Structure of the lightest tin isotopes

Structure of the lightest tin isotopes

T. D. Morris Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    J. Simonis Institut für Kernphysik, TU Darmstadt, Schlossgartenstr. 2, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany    S. R. Stroberg TRIUMF 4004 Wesbrook Mall, Vancouver BC V6T 2A3, Canada Physics Department, Reed College, Portland OR, 97202, USA    C. Stumpf Institut für Kernphysik, TU Darmstadt, Schlossgartenstr. 2, 64289 Darmstadt, Germany    G. Hagen Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA    J. D. Holt TRIUMF 4004 Wesbrook Mall, Vancouver BC V6T 2A3, Canada    G. R. Jansen National Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    T. Papenbrock Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    R. Roth Institut für Kernphysik, TU Darmstadt, Schlossgartenstr. 2, 64289 Darmstadt, Germany    A. Schwenk Institut für Kernphysik, TU Darmstadt, Schlossgartenstr. 2, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
Abstract

We link the structure of nuclei around Sn, the heaviest doubly magic nucleus with equal neutron and proton numbers (), to nucleon-nucleon () and three-nucleon () forces constrained by data of few-nucleon systems. Our results indicate that Sn is doubly magic, and we predict its quadrupole collectivity. We present precise computations of Sn based on three-particle–two-hole excitations of Sn, and reproduce the small splitting between the lowest and states. Our results are consistent with the sparse available data.

IntroductionSn is a nucleus of superlatives: It is the heaviest self-conjugate () nucleus Lewitowicz et al. (1994), exhibits the largest strength in allowed decay Hinke et al. (2012), is close to the proton dripline Erler et al. (2012), and is the endpoint of a region of nuclei with enhanced decays Liddick et al. (2006); Seweryniak et al. (2006). While these properties make Sn the cornerstone of a most interesting region of the nuclear chart, our understanding of this nucleus and its neighbors is still rather limited, see Faestermann et al. (2013) for a review. No data exist regarding the spectrum of Sn, and the spin assignments for low-lying states in Sn are controversial Seweryniak et al. (2007); Darby et al. (2010). Likewise, the evolution of collective observables towards neutron number is experimentally unclear at present Banu et al. (2005); Vaman et al. (2007); Ekström et al. (2008); Guastalla et al. (2013); Bader et al. (2013); Doornenbal et al. (2014). On the other hand, with naively expected shell closures for both protons and neutrons, and the stabilizing effects of the Coulomb and centrifugal barriers, Sn should be particularly suitable for a reliable theoretical treatment.

In this Letter, we calculate properties of Sn and neighboring nuclei using realistic interactions between protons and neutrons. This is in contrast to large-scale shell-model (LSSM) calculations Lipoglavšek et al. (2002); Blazhev et al. (2004); Boutachkov et al. (2011); Brock et al. (2010); Caurier et al. (2010) in this region of the nuclear chart that employ Zr or Sr cores and phenomenologically adjusted interactions based on the -matrix approach Hjorth-Jensen et al. (1995). The strong nuclear force is rooted in the fundamental theory of strong interactions, quantum chromodynamics (QCD), and is manifested in dominant two-nucleon () forces and weaker but pivotal three-nucleon () forces between protons and neutrons. Effective field theories (EFTs) of QCD provide us with a systematically improvable low-momentum expansion of these interactions Kolck (1999); Epelbaum et al. (2009); Machleidt and Entem (2011). So far, interactions derived from the EFT framework have been applied to light and medium-mass nuclei (see Navrátil et al. (2009); Barrett et al. (2013); Hagen et al. (2014); Hergert et al. (2016) for recent reviews).

The extension of ab initio computations from light Pieper and Wiringa (2001); Navrátil et al. (2009); Barrett et al. (2013) to heavier nuclei is based on the development and application of quantum many-body methods that exhibit a polynomial scaling in mass number Mihaila and Heisenberg (2000); Dean and Hjorth-Jensen (2004); Dickhoff and Barbieri (2004); Hagen et al. (2008, 2010); Tsukiyama et al. (2011); Roth et al. (2012); Hergert et al. (2013); Somà et al. (2014); Binder et al. (2013); Lähde et al. (2014). Medium-mass and heavy nuclei can technically be computed with these methods, but most interactions developed thus far considerably overbind heavier nuclei Binder et al. (2014). In the quest for nuclear interactions with more acceptable saturation properties Hebeler et al. (2011); Ekström et al. (2013, 2015, 2017), one interaction labeled 1.8/2.0(EM) has emerged that describes binding energies, two-neutron separation energies, and the first excited state in nuclei up to neutron-rich nickel isotopes remarkably well, while charge radii are too small Hagen et al. (2016a, b, 2016); Simonis et al. (2016); Garcia Ruiz et al. (2016); Simonis et al. (2017). It is primarily this interaction from Ref. Hebeler et al. (2011) that we will employ in the computation of Sn and its neighbors.

This Letter is organized as follows. First, we briefly describe the Hamiltonian, the employed model spaces, and computational methods. Then, we validate the interactions in the tin region by computing known binding energies and level splittings, and address method uncertainties by employing the coupled-cluster method Bartlett and Musiał (2007); Hagen et al. (2014) and the valence-space in-medium similarity-renormalization-group method (VS-IMSRG) Stroberg et al. (2017) coupled with the importance-truncated large-scale shell model Stumpf et al. (2016). Finally, we present results for the structure of the lightest isotopes of tin.

Hamiltonian and model space – We employ the intrinsic Hamiltonian

(1)

where the and potentials ( and , respectively) are the interactions 1.8/2.0(EM), 2.0/2.0(EM), 2.2/2.0(EM), and 2.0/2.0(PWA) from Ref. Hebeler et al. (2011). These interactions result from a similarity-renormalization-group (SRG) Bogner et al. (2007) evolution of the chiral interaction of Entem and Machleidt (2003) to cutoffs and  fm, respectively. The potential is not evolved but rather taken as the leading forces from chiral EFT van Kolck (1994); Epelbaum et al. (2002); Hebeler et al. (2015) and has a cutoff  fm. The two low-energy constants of the short-range part of the forces are adjusted to binding energy of the triton and the radius of the particle, following Ref. Nogga et al. (2004). These interactions are quite soft (due to the relatively small cutoffs), which allows us to achieve reasonably well converged binding energies and spectra in nuclei up to neutron-rich Ni Hagen et al. (2016); Simonis et al. (2017), and in the neutron-deficient tin isotopes considered in this work.

Figure 1: (Color online) Ground-state energies per nucleon for selected closed-shell nuclei computed with the closed-shell IMSRG Tsukