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The generator coordinate method begins with the variational construction of a set of non-orthogonal mean-field states that span a subspace of the full many-body Hilbert space. These states are then often projected onto states with good quantum numbers to restore symmetries, leading to a set with members that can be similar to one another, and it is sometimes possible to reduce this set without greatly affecting results. Here we propose a greedy algorithm that we call the energy-transition-orthogonality procedure (ENTROP) to select subsets of important states. As applied here, the approach selects on the basis of diagonal energy, orthogonality, and contribution to the matrix element that governs neutrinoless double-$\beta$ decay. We present both shell-model and preliminary ab initio calculations of this matrix element for the decay of $^{76}$Ge, with quadrupole deformation parameters and the isoscalar pairing strength as generator coordinates. ENTROP converges quickly, reducing significantly the number of basis states needed for an accurate calculation., Comment: 7 pages, 9 figures, authors added, some details clarified

The potential sub-shell closures N = 32 and N = 34 are analyzed at high spins in the titanium isotopes within the generalized beyond mean field theory considering triaxial deformations and the angular frequency as generator coordinates together with the particle number and the angular momentum conservation. A good description of bulk properties, high angular momenta spectra and transition probabilities is obtained. The outcome at high spin in these nuclei is consistent with the magic number character of N = 32 but not of N = 34., Published: Phys. Lett. B804 (2020) 135359

Background: Shape coexistence in the Z≈82 region has been established in mercury, lead, and polonium isotopes. For even-even mercury isotopes with 100≤N≤106 multiple fingerprints of this phenomenon are observed, which seems to be no longer present for N≥110. According to a number of theoretical calculations, shape coexistence is predicted in the Hg188 isotope. Purpose: The aim of this work was to measure lifetimes of excited states in Hg188 to infer their collective properties, such as the deformation. Extending the investigation to higher-spin states, which are expected to be less affected by band-mixing effects, can provide additional information on the coexisting structures. Methods: The Hg188 nucleus was populated using two different fusion-evaporation reactions with two targets, Gd158 and Gd160, and a beam of S34 provided by the Tandem-ALPI accelerator complex at the Laboratori Nazionali di Legnaro. The channels of interest were selected using the information from the Neutron Wall array, while the γ rays were detected using the GALILEO γ-ray spectrometer. Lifetimes of excited states were determined using the recoil-distance Doppler-shift method, employing the dedicated GALILEO plunger device. Results: Lifetimes of the states up to spin 16ℏ were measured and the corresponding reduced transition probabilities were calculated. Assuming two-band mixing and adopting, as done commonly, the rotational model, the mixing strengths and the deformation parameters of the unperturbed structures were obtained from the experimental results. In order to shed light on the nature of the observed configurations in the Hg188 nucleus, the extracted transition strengths were compared with those resulting from state-of-the-art beyond-mean-field calculations using the symmetry-conserving configuration-mixing approach, limited to axial shapes, and the five-dimensional collective Hamiltonian, including the triaxial degree of freedom. Conclusions: The first lifetime measurement for states with spin ≥6 suggested the presence of an almost spherical structure above the 121+ isomer and allowed elucidating the structure of the intruder band. The comparison of the extracted B(E2) strengths with the two-band mixing model allowed the determination of the ground-state band deformation. Both beyond-mean-field calculations predict coexistence of a weakly deformed band with a strongly prolate-deformed one, characterized by elongation parameters similar to those obtained experimentally, but the calculated relative position of the bands and their mixing strongly differ.

7 pags., 4 figs., 1 tab., Excited states in the N=40 isotone Ti were populated via the V(p,2p)Ti reaction at ∼200 MeV/nucleon at the Radioactive Isotope Beam Factory and studied using γ-ray spectroscopy. The energies of the 2 →0 and 4 →2 transitions, observed here for the first time, indicate a deformed Ti ground state. These energies are increased compared to the neighboring Cr and Fe isotones, suggesting a small decrease of quadrupole collectivity. The present measurement is well reproduced by large-scale shell-model calculations based on effective interactions, while ab initio and beyond mean-field calculations do not yet reproduce our findings. The shell-model calculations for Ti show a dominant configuration with four neutrons excited across the N=40 gap. Likewise, they indicate that the N=40 island of inversion extends down to Z=20, disfavoring a possible doubly magic character of the elusive Ca., We thank the RIKEN Nishina Center accelerator staff and the Bi-gRIPS team for the stable operation of the high-intensity Zn beam and for the preparation of the secondary beam setting. K.O. ac-knowledges the support by Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (JSPS) JP16K05352. A.P. is supported in part by the Ministerio de Ciencia, Innovación y Universidades (Spain), Severo Ochoa Programme SEV-2016-0597 and grant PGC-2018-94583. F.B. is supported by the RIKEN Spe-cial Postdoctoral Researcher Program. L.X.C. and B.D.L. would like to thank the Vietnam Ministry of Science and Technology (MOST) for its support through the Physics Development Program Grant No. ÐTÐLCN.25/18. I.G. has been supported by HIC for FAIR and Croatian Science Foundation under projects no. 1257 and 7194. D. So. was supported by the the European Regional Develop-ment Fund contract No. GINOP-2.3.3-15-2016-00034 and the National Research, Development and Innovation Fund of Hungary via Project No. K128947. V.V. acknowledges support from the Span-ish Ministerio de Economía y Competitividad under Contract No. FPA2017-84756-C4-2-P. K.I.H., D.K. and S.Y.P. acknowledge the sup-port from the National Research Foundation of Korea grant No. 2018R1A5A1025563 and 2019M7A1A1033186. The development of MINOS was supported by the European Research Council through the ERC Grant No. MINOS-258567. This work was also supported by the JSPS KAKENHI Grant No. 18K03639, MEXT as “Priority is-sue on post-K computer” (Elucidation of the fundamental laws and evolution of the universe), the Joint Institute for Computational Fundamental Science (JICFuS), the CNS-RIKEN joint project for large-scale nuclear structure calculations, Natural Sciences and Engineering Research Council (NSERC) of Canada, the Deutsche Forschungsgemeinschaft – Projektnummer 279384907 – SFB 1245, the PRISMA Cluster of Excellence, and the BMBF under Contracts No. 05P18RDFN1 and 05P19RDFN1. TRIUMF receives funding via a contribution through the National Research Council Canada. Com-putations were performed at the Jülich Supercomputing Center (JURECA)

12 pags., 11 figs., 2 tabs., The N= Z= 36 nucleus Kr has been studied by inelastic scattering at intermediate energies. Two targets, Be and Au, were used to extract the nuclear deformation length, δ, and the reduced E2 transition probability, B(E2). The previously unknown non-yrast 2 and 4 states as well as a new candidate for the octupole 3 state have been observed in the scattering on the Be target and placed in the level scheme based on γ- γ coincidences. The second 2 state was also observed in the scattering on the Au target and the B(E2;22+→01+) value could be determined for the first time. Analyzing the results in terms of a two-band mixing model shows clear evidence for a oblate-prolate shape coexistence and can be explained by a shape change from an oblate ground state to prolate deformed yrast band from the first 2 state. This interpretation is corroborated by beyond mean field calculations using the Gogny D1S interaction., We would like to thank the RIKEN accelerator and BigRIPS teams for providing the high intensity beams. We thank T. Furumoto for providing us with the optical potentials and A. Moro for giving access to an unpublished version of the FRESCO code. This work has been supported by UK STFC under Grant numbers ST/L005727/1 and ST/P003885/1, the Spanish Ministerio de Economía y Competitividad under Grants FPA2011-24553, FPA2014-52823-C2- 1-P, and PGC2018-094583-B-I00, the Program Severo Ochoa (SEV2014-0398), the European Research Council through the ERC Grant No. MINOS-258567, NKFIH (NN128072), and by the ÚNKP-19-4- DE-65 New National Excellence Program of the Ministry of Human Capacities of Hungary. G. K. acknowledges support from the János Bolyai research fellowship of the Hungarian Academy of Sciences. K. W. acknowledges the support from the Spanish Ministerio de Economía y Competitividad RYC-2017-22007.

We study the performance of self-consistent mean-field and beyond-mean-field approximations in shell-model valence spaces. In particular, Hartree-Fock-Bogolyubov, particle-number variation after projection and projected generator coordinate methods are applied to obtain ground-state and excitation energies for even-even and odd-even Calcium isotopes in the pf-shell. The standard (and non-trivial) KB3G nuclear effective interaction has been used. The comparison with the exact solutions -- provided by the full diagonalization of the Hamiltonian -- shows an outstanding agreement when particle-number and angular-momentum restorations are performed and both quadrupole and neutron-neutron pairing degrees of freedom are explicitly explored as collective coordinates., 14 pages, 10 figures

7 pags., 4 figs. -- XXII International School on Nuclear Physics, Neutron Physics and Applications. -- Open Access funded by Creative Commons Atribution Licence 3.0, Low-spin states of neutron-rich Ge-84,Ge-86,Ge-88 were measured by in-flight gamma-ray spectroscopy at 270 MeV/u at the RIKEN-RIBF facility. The exotic beams have been produced by primary U-238 in-flight fission reactions and impinged on the MINOS device. MINOS combines a 10-cm long LH2 target with a Time Projection Chamber (TPC) to reconstruct the reaction vertices. The reactions were selected by the BigRIPS and the ZeroDegree spectrometers for the incoming and outgoing channels, respectively. Emitted gamma radiation was detected by the NaI-array DALI2. De-excitations from the 6(1)(+), 4(1,2)(+), and 2(1,2)(+) states of Ge-84,Ge- 86 and 4(1)(+) and 26 states of Ge-88 were observed. The data are compared to state-of-the-art shell model and beyond-mean-field calculations. Furthermore, a candidate for a 3(1)(+) state of Ge-86 was identified. This state plays a key role in the discussion of ground-state triaxiality of Ge-86, along with other features of the low-energy level scheme., We acknowledge support from the German BMBF Grant Nos. 05P12RDFN1, 05P12RDFN8 and 05P15PKFNA, the ERC Grant No. MINOS258567, the Spanish Ministerio de Economıa y Competitividad under contracts FPA2014-57196-C5-4-P and FIS−2014−53434, the Vietnam Ministry of Science and Technology, as well as from the Science and Technology Facilities Council (STFC). We further thank GSI for providing computing facilities

D. Ralet et al. -- 11 pags., 10 figs., 3 tabs., Background: In the neutron-rich A≈100 mass region, rapid shape changes as a function of nucleon number as well as coexistence of prolate, oblate, and triaxial shapes are predicted by various theoretical models. Lifetime measurements of excited levels in the molybdenum isotopes allow the determination of transitional quadrupole moments, which in turn provides structural information regarding the predicted shape change. Purpose: The present paper reports on the experimental setup, the method that allowed one to measure the lifetimes of excited states in even-even molybdenum isotopes from mass A=100 up to mass A=108, and the results that were obtained. Method: The isotopes of interest were populated by secondary knock-out reaction of neutron-rich nuclei separated and identified by the GSI fragment separator at relativistic beam energies and detected by the sensitive PreSPEC-AGATA experimental setup. The latter included the Lund-York-Cologne calorimeter for identification, tracking, and velocity measurement of ejectiles, and AGATA, an array of position sensitive segmented HPGe detectors, used to determine the interaction positions of the γ ray enabling a precise Doppler correction. The lifetimes were determined with a relativistic version of the Doppler-shift-attenuation method using the systematic shift of the energy after Doppler correction of a γ-ray transition with a known energy. This relativistic Doppler-shift-attenuation method allowed the determination of mean lifetimes from 2 to 250 ps. Results: Even-even molybdenum isotopes from mass A=100 to A=108 were studied. The decays of the low-lying states in the ground-state band were observed. In particular, two mean lifetimes were measured for the first time: τ=29.7-9.1+11.3 ps for the 4+ state of Mo108 and τ=3.2-0.7+0.7 ps for the 6+ state of Mo102. Conclusions: The reduced transition strengths B(E2), calculated from lifetimes measured in this experiment, compared to beyond-mean-field calculations, indicate a gradual shape transition in the chain of molybdenum isotopes when going from A=100 to A=108 with a maximum reached at N=64. The transition probabilities decrease for Mo108 which may be related to its well-pronounced triaxial shape indicated by the calculations., This work was supported by the BMBF under Grants No. 05P09RDFN4, No. 05P12RDFN8, and No. 05P12PKFNE and by the LOEWE center HIC for FAIR. Part of this work was supported by the P2IO excellence center. This work was also supported by the European Community FP7–Capacities, ENSAR Contract No. 262010, by the Swedish Research Council under Contracts No. 2010-147, No. 2010-3694, No. 2011-5253, No. 2011-6127, and No. 2014-5558, by the IN2P3- GSI No. 02-42 collaboration, by the Science and Technology Facilities Council (STFC) and by MINECO, Spain, under Grant No. FPA2014-57196-C5, and Generalitat Valenciana, Spain, under Grant No. PROMETEOII/2014/019

