − + + − +0 . − . * 0.713(18) 6.444(469)4450.2(2) (1/2 − ,3/2 − ) 2578.3(1) 3/2 + − − * 2875.6(3) 3/2 − + − +0 . − . * 0.861(20) 1.591(106)7275.4(2) 1/2 + − − ,3/2 − ) 2825.1(2)* 0.0040(4) 0.398(29)4057.3(2) 3/2 − − + + − C oun t s E [keV] gate 2023 st × (cid:1)(cid:2) (cid:1)(cid:2) (cid:1) / (cid:2) - (cid:3)(cid:4) / (cid:2) - (cid:2)(cid:3)(cid:2)(cid:1)(cid:1)(cid:5)(cid:6)(cid:4)(cid:1) / (cid:2) - (cid:7)(cid:3)(cid:8)(cid:2)(cid:4) / (cid:2) - (cid:7)(cid:2)(cid:6)(cid:4)(cid:4) / (cid:2) - (cid:7)(cid:2)(cid:8)(cid:2)(cid:4) / (cid:2) + (cid:9)(cid:4)(cid:7)(cid:6) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6)(cid:5)(cid:2)(cid:6)(cid:7)(cid:4)(cid:2)(cid:8)(cid:9)(cid:3)(cid:7)(cid:7)(cid:1)(cid:7)(cid:9)(cid:3)(cid:3)(cid:6)(cid:9)(cid:6)(cid:3)(cid:6)(cid:4)(cid:2)(cid:3)(cid:8)(cid:9)(cid:6)(cid:6)(cid:8)(cid:3)(cid:10)(cid:5)(cid:7)(cid:4)(cid:2)(cid:6)(cid:8)(cid:6)(cid:5) (1/2 - ,3/2 - ) (cid:1) (cid:2)(cid:3)(cid:4) (cid:5) (cid:6) (cid:7)(cid:8)(cid:7)(cid:7)(cid:9)(cid:7)(cid:7)(cid:7)(cid:9)(cid:8)(cid:7)(cid:7)(cid:10)(cid:7)(cid:7)(cid:7) (cid:1) (cid:11) (cid:12) (cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18) (cid:7) (cid:9)(cid:7)(cid:7)(cid:7) (cid:10)(cid:7)(cid:7)(cid:7) (cid:19)(cid:7)(cid:7)(cid:7) (cid:20)(cid:7)(cid:7)(cid:7) (cid:1) (cid:21)(cid:22)(cid:5)(cid:16)(cid:13)(cid:10)(cid:7)(cid:10)(cid:19) (cid:19)(cid:9)(cid:10)(cid:19) (cid:9) (cid:6)(cid:5) (cid:10)(cid:7)(cid:20)(cid:23)(cid:9)(cid:10)(cid:24)(cid:25)(cid:9)(cid:7)(cid:26)(cid:20) (cid:9)(cid:7)(cid:27) (cid:1)(cid:2) (cid:1)(cid:2) (cid:1) / (cid:2) - (cid:3)(cid:4) / (cid:2) - (cid:2)(cid:3)(cid:2)(cid:1)(cid:1)(cid:5)(cid:6)(cid:4)(cid:1) / (cid:2) - (cid:7)(cid:3)(cid:8)(cid:2)(cid:4) / (cid:2) - (cid:7)(cid:2)(cid:6)(cid:4)(cid:4) / (cid:2) - (cid:7)(cid:2)(cid:8)(cid:2)(cid:4) / (cid:2) + (cid:9)(cid:4)(cid:7)(cid:6) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6)(cid:5)(cid:2)(cid:6)(cid:7)(cid:4)(cid:2)(cid:8)(cid:9)(cid:3)(cid:7)(cid:7)(cid:1)(cid:7)(cid:9)(cid:3)(cid:3)(cid:6)(cid:9)(cid:6)(cid:3)(cid:6)(cid:4)(cid:2)(cid:3)(cid:8)(cid:9)(cid:6)(cid:6)(cid:8)(cid:3)(cid:10)(cid:5)(cid:7)(cid:4)(cid:2)(cid:6)(cid:8)(cid:6)(cid:5) (1/2 - ,3/2 - ) FIG. 7. (Left) Level scheme of Ca, as obtained in this work, with new transitions displayed in red. (Right) Projection of the γ - γ coincidence matrix gated on the 2023-keV transition in Ca, showing two new γ rays at 1074 keV and 2049 keV. γ rayscoming from (n, γ ) reactions on target contaminants are marked by circles (see Sec. III C for details). The first escape peak forthe 3123-keV transition is marked by 1 st .TABLE V. Initial and final states, γ -ray energies, branching ratios, and γ -ray intensities normalized to the 5145.9-keV transition(100 units) of Ca, as observed in this work. New findings are marked by a star. Multipolarities are taken from [48, 51].E i [keV] J πi E f [keV] J πf E γ [keV] Multipolarity δ BR γ I γ − − − ,3/2 − ) 0 3/2 − − − − − − − − + − − − − ,3/2 − ) 1286.1(2)* 0.0042(9) 0.566(118)2023.2(1) 1/2 − − IV. COMPARISON WITH THEORY
The experimental excitation energy spectra of Ca, Ca and Ca have been partially compared with theo-retical calculations performed in the framework of theHybrid Configuration Mixing Model (HCM) [35, 36].The model was designed to microscopically describe one-valence-particle/hole nuclei with respect to an even-even,doubly-magic “core” with mass A and it is based on aHamiltonian of Skyrme type, which for the particle-corecoupling case reads H = H + V,H = (cid:88) jm (cid:15) j a † jm a jm + (cid:88) NJM ¯ hω NJ Γ † JM Γ JM , (2) V = (cid:88) jmj (cid:48) m (cid:48) (cid:88) NJM h ( jm ; j (cid:48) m (cid:48) , N JM ) a jm [ a † j (cid:48) m (cid:48) ⊗ Γ † JM ] jm .H is the mean-field solution corresponding to Hartree-Fock (HF) particle states and Random Phase Approxi-mation (RPA) excitations of the core calculated accord-ing to Ref. [53], with a † and Γ † being the usual fermion-creator and boson-creator operators, respectively. V isthe coupling between single-particle states and core ex-citations (see Ref. [54] for details). A similar expres-sion holds for the hole-core coupling case [55]. Conse-quently, the model accounts for both single-particle/holestates and couplings with core excitations, predicting ei-ther particle/hole-phonon coupled states or 2p-1h and2h-1p shell-model-like excitations, or hybrid mixtures, forthe A+1 and
A-1 systems, respectively. It is importantto note that the orthonormality and the completeness ofbasis states are properly taken into account by eliminat-ing, from the model space, those spurious configurationswhich violate the Pauli principle. The wave functions | Ψ n (cid:105) for each state are then written in terms of the orig-inal basis | α (cid:105) as | Ψ n (cid:105) = (cid:88) α ξ n ( α ) | α (cid:105) , (3)where ξ n are the amplitudes of each component. Inthe case of pure single-particle/hole states, ξ n can beinterpreted as the spectroscopic factor. Particular atten-tion was given to the choice of the Skyrme interaction.In this work, calculations performed with the SkX TABLE VI. Experimental energy and B(E3; 3 − → + ) valueof the 3 − phonon in the Ca nucleus [29], compared withRPA calculations performed with the SkX [56] and SLy5 [57]Skyrme interactions. Calculations are done according to [53].E − [MeV] B(E3; 3 − → + ) [W.u.]EXP 3.74 27.7(30)SkX 2.95 17.0SLy5 3.67 21.5 parametrization [56] are presented for all nuclei. Thisinteraction was fitted on binding energies, charge radiibut also single-particle energies of many doubly-magicisotopes, resulting in an effective mass m*/m ≈
1. Yet,different parametrizations were also tested. Calculationsdone with the SLy5 interaction (m*/m ≈ − phononin the Ca nucleus (see Tab VI). As explained inRef. [56], single-particle energies predicted by the SkXinteraction specifically for this nucleus quantitativelydiffer from the experimental values. Such an effect canbe ascribed to proton-neutron correlations, which areparticularly enhanced in
N=Z systems. In Ca, 4p-4hand 8p-8h excitations start playing a crucial role even atlow energies [30], thus affecting the shell structure. Thisaspect is less pronounced in neutron-rich systems, whereexcitations of the neutron excess dominate over thoseof the symmetric core. The Ca and Ca, RPA coreexcitations, used in this work for HCM calculations, arereported in Tabs. VII and VIII. Along with spins andenergies, the main components of the wave functionsand the B(E λ ; J πn → +g.s. ) values for low-spin states arepresented for both collective phonons and non-collectiveexcitations. Calculations for the Ca nucleus wereperformed by assuming a Ca core and includingneutron single-particle states of the pf g / shell and the sd levels above the N =50 shell gap. In the cases of the Ca and Ca isotopes, a Ca core was taken withthe full hole space for the former and by including the pf / g / orbitals for the latter. In the first case, Cacore excitations up to 8 MeV and angular momentumL=8 were considered, while for Ca core excitationsup to 6 MeV and L=8 were taken into account. Coreexcitations in Ca are located, in general, at higherexcitation energy than in Ca. We note that the 8MeV, L=8 cutoffs in Ca select only negative-paritystates, as positive-parity ones are predicted to be evenhigher.Fig. 8 shows a comparison between predictions fromthe HCM model, with the SkX Skyrme interaction, andexperimental low-spin yrast states of Ca (left), Ca(middle) and Ca (right), obtained in this work. In thefollowing, the comparison will be limited to the energy ofthe states, since very limited mixing ratios informationis experimentally available for transitions depopulatingsuch states. For detailed comparison in terms of selectedB(E3) values we refer to Ref. [5]. The results of thecalculations are also summarized in Tab. IX, for whatconcerns state energies and dominant wave functioncomponents. It is important to stress that similarpredictions are obtained by using the SLy5 interaction:small differences are observed in the energies of thelevels (within ≈
300 keV on average), whereas wavefunction compositions are almost independent from thechoice of the interaction.The Ca ground state is predicted to have a pure f / ,single-neutron nature. In the case of excited states, thecomparison between experimental results and theoretical4 TABLE VII. RPA results for Ca core excitations used in theHCM calculations (see text for details), showing spins, ener-gies and main composition of the wave function, along withthe squared X RPA forward amplitudes. Only componentswith X ≥ λ ; J πn → +g.s. ) values for the1 − and 3 − states are also reported.J πn E [keV] main w. f. B(E λ ; J πn → +g.s. )composition [W.u.] Ca − πd − / f / (0.11) 2.06 · − πd − / p / (0.33) νd − / f / (0.10) νd − / p / (0.24)2 − πd − / f / (0.52) νd − / f / (0.45)2 − νd − / f / (0.51) πd − / f / (0.44)3 − πd − / f / (0.30) 16.95 νd − / f / (0.27) πs − / f / (0.23) νs − / f / (0.21)3 − πd − / f / (0.62) 0.52 πs − / f / (0.32)3 − νd − / f / (0.60) 1.07 νs − / f / (0.32)3 − νs − / f / (0.44) 0.28 πs − / f / (0.36)4 − πd − / f / (0.89)4 − νd − / f / (0.87)4 − πs − / f / (0.65) νs − / f / (0.34)4 − νs − / f / (0.63) πs − / f / (0.30)5 − πd − / f / (0.59) νd − / f / (0.41)5 − νd − / f / (0.40) πd − / f / (0.40) predictions is limited to positive-parity states below 4MeV (see Fig 8). In this region, the HCM model predictsa multiplet of states with spin 1/2 + , 3/2 + and 5/2 + at2468 keV, 2608 keV, and 2429 keV, respectively. Thesestates arise mainly from the coupling between a f / neutron and the octupole 3 − vibration of the Ca core,with contributions from couplings with other phonons(see Tab. IX). A good correspondence in terms of levelordering and energy spacing with the lowest experimen-
TABLE VIII. RPA results for Ca core excitations used inthe HCM calculations (see text for details), showing spins, en-ergies and main composition of the wave function, along withthe squared X RPA forward amplitudes. Only componentswith X ≥ λ ; J πn → +g.s. ) values for the2 + and 3 − states are also reported.J πn E [keV] main w. f. B(E λ ; J πn → +g.s. )composition [W.u.] Ca +1 νf − / p / (0.98) 1.313 +1 νf − / p / (0.99)3 +2 νf − / p / (0.98)3 − πs − / f / (0.76) 6.77 πd − / f / (0.19)3 − πd − / f / (0.79) πs − / f / (0.20)4 +1 νf − / p / (1.00)4 +2 νf − / p / (0.98)4 − πs − / f / (0.92)5 +1 νf − / p / (1.00) tal 1/2 + , 3/2 + and 5/2 + states is found. However, thecalculated multiplet is located ≈ − octupole vibration of Ca, whichis predicted ≈
800 keV below the experimental value.Calculations for the Ca nucleus are presented in themiddle of Fig. 8. These are the first results obtainedwith the HCM model for a valence-hole system in thismass region. The ground state of Ca is calculatedas a neutron f − / configuration. Concerning the 3/2 − negative-parity state, it is predicted to be the couplingbetween a f − / neutron hole and non-collective 1p-1hexcitations of the Ca core, namely (f − / p / ) +1 and(f − / p / ) +1 . It is interesting to note that the wavefunction composition is similar to the one obtainedby shell model calculations in the full fpg space, i.e.,[ νf − / p / ] [58]. The positive-parity 1/2 + and 3/2 + states are instead suggested to be members of the ν f − / ⊗ − multiplet and are predicted about 1 MeV higherthan in experiments. These results indicate that thelow-spin structure of the Ca nucleus is more complexand probably contains more configurations than thoseincluded in the present HCM model.Finally, the level scheme of the Ca nucleus is presentedon the right side of Fig. 8. Its ground state is predictedto be pure, with a ν p / configuration, as well asthe 1/2 − state which is suggested to be a p / , pure5 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / C a C a C a E X P H C M E X P H C M E X P H C M ( / ± , / ± )( / + , / + )( / + , / + ) / + , / + , / + / + , / + / - ( / - , / - ) FIG. 8. Experimental low-spin states in the Ca, Ca, and Ca nuclei, compared with theoretical calculations performedwith the HCM model, using the SkX Skyrme interaction. Higher-spin levels observed in Ref. [5] are shown as dashed lines (seetext for details) . TABLE IX. Results of the Hybrid Configuration Mixing Model calculations for different states in Ca, Ca, and Ca. Themain components | α (cid:105) of the wave functions are reported, along with the corresponding squared amplitudes ξ n ( α ), consideringonly contributions with ξ n ( α ) ≥ π E [keV] | α (cid:105) ξ n ( α ) Ca − ν f / + ν f / ⊗ − + ν f / ⊗ − ν f / ⊗ − + ν f / ⊗ − + ν f / ⊗ − ν f / ⊗ − + ν f / ⊗ − + ν f / ⊗ − ν f / ⊗ − + ν f / ⊗ − ν f / ⊗ − Ca − ν f − / − ν f − / ⊗ ( ν f − / p / ) +1 ν f − / ⊗ ( ν f − / p / ) +1 + ν s − / ν f − / ⊗ − + ν d − / ν f − / ⊗ − + ν f − / ⊗ − + ,9/2 + ν f − / ⊗ − + ν f − / ⊗ − + ν f − / ⊗ − Ca − ν p / − ν p / + ν g / ν p / ⊗ − + ,5/2 + ,7/2 + ν p / ⊗ − + state is insteadcalculated as the lowest member of the ν p3 / ⊗ − multiplet.All the calculated members of the ν f / ⊗ − , ν f − / ⊗ − and ν p3 / ⊗ − multiplets in Ca, Ca and Ca, respectively, are presented in Fig. 8 and theirwave function compositions are given in Tab. IX. Thesestates in Ca and Ca were already investigated bythis collaboration in Ref. [5] and interpreted using theperturbative particle-vibration coupling approach [32].Similar studies were also performed in the case of the Cu and Cu nuclei, where Ni cores were consid-ered [59, 60]. This neutron-rich region around Z =28is characterized by the coexistence of different nuclearshapes [61–66], the emergence of which is intimately re-lated to proton-neutron correlations and shell structure.In this sense, a microscopic description of the manyfacets of nuclear excitations is much desirable and theHybrid Configuration Mixing model presented in thiswork provides a step forward in this direction.In the case of Ca, low-lying positive-parity states upto 13/2 + are predicted to be members of the multiplet,located around 2500 keV of excitation energy, in arange of ≈
400 keV. Despite the fact that the ν f / ⊗ − coupling is the dominant component in their wavefunction, contributions from couplings with other coreexcitations are significantly present. In the case of Ca,in addition to the 1/2 + and 3/2 + states discussed above,the higher spin states between 5/2 + and 13/2 + are dis-played and compared to experimental energies obtainedin previous works [5] and shown by dashed lines. TheHCM model also predicts the B(E3; (13/2 + ,11/2 + ) → − ) = 6.7 W.u., which is in agreement, withinthe error, with the 7.4(19) W.u. experimental valuereported in Ref. [5]. For Ca, the 3/2 + , 5/2 + , 7/2 + ,and 9/2 + states, members of the ν p3 / ⊗ − multiplet,are also shown, and compared to the 9/2 + state locatedat 4017 keV (dashed line), which is the only one knownexperimentally, as reported in several works [3, 5, 8, 11].On the other hand, the HCM model predicts the 3/2 + ,5/2 + , 7/2 + to be degenerate at 4652 keV. This is dueto the absence of the d / , d / , and g / orbitals inthe configuration space. Nevertheless, the B(E3; 9/2 + → − ) is calculated to be 5.2 W.u., which is infair agreement with the experimental value of 7.9(20)W.u. obtained by lifetime measurements, as reported inRef. [5]The Ca and Ca isotopes were also recently inves-tigated in neutron knockout [11] and neutron pickupexperiments [8], where the strength of the ν / in Ca and the relative strength of the ν g/ and ν / in Ca were extracted. Experimental data were com-pared with large-scale shell-model calculations usingthe GXPF1 effective interaction in the sd+fp+sdg model space, as well as NN+3N, ab initio calculationsin the pf and pfg / model space (see [8, 11] and references therein). In the case of Ca, the strengthof the ν / orbital is found to be concentrated in theground state, with a measured (2J+1)C S spectroscopicfactor of 9.3( +1 . − . ) stat ( ± sys . This is qualitativelyreproduced by shell-model calculations, which predict(2J+1)C S=7.7 and (2J+1)C S=6.7-7.0 for the GXPF1and NN+3N interactions, respectively. Similar resultsare obtain in the present work with the HCM model,which estimates (2J+1)C S=7.9.For Ca, results for the C S spectroscopic factor for the9/2 + state are summarized in Tab. X. It can be seenthat the experimental results point to a rather smallvalue for the strength of the ν g/ orbital at 4296 keV,consistently with the complex octupole-coupled natureof the 9/2 + state. This quenching is more pronouncedin the case of the (d,p) measurement [67], which is wellreproduced by the HCM calculations here presented. Onthe other hand, shell-model calculations (see [8]) predicta larger spectroscopic factor, which sits in between thetwo experimental results. TABLE X. Experimental and theoretical (see [8] and refer-ences therein) C S spectroscopic factors for the 9/2 + at 4296keV in Ca.