International audience; Octupole correlations are described in a microscopic framework involving angular momentum, parity and particle number projected intrinsic Hartree–Fock–Bogoliubov states. Linear combinations of those symmetry restored states are considered to account for collective positive and negative parity states in the nucleus 144Ba. The well-known Gogny D1S interaction is used in the calculations. A strong octupole collectivity is observed in the negative parity states justifying the assignment of 144Ba as an octupole deformed nucleus. Higher lying excited states are studied and its structure is identified by looking at the collective wave functions obtained in the calculations. An oblate-spherical 0 + shape isomer and a two-phonon octupole multiplet are described in detail.

K. Hadyńska-Kl¸ek et al. ; 7 pags., 3 figs., 2 tabs., Shape parameters of a weakly deformed ground-state band and highly deformed slightly triaxial sideband in Ca42 were determined from E2 matrix elements measured in the first low-energy Coulomb excitation experiment performed with AGATA. The picture of two coexisting structures is well reproduced by new state-of-the-art large-scale shell model and beyond-mean-field calculations. Experimental evidence for superdeformation of the band built on 02+ has been obtained and the role of triaxiality in the A∼40 mass region is discussed. Furthermore, the potential of Coulomb excitation as a tool to study superdeformation has been demonstrated for the first time. © 2016 American Physical Society, Special gratitude goes to the INFN LNL and HIL Warsaw technical staff for their support and help, in particular, the accelerator crews for providing intense and good-quality 42Ca and 32S beams. K. H.-K. acknowledges support from the Research Council of Norway under the project Grant No. 213442. T. R. R. acknowledges computing time at GSI-Darmstadt and support from Spanish MINECO under Programa Ramon y Cajal 11420 and FIS-2014-53434-P, H. N. acknowledges support from Helmholtz Association through the Nuclear Astrophysics Virtual Institute NAVI (No. VH-VI-417). We also acknowledge the support by Generalitat Valenciana, Spain, under the grant PROMETEOII/2014/019 and by the FEDER funds of the European Commission (A. Gadea), Spanish Ministerio de Economía y Competitividad under Contract No. FPA2014-57196-C5 (A. J., A. Gadea), German Bundesministerium für Bildung und Forschung (BMBF) under Contract No. 05P12PKFNE TP4 (B. Birkenbach), the Polish National Science Centre under Projects No. DEC-2013/10/M/ST2/00427, No. DPN/N190/ AGATA/2009, No. 2011/03/B/ST2/01894, No. UMO- 2014/14/M/ST2/00738 (COPIN-INFN Collaboration), and No. 2015/17/B/ST2/01534.