Experiment C S (d,p) [67] 0.14 C+ Ca [8] 0.27(1)
Theory C S HCM 0.11GXPF1 [8] 0.42
V. CONCLUSIONS
In conclusion, the low-spin structure of the Ca, Ca, and Ca nuclei was investigated in the EXILLexperimental campaign, following γ -ray spectroscopy ofneutron-capture reactions on Ca targets. New levels, γ -ray transitions and γ -ray branching ratios and intensitieswere reported and a number of transitions were charac-terized by γ -ray angular correlations, enabling to extractmultiploraties and mixing ratios and to assign spins andparities to the states involved in the decays.Portions of the level schemes below 5 MeV were com-pared with theoretical calculations performed with theHybrid Configuration Mixing Model. Despite some dis-crepancies, the model indicates the coexistence, at lowenergy, of single-particle/hole states and coupled con-figurations with collective and non-collective excitationsof the doubly-magic core for all the nuclei studied inthis work. Moreover, experimental results and beyond-mean-filed calculations obtained in this work by the HCMmodel for the Ca and Ca nuclei were compared withother measurements, as well as shell model and ab intio calculations. Similar results are observed, although the8HCM model better reproduces the quenching of spectro-scopic factor for the 9/2 + state in Ca, pointing to theimpact of long-range correlations, such as couplings withphonons, upon the structure of nuclear excitations.Overall, it appears that Ca isotopes provide a funda-mental play ground for state-of-the-art theories, which,in this mass region, tend to converge to similar results,making Ca nuclei a cornerstone for a comprehensive de-scription of nuclear structure.This work is also an important benchmark for the Hy-brid Configuration Mixing Model, here discussed, whichbecomes a powerful tool to compute the complex struc-ture of isotopes in heavier mass regions, such as theneutron-rich region around the doubly-magic
Sn nu-cleus [68, 69]. Indeed, in these heavy systems, shell modelcalculations and ab initio methods have severe difficul- ties in dealing with collective excitations of the core, dueto the diverging dimension of the model space, thus re-sulting, up to now, in a limited description of complexexcitations.
VI. ACKNOWLEDGMENTS
This work was supported by the Italian IstitutoNazionale di Fisica Nucleare and by the Pol-ish National Science Centre under Contract Nos.2014/14/M/ST2/00738 and 2013/08/M/ST2/00257.The authors would like to thank the technical services ofthe ILL, LPSC and GANIL for supporting the EXILLcampaign, as well as the EXOGAM collaboration andthe INFN Legnaro for providing HPGe detectors. [1] Y. Yasuda, H. Sakaguchi, S. Asaji, K. Fujita, Y. Hag-ihara, K. Hatanaka, T. Ishida, M. Itoh, T. Kawabata,S. Kishi, T. Noro, Y. Sakemi, Y. Shimizu, H. Takeda,Y. Tameshige, S. Terashima, M. Uchida, T. Wakasa,T. Yonemura, H. P. Yoshida, M. Yosoi, and J. Zenihiro,Phys. Rev. C , 044315 (2010).[2] Y.-W. Lui, D. H. Youngblood, S. Shlomo, X. Chen,Y. Tokimoto, Krishichayan, M. Anders, and J. Button,Phys. Rev. C , 044327 (2011).[3] D. Montanari, S. Leoni, D. Mengoni, G. Benzoni,N. Blasi, G. Bocchi, P. Bortignon, A. Bracco, F. Cam-era, G. Col, A. Corsi, F. Crespi, B. Million, R. Nicolini,O. Wieland, J. Valiente-Dobon, L. Corradi, G. de An-gelis, F. D. Vedova, E. Fioretto, A. Gadea, D. Napoli,R. Orlandi, F. Recchia, E. Sahin, R. Silvestri, A. Ste-fanini, R. Singh, S. Szilner, D. Bazzacco, E. Farnea,R. Menegazzo, A. Gottardo, S. Lenzi, S. Lunardi,G. Montagnoli, F. Scarlassara, C. Ur, G. L. Bianco,A. Zucchiatti, M. Kmiecik, A. Maj, W. Meczynski,A. Dewald, T. Pissulla, and G. Pollarolo, Phys. Lett.B , 288 (2011).[4] D. Montanari, S. Leoni, L. Corradi, G. Pollarolo, G. Ben-zoni, N. Blasi, S. Bottoni, A. Bracco, F. Camera,A. Corsi, F. C. L. Crespi, B. Million, R. Nicolini,O. Wieland, G. de Angelis, F. Della Vedova, E. Fioretto,A. Gadea, B. Guiot, D. Mengoni, D. R. Napoli, R. Or-landi, F. Recchia, A. M. Stefanini, R. P. Singh, J. J.Valiente-Dobon, D. Bazzacco, E. Farnea, S. M. Lenzi,S. Lunardi, G. Montagnoli, F. Scarlassara, C. Ur,G. Lo Bianco, A. Zucchiatti, S. Szilner, M. Kmiecik,A. Maj, and W. Meczynski, Phys. Rev. C , 054613(2011).[5] D. Montanari, S. Leoni, D. Mengoni, J. J. Valiente-Dobon, G. Benzoni, N. Blasi, G. Bocchi, P. F. Bor-tignon, S. Bottoni, A. Bracco, F. Camera, P. Casati,G. Col`o, A. Corsi, F. C. L. Crespi, B. Million, R. Nicol-ini, O. Wieland, D. Bazzacco, E. Farnea, G. Germogli,A. Gottardo, S. M. Lenzi, S. Lunardi, R. Menegazzo,G. Montagnoli, F. Recchia, F. Scarlassara, C. Ur, L. Cor-radi, G. de Angelis, E. Fioretto, D. R. Napoli, R. Orlandi,E. Sahin, A. M. Stefanini, R. P. Singh, A. Gadea, S. Szil-ner, M. Kmiecik, A. Maj, W. Meczynski, A. Dewald, T. Pissulla, and G. Pollarolo, Phys. Rev. C , 044301(2012).[6] R. F. Garcia Ruiz, M. L. Bissell, K. Blaum,N. Fr¨ommgen, M. Hammen, J. D. Holt, M. Kowal-ska, K. Kreim, J. Men´endez, R. Neugart, G. Neyens,W. N¨ortersh¨auser, F. Nowacki, J. Papuga, A. Poves,A. Schwenk, J. Simonis, and D. T. Yordanov, Phys. Rev.C , 041304 (2015).[7] S. Noji, R. G. T. Zegers, S. M. Austin, T. Baugher,D. Bazin, B. A. Brown, C. M. Campbell, A. L. Cole,H. J. Doster, A. Gade, C. J. Guess, S. Gupta, G. W.Hitt, C. Langer, S. Lipschutz, E. Lunderberg, R. Meharc-hand, Z. Meisel, G. Perdikakis, J. Pereira, F. Rec-chia, H. Schatz, M. Scott, S. R. Stroberg, C. Sullivan,L. Valdez, C. Walz, D. Weisshaar, S. J. Williams, andK. Wimmer, Phys. Rev. C , 024312 (2015).[8] A. Gade, J. A. Tostevin, V. Bader, T. Baugher, D. Bazin,J. S. Berryman, B. A. Brown, D. J. Hartley, E. Lunder-berg, F. Recchia, S. R. Stroberg, Y. Utsuno, D. Weis-shaar, and K. Wimmer, Phys. Rev. C , 031601(R)(2016).[9] K. Hady´nska-Kl¸ek, P. J. Napiorkowski, M. Zieli´nska,J. Srebrny, A. Maj, F. Azaiez, J. J. Valiente Dob´on,M. Kici ´nska Habior, F. Nowacki, H. Na¨ıdja, B. Boun-thong, T. R. Rodr´ıguez, G. de Angelis, T. Abraham,G. Anil Kumar, D. Bazzacco, M. Bellato, D. Borto-lato, P. Bednarczyk, G. Benzoni, L. Berti, B. Birken-bach, B. Bruyneel, S. Brambilla, F. Camera, J. Chavas,B. Cederwall, L. Charles, M. Ciema(cid:32)la, P. Cocconi,P. Coleman-Smith, A. Colombo, A. Corsi, F. C. L.Crespi, D. M. Cullen, A. Czermak, P. D´esesquelles,D. T. Doherty, B. Dulny, J. Eberth, E. Farnea, B. For-nal, S. Franchoo, A. Gadea, A. Giaz, A. Gottardo,X. Grave, J. Gr¸ebosz, A. G¨orgen, M. Gulmini, T. Haber-mann, H. Hess, R. Isocrate, J. Iwanicki, G. Jaworski,D. S. Judson, A. Jungclaus, N. Karkour, M. Kmiecik,D. Karpi´nski, M. Kisieli´nski, N. Kondratyev, A. Ko-richi, M. Komorowska, M. Kowalczyk, W. Korten,M. Krzysiek, G. Lehaut, S. Leoni, J. Ljungvall,A. Lopez-Martens, S. Lunardi, G. Maron, K. Mazurek,R. Menegazzo, D. Mengoni, E. Merch´an, W. M¸eczy´nski,C. Michelagnoli, J. Mierzejewski, B. Million, S. Myal- ski, D. R. Napoli, R. Nicolini, M. Niikura, A. Obertelli,S. F. ¨Ozmen, M. Palacz, L. Pr´ochniak, A. Pullia,B. Quintana, G. Rampazzo, F. Recchia, N. Redon,P. Reiter, D. Rosso, K. Rusek, E. Sahin, M.-D. Sal-sac, P.-A. S¨oderstr¨om, I. Stefan, O. St´ezowski, J. Sty-cze´n, C. Theisen, N. Toniolo, C. A. Ur, V. Vandone,R. Wadsworth, B. Wasilewska, A. Wiens, J. L. Wood,K. Wrzosek-Lipska, and M. Zi¸ebli´nski, Phys. Rev. Lett. , 062501 (2016).[10] L. A. Riley, D. M. McPherson, M. L. Agiorgousis, T. R.Baugher, D. Bazin, M. Bowry, P. D. Cottle, F. G. De-Vone, A. Gade, M. T. Glowacki, S. D. Gregory, E. B.Haldeman, K. W. Kemper, E. Lunderberg, S. Noji,F. Recchia, B. V. Sadler, M. Scott, D. Weisshaar, andR. G. T. Zegers, Phys. Rev. C , 044327 (2016).[11] H. L. Crawford, A. O. Macchiavelli, P. Fallon, M. Albers,V. M. Bader, D. Bazin, C. M. Campbell, R. M. Clark,M. Cromaz, J. Dilling, A. Gade, A. Gallant, J. D. Holt,R. V. F. Janssens, R. Kr¨ucken, C. Langer, T. Lauritsen,I. Y. Lee, J. Men´endez, S. Noji, S. Paschalis, F. Rec-chia, J. Rissanen, A. Schwenk, M. Scott, J. Simonis,S. R. Stroberg, J. A. Tostevin, C. Walz, D. Weisshaar,A. Wiens, K. Wimmer, and S. Zhu, Phys. Rev. C ,064317 (2017).[12] R. Talwar, M. J. Bojazi, P. Mohr, K. Auranen, M. L.Avila, A. D. Ayangeakaa, J. Harker, C. R. Hoffman,C. L. Jiang, S. A. Kuvin, B. S. Meyer, K. E. Rehm,D. Santiago-Gonzalez, J. Sethi, C. Ugalde, and J. R.Winkelbauer, Phys. Rev. C , 055801 (2018).