We investigate the extent to which theories of collective motion can capture the physics that determines the nuclear matrix elements governing neutrinoless double-β decay. To that end we calculate the matrix elements for a series of isotopes in the full pf shell, omitting no spin-orbit partners. With the inclusion of isoscalar pairing, a separable collective Hamiltonian that is derived from the shell model effective interaction reproduces the full shell-model matrix elements with good accuracy. A version of the generator coordinate method that includes the isoscalar pairing amplitude as a coordinate also reproduces the shell model results well, an encouraging result for theories of collective motion, which can include more single-particle orbitals than the shell model. We briefly examine heavier nuclei relevant for experimental double-β decay searches, in which shell-model calculations with all spin-orbit partners are not feasible; our estimates suggest that isoscalar pairing also plays a significant role in these nuclei, though one we are less able to quantify precisely., This work was supported in part by an International Research Fellowship from the Japan Society for the Progress of Science (JSPS), and JSPS KAKENHI Grant No. 26·04323, by the Deutsche Forschungsgemeinschaft through Contract No. SFB 634, by the Helmholtz Association through the Helmholtz Alliance Program, Contract No. HA216/EMMI “Extremes of Density and Temperature: Cosmic Matter in the Laboratory”, by the European Research Council under Grant No. 307986 STRONGINT, by the U.S. Department of Energy through Contract No. DE-FG02-97ER41019, and by the Spanish MINECO under Programa Ramón y Cajal 11420 and FIS-2014-53434-P

We present a method to compute the number of particles occupying spherical single-particle (SSP) levels within the energy density functional (EDF) framework. These SSP levels are defined for each nucleus by performing self-consistent mean-field calculations. The nuclear many-body states, in which the occupation numbers are evaluated, are obtained with a symmetry conserving configuration mixing (SCCM) method based on the Gogny EDF. The method allows a closer comparison between EDF and shell model with configuration mixing in large valence spaces (SM-CI) results, and can serve as a guidance to define physically sound valence spaces for SM-CI calculations. As a first application of the method, we analyze the onset of deformation in neutron-rich $N=40$ isotones and the role of the SSP levels around this harmonic oscillator magic number, with particular emphasis in the structure of $^{64}$Cr., Comment: 11 pages, 5 Figures

Shape transitions and shape coexistence in the $^{70-98}$Kr region are studied in a unified view with state-of-the-art beyond self-consistent mean field methods based on the Gogny D1S interaction. Beyond mean field effects are taken into account through the exact angular momentum and particle number restoration and the possibility of axial and non-axial shape mixing. The results of the low-lying properties of these isotopes are in good agreement with the experimental data when the triaxial degree of freedom is included. Shape transitions from axial-oblate ($^{70-72}$Kr) to triaxial-prolate ($^{74-78}$Kr) and from spherical-triaxial ($^{86-92}$Kr) to axial-oblate ($^{94-98}$Kr) ground states are obtained. Additionally, low-lying $0^{+}$ excited states and quasi-gamma bands are found showing the richness of the collective structure in this region., 15 pages, 9 figures, accepted in Physical Review C

We use energy density functional methods to compute the nuclear matrix elements (NME) of neutrinoless double beta decay (0νββ) in the Cadmium isotopic chain. The interconnected role of deformation, pairing, configuration mixing in the NMEs is discussed in the candidate 116Cd and extended to the whole isotopic chain. In addition, strong shell effects are found and compared to a generalized seniority model.

4 pags., 2 figs. -- INPC 2013 – International Nuclear Physics Conference, The investigation of the excitation energies of the 21+ –states in the neutron-rich Cd isotopes shows an irregular behaviour when approaching the neutron shell-closure at N = 82. The energy of the 21+–state in 128Cd is lower than the one in 126Cd. The transition strength B(E2, 0gs+ → 21+) in the even isotopes 122−128Cd was measured in Coulomb excitation experiments with the high-purity germanium detector array MINIBALL at REXISOLDE (CERN). The values for 122,124Cd coincide with beyond-mean-field calculations with a resultant prolate deformation, whereas 126,128Cd are better described by shell-model calculations., This project is supported by BMBF (No. 06 DA 9036I, No. 05 P12 RDCIA, No. 05 P12 RDCIB and No. 05 P12 PKFNE), HIC for FAIR, EU through EURONS (No. 506065) and ENSAR (No. 262010) and the MINIBALL and REX-ISOLDE collaborations.