[13] K. Hady´nska-Kl¸ek, P. J. Napiorkowski, M. Zieli´nska,J. Srebrny, A. Maj, F. Azaiez, J. J. Valiente Dob´on,M. Kici ´nska Habior, F. Nowacki, H. Na¨ıdja, B. Boun-thong, T. R. Rodr´ıguez, G. de Angelis, T. Abraham,G. Anil Kumar, D. Bazzacco, M. Bellato, D. Borto-lato, P. Bednarczyk, G. Benzoni, L. Berti, B. Birkenbach,B. Bruyneel, S. Brambilla, F. Camera, J. Chavas, B. Ced-erwall, L. Charles, M. Ciema(cid:32)la, P. Cocconi, P. Coleman-Smith, A. Colombo, A. Corsi, F. C. L. Crespi, D. M.Cullen, A. Czermak, P. D´esesquelles, D. T. Doherty,B. Dulny, J. Eberth, E. Farnea, B. Fornal, S. Franchoo,A. Gadea, A. Giaz, A. Gottardo, X. Grave, J. Gr¸ebosz,A. G¨orgen, M. Gulmini, T. Habermann, H. Hess,R. Isocrate, J. Iwanicki, G. Jaworski, D. S. Judson,A. Jungclaus, N. Karkour, M. Kmiecik, D. Karpi´nski,M. Kisieli´nski, N. Kondratyev, A. Korichi, M. Ko-morowska, M. Kowalczyk, W. Korten, M. Krzysiek,G. Lehaut, S. Leoni, J. Ljungvall, A. Lopez-Martens,S. Lunardi, G. Maron, K. Mazurek, R. Menegazzo,D. Mengoni, E. Merch´an, W. M¸eczy´nski, C. Michelagnoli,B. Million, S. Myalski, D. R. Napoli, M. Niikura,A. Obertelli, S. F. ¨Ozmen, M. Palacz, L. Pr´ochniak,A. Pullia, B. Quintana, G. Rampazzo, F. Recchia, N. Re-don, P. Reiter, D. Rosso, K. Rusek, E. Sahin, M.-D. Sal-sac, P.-A. S¨oderstr¨om, I. Stefan, O. St´ezowski, J. Sty-cze´n, C. Theisen, N. Toniolo, C. A. Ur, R. Wadsworth,B. Wasilewska, A. Wiens, J. L. Wood, K. Wrzosek-Lipska, and M. Zi¸ebli´nski, Phys. Rev. C , 024326(2018).[14] F. D. Bodansky, D. Clayton and A. Fowler, Astrophys.J. Suppl. S. , 299 (1968).[15] J. W. Truran, Ap. Sapce. Sci. , 306 (1972).[16] F. K¨appeler, G. Walter, and J. Mathews, Astrophys. J. , 319 (1985). [17] D. S. E. Woosley, W. David Arnett and D. Clayton, As-trophys. J. Suppl. S. , 231 (1973).[18] F. K¨appeler, H. Beer, K. Wisshak, and D. D. Clayton,Astrophys. J. Suppl. S. , 821 (1982).[19] A. G. W. Cameron, Astrophys. J. , 53 (1979).[20] P. W. Kubik, D. Elmore, N. J. Conard, K. Nishiizumi,and J. R. Arnold, Nature , 568 (1986).[21] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki,Phys. Rev. C , 061301 (2002).[22] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki,Eur. Phys. Journ. A. , 499 (2005).[23] Y. Utsuno, T. Otsuka, B. A. Brown, M. Honma,T. Mizusaki, and N. Shimizu, Progr. Theor. Phys. Suppl. , 304 (2012).[24] N. Tsunoda, K. Takayanagi, M. Hjorth-Jensen, andT. Otsuka, Phys. Rev. C , 024313 (2014).[25] K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga, andA. Schwenk, Phys. Rev. C , 031301 (2011).[26] J. D. Holt, T. Otsuka, A. Schwenk, and T. Suzuki, Journ.Phys. G: Nucl. and Part. Phys. , 085111 (2012).[27] J. D. Holt, J. Men´endez, J. Simonis, and A. Schwenk,Phys. Rev. C , 024312 (2014).[28] J. Simonis, K. Hebeler, J. D. Holt, J. Men´endez, andA. Schwenk, Phys. Rev. C , 011302 (2016).[29] T. Kibedi and R.H.Spear, Atom. Nucl. Data Tables ,35 (2002).[30] E. Ideguchi, D. G. Sarantites, W. Reviol, A. V.Afanasjev, M. Devlin, C. Baktash, R. V. F. Janssens,D. Rudolph, A. Axelsson, M. P. Carpenter, A. Galindo-Uribarri, D. R. LaFosse, T. Lauritsen, F. Lerma, C. J.Lister, P. Reiter, D. Seweryniak, M. Weiszflog, and J. N.Wilson, Phys. Rev. Lett. , 222501 (2001).[31] T. W. Burrows, Nucl. Data Sheet , 1747 (2006).[32] A. Bohr and B. M. Mottelson, Nuclear Structure. VolumeII: Nuclear Deformations (W. A. Benjamin, New York,1980).[33] I. Hamamoto, Nucl. Phys. A , 545 (1969).[34] I. Hamamoto, Phys. Rep. , 63 (1974).[35] G. Col`o, P. F. Bortignon, and G. Bocchi, Phys. Rev. C , 034303 (2017).[36] S. Bottoni, G. Col`o, Y. Niu, and P. F. Bortignon, to bepublished.[37] M. Jentschel, A. Blanc, G. de France, U. K¨oster, S. Leoni,P. Mutti, G. Simpson, T. Soldner, C. Ur, W. Urban,S. Ahmed, A. Astier, L. Augey, T. Back, P. Baczyk,A. Bajoga, D. Balabanski, T. Belgya, G. Benzoni,C. Bernards, D. Biswas, G. Bocchi, S. Bottoni, R. Brit-ton, B. Bruyneel, J. Burnett, R. Cakirli, R. Carroll,W. Catford, B. Cederwall, I. Celikovic, N. Cieplicka-Ory´nczak, E. Clement, N. Cooper, F. Crespi, M. Csatlos,D. Curien, M. Czerwi´nski, L. Danu, A. Davies, F. Di-dierjean, F. Drouet, G. Duchˆene, C. Ducoin, K. Eber-hardt, S. Erturk, L. Fraile, A. Gottardo, L. Grente,L. Grocutt, C. Guerrero, D. Guinet, A.-L. Hartig,C. Henrich, A. Ignatov, S. Ilieva, D. Ivanova, B. John,R. John, J. Jolie, S. Kisyov, M. Krticka, T. Kon-stantinopoulos, A. Korgul, A. Krasznahorkay, T. Kr¨oll,J. Kurpeta, I. Kuti, S. Lalkovski, C. Larijani, R. Leguil-lon, R. Lica, O. Litaize, R. Lozeva, C. Magron, C. Man-cuso, E. R. Martinez, R. Massarczyk, C. Mazzocchi,B. Melon, D. Mengoni, C. Michelagnoli, B. Million,C. Mokry, S. Mukhopadhyay, K. Mulholland, A. Nannini,D. Napoli, B. Olaizola, R. Orlandi, Z. Patel, V. Paziy,C. Petrache, M. Pfeiffer, N. Pietralla, Z. Podolyak, M. Ramdhane, N. Redon, P. Regan, J. Regis, D. Regnier,R. J. Oliver, M. Rudigier, J. Runke, T. Rzaca-Urban,N. Saed-Samii, M. Salsac, M. Scheck, R. Schwengner,L. Sengele, P. Singh, J. Smith, O. Stezowski, B. Szpak,T. Thomas, M. Th¨urauf, J. Timar, A. Tom, I. Tomandl,T. Tornyi, C. Townsley, A. Tuerler, S. Valenta, A. Van-craeyenest, V. Vandone, J. Vanhoy, V. Vedia, N. Warr,V. Werner, D. Wilmsen, E. Wilson, T. Zerrouki, andM. Zielinska, J. Instrum. , 11003 (2017).[38] E. Moll, The Franco-German High Flux Reactor and itsFacilities for Nuclear Research, Nuclear Structure Studywith Neutrons (Springer, Boston, MA) (1974).[39] H. Abele, D. Dubbers, H. Hse, M. Klein, A. Knapfler,M. Kreuz, T. Lauer, B. Mrkisch, D. Mund,V. Nesvizhevsky, A. Petoukhov, C. Schmidt, M. Schu-mann, and T. Soldner, Nucl. Instrum. Meth. A ,407 (2006).[40] J. Simpson, F. Azaiez, G. DeFrance, J. Fouan, J. Gerl,R. Julin, W. Korten, P. Nolan, B. Nyak, G. Sletten, andP. Walker, Acta Phys. Hung., New Ser. Heavy Ion Phys. , 159 (2000).[41] C. R. Alvarez, Nucl. Phys. News , 3 (1993).[42] O. J. Roberts, A. M. Bruce, P. H. Regan, Z. Podolyak,C. M. Townsley, J. F. Smith, K. F. Mulholland, andA. Smith, Nucl. Instrum. Meth. A , 91 (2014).[43] J.-M. Regis, H. Mach, G. Simpson, J. Jolie, G. Pas-covici, N. Saed-Samii, N. Warr, A. Bruce, J. De-genkolb, L. Fraile, C. Fransen, D. Ghita, S. Kisyov,U. Koester, A. Korgul, S. Lalkovski, N. Marginean,P. Mutti, B. Olaizola, Z. Podolyak, P. Regan, O. Roberts,M. Rudigier, L. Stroe, W. Urban, and D. Wilmsen, Nucl.Instrum. Meth. A , 191 (2013).[44] H. Morinaga, In-beam gamma-ray spectroscopy (North-Holland; Amsterdam, Netherlands) (1976).[45] S. F. Mughabghab, Atlas of Neutron Resonances 6 th edn.(Elsevier Upton, 2018) .[46] H. Gruppelaar and P. Spilling, Nucl. Phys. A , 226(1967).[47] F. Cranston, R. Birkett, D. White, and J. Hughes, Nucl.Phys. A , 413 (1970).[48] S. E. Arnell, R. Hardell, O. Skeppstedt, and E. Wal-lander, Proc.Intern.Symp.Neutron Capture Gamma-RaySpectroscopy , (1969).[49] C. Nesaraja and E. McCutchan, Nucl. Data Sheets ,1 (2016).[50] T. Burrows, Nucl. Data Sheets , 923 (2007).[51] T. Burrows, Nucl. Data Sheets , 142 (2013).[54] G. Col`o, H. Sagawa, and P. F. Bortignon, Phys. Rev. C , 064307 (2010).[55] S. Bottoni, N. Cieplicka-Ory´nczak, G. Bocchi, S. Leoni,B. Fornal, G. Col`o, P. Bortignon, G. Benzoni, A. Blanc,A. Bracco, F. Crespi, M. Jentschel, U. K¨oster,C. Michelagnoli, B. Million, P. Mutti, T. Soldner,A. T¨urler, C. Ur, and W. Urban, EPJ Web Conf. ,05001 (2018).[56] B. Alex Brown, Phys. Rev. C , 220 (1998).[57] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, andR. Schaeffer, Nucl. Phys. A , 231 (1998). [58] A. Poves, J. Sanchez-Solano, E. Caurier, and F. Nowacki,Nucl. Phys. A , 157 (2001).[59] G. Bocchi, S. Leoni, S. Bottoni, G. Benzoni, A. Bracco,P. F. Bortignon, G. Col`o, B. Belvito, C. R. Nit¸˘a,N. Marginean, D. Filipescu, D. Ghita, T. Glodariu,R. Lica, R. Marginean, C. Mihai, A. Negret, T. Sava,L. Stroe, S. Toma, D. Bucurescu, I. Georghe, R. Suv˘ail˘a,D. Deleanu, C. A. Ur, and S. Aydin, Phys. Rev. C ,054302 (2014).[60] C. R. Nit¸˘a, D. Bucurescu, N. M˘arginean, M. Avrigeanu,G. Bocchi, S. Bottoni, A. Bracco, A. M. Bruce, G. C ˘ataDanil, G. Col´o, D. Deleanu, D. Filipescu, D. G. Ghit¸˘a,T. Glodariu, S. Leoni, C. Mihai, P. J. R. Mason,R. M˘arginean, A. Negret, D. Pantelic˘a, Z. Podolyak,P. H. Regan, T. Sava, L. Stroe, S. Toma, C. A. Ur, andE. Wilson, Phys. Rev. C , 064314 (2014).[61] B. Crider, C. Prokop, S. Liddick, M. Al-Shudifat,A. Ayangeakaa, M. Carpenter, J. Carroll, J. Chen,C. Chiara, H. David, A. Dombos, S. Go, R. Grzywacz,J. Harker, R. Janssens, N. Larson, T. Lauritsen,R. Lewis, S. Quinn, F. Recchia, A. Spyrou, S. Suchyta,W. Walters, and S. Zhu, Phys. Lett. B , 108 (2016).