Nuclear matrix elements (NME) for the most promising candidates to detect neutrinoless double beta decay have been computed with energy density functional methods including deformation and pairing fluctuations explicitly on the same footing. The method preserves particle number and angular momentum symmetries and can be applied to any decay without additional fine tunings. The finite range density dependent Gogny force is used in the calculations. An increase of $10\%-40\%$ in the NME with respect to the ones found without the inclusion of pairing fluctuations is obtained, reducing the predicted half-lives of these isotopes., 5 pages, 3 Figures

We study the development of collectivity in neutron rich nuclei in the close vicinity of ${}^{78}$Ni. We report on the large scale shell model calculations in the $N=52--54$ even-even nuclei with $Z=30--36$. We predict maximum of triaxiality in ${}^{86}$Ge and explain this phenomenon on the basis of a pseudo-SU(3) symmetry interpretation. For the cases where signs for nonaxial shapes appear, we perform the triaxial Gogny calculations with particle number and angular momentum projections. The comparison of results obtained in the laboratory and intrinsic frames provides a comprehensive and complete picture of nuclear deformation in this region.

Background: Quadrupole moments of excited nuclear states are important observables for geometrically interpreting nuclear structure in terms of deformed shapes, although data are scarce and sometimes ambiguous, in particular, in neutron-rich nuclides. Purpose: A measurement was performed for determining the spectroscopic quadrupole moment of the 21+ state of 140Ba in order to clarify the character of quadrupole deformation (prolate or oblate) of the state in its yrast sequence of levels. Method: We have utilized a new combined technique of lifetime measurement at REX-ISOLDE and MINIBALL using the Doppler-shift attenuation method (DSAM) and a reorientation analysis of Coulomb-excitation yields. Results: On the basis of the new lifetime of τ(21+)=10.4-0.8+2.2 ps the electric quadrupole moment was determined to be Q(21+)=-0.52(34) eb, indicating a predominant prolate deformation. Conclusions: This finding is in agreement with beyond-mean-field calculations using the Gogny D1S force and with results from the Monte Carlo shell-model approach, This experiment was supported by the BMBF (06DA9036I, 06DA9041I, 06DA7046, 06MT238, 06MT9156, 06ML234, 06KY205I, 06KY9136I), RII3-EURONS (Contract No. 506065), HIC for FAIR, Grants-in-Aid for Young Scientists (20740127) from JSPS, the SPIRE Field 5 from MEXT (Japan), and the MINIBALL and ISOLDE Collaborations

We compute nuclear matrix elements for neutrinoless double electron capture on ${}^{152}$Gd, ${}^{164}$Er, and ${}^{180}$W nuclei. Recent precise mass measurements for these nuclei have shown a large resonance enhancement factor that makes them the most promising candidates for observing this decay mode. We use an advanced energy density functional method which includes beyond mean-field effects such as symmetry restoration and shape mixing. Our calculations reproduce experimental charge radii and $B(E2)$ values predicting a large deformation for all these nuclei. This fact reduces significantly the values of the NMEs leading to half-lives larger than ${10}^{29}$ years for the three candidates.

We study the spherical to prolate-deformed shape transition in $^{144-158}$Sm and $^{146-160}$Gd isotopes with modern calculations beyond the mean field with the Gogny D1S force. We compare the results with the shape-phase transition predicted by the collective hamiltonian model and with the experimental data. Our calculations do not support the existence of a first order phase transition in these isotopic chains in the viewpoint of the Bohr hamiltonian neither the interpretation of the nuclei N=90 as critical points., Proceeding of the XIII Conference on Capture Gamma-ray Spectroscopy and Related Topics, Cologne, 2008