[62] A. I. Morales, G. Benzoni, H. Watanabe, S. Nishimura,F. Browne, R. Daido, P. Doornenbal, Y. Fang,G. Lorusso, Z. Patel, S. Rice, L. Sinclair, P.-A.S¨oderstr¨om, T. Sumikama, J. Wu, Z. Y. Xu, A. Yagi,R. Yokoyama, H. Baba, R. Avigo, F. L. Bello Gar-rote, N. Blasi, A. Bracco, F. Camera, S. Ceruti,F. C. L. Crespi, G. de Angelis, M.-C. Delattre, Z. Dom-bradi, A. Gottardo, T. Isobe, I. Kojouharov, N. Kurz,I. Kuti, K. Matsui, B. Melon, D. Mengoni, T. Miyazaki,V. Modamio-Hoyborg, S. Momiyama, D. R. Napoli,M. Niikura, R. Orlandi, H. Sakurai, E. Sahin, D. Sohler,H. Shaffner, R. Taniuchi, J. Taprogge, Z. Vajta, J. J.Valiente-Dob´on, O. Wieland, and M. Yalcinkaya, Phys.Rev. C , 034328 (2016).[63] A. Morales, G. Benzoni, H. Watanabe, Y. Tsunoda,T. Otsuka, S. Nishimura, F. Browne, R. Daido, P. Door-nenbal, Y. Fang, G. Lorusso, Z. Patel, S. Rice, L. Sin-clair, P.-A. Sderstrm, T. Sumikama, J. Wu, Z. Xu,A. Yagi, R. Yokoyama, H. Baba, R. Avigo, F. B. Garrote,N. Blasi, A. Bracco, F. Camera, S. Ceruti, F. Crespi,G. de Angelis, M.-C. Delattre, Z. Dombradi, A. Got-tardo, T. Isobe, I. Kojouharov, N. Kurz, I. Kuti, K. Mat-sui, B. Melon, D. Mengoni, T. Miyazaki, V. Modamio-Hoybjor, S. Momiyama, D. Napoli, M. Niikura, R. Or-landi, H. Sakurai, E. Sahin, D. Sohler, H. Schaffner,R. Taniuchi, J. Taprogge, Z. Vajta, J. Valiente-Dobn,O. Wieland, and M. Yalcinkaya, Phys. Lett. B , 328(2017).[64] S. Leoni, B. Fornal, N. M˘arginean, M. Sferrazza,Y. Tsunoda, T. Otsuka, G. Bocchi, F. C. L. Crespi,A. Bracco, S. Aydin, M. Boromiza, D. Bucurescu,N. Cieplicka-Ory`nczak, C. Costache, S. C˘alinescu,N. Florea, D. G. Ghit¸˘a, T. Glodariu, A. Ionescu, L. Iskra,M. Krzysiek, R. M˘arginean, C. Mihai, R. E. Mihai,A. Mitu, A. Negret¸, C. R. Nit¸˘a, A. Ol˘acel, A. Oprea,S. Pascu, P. Petkov, C. Petrone, G. Porzio, A. S¸erban,C. Sotty, L. Stan, I. S¸tiru, L. Stroe, R. S¸uv˘ail˘a, S. Toma,A. Turturic˘a, S. Ujeniuc, and C. A. Ur, Phys. Rev. Lett. , 162502 (2017).[65] N. M˘arginean, D. Little, Y. Tsunoda, S. Leoni, R. V. F.Janssens, B. Fornal, T. Otsuka, C. Michelagnoli, L. Stan,F. C. L. Crespi, C. Costache, R. Lica, M. Sfer- razza, A. Turturica, A. D. Ayangeakaa, K. Auranen,M. Barani, P. C. Bender, S. Bottoni, M. Boromiza,A. Bracco, S. C˘alinescu, C. M. Campbell, M. P. Carpen-ter, P. Chowdhury, M. Ciema(cid:32)la, N. Cieplicka-Ory`nczak,D. Cline, C. Clisu, H. L. Crawford, I. E. Dinescu, J. Du-douet, D. Filipescu, N. Florea, A. M. Forney, S. Fra-cassetti, A. Gade, I. Gheorghe, A. B. Hayes, I. Harca,J. Henderson, A. Ionescu, L. W. Iskra, M. Jentschel,F. Kandzia, Y. H. Kim, F. G. Kondev, G. Korschinek,U. K¨oster, Krishichayan, M. Krzysiek, T. Lauritsen,J. Li, R. M˘arginean, E. A. Maugeri, C. Mihai, R. E. Mi-hai, A. Mitu, P. Mutti, A. Negret, C. R. Nit¸˘a, A. Ol˘acel,A. Oprea, S. Pascu, C. Petrone, C. Porzio, D. Rhodes,D. Seweryniak, D. Schumann, C. Sotty, S. M. Stolze,R. S¸uv˘ail˘a, S. Toma, S. Ujeniuc, W. B. Walters, C. Y.Wu, J. Wu, S. Zhu, and S. Ziliani, Phys. Rev. Lett. ,102502 (2020).[66] C. Porzio, C. Michelagnoli, N. Cieplicka-Ory´nczak,M. Sferrazza, S. Leoni, B. Fornal, Y. Tsunoda, T. Ot-suka, S. Bottoni, C. Costache, F. C. L. Crespi, L. W.Iskra, M. Jentschel, F. Kandzia, Y.-H. Kim, U. K¨oester,N. M˘arginean, C. Mihai, P. Mutti, and A. Turturica, accepted in Phys. Rev. C.[67] Y. Uozumi, O. Iwamoto, S. Widodo, A. Nohtomi,T. Sakae, M. Matoba, M. Nakano, T. Maki, andN. Koori, Nucl. Phys. A , 123 (1994).[68] G. Bocchi, S. Leoni, B. Fornal, G. Col`o, P. Bortignon,S. Bottoni, A. Bracco, C. Michelagnoli, D. Bazza-cco, A. Blanc, G. de France, M. Jentschel, U. K¨oster,P. Mutti, J. Regis, G. Simpson, T. Soldner, C. Ur,W. Urban, L. Fraile, R. Lozeva, B. Belvito, G. Benzoni,A. Bruce, R. Carroll, N. Cieplicka-Orynczak, F. Crespi,F. Didierjean, J. Jolie, W. Korten, T. Kr¨oll, S. Lalkovski,H. Mach, N. Marginean, B. Melon, D. Mengoni, B. Mil-lion, A. Nannini, D. Napoli, B. Olaizola, V. Paziy,Z. Podolyak, P. Regan, N. Saed-Samii, B. Szpak, andV. Vedia, Phys. Lett. B , 273 (2016).[69] S. Bottoni, L. Iskra, S. Leoni, B. Fornal, G. Col`o,D. Bazzacco, L. Gatti, G. Benzoni, A. Blanc, G. Boc-chi, A. Bracco, N. Cieplicka-Ory´nczak, F. Crespi,M. Jentschel, U. K¨oster, C. Michelagnoli, B. Million,P. Mutti, T. Soldner, C. Ur, and W. Urban, Acta. Phys.Pol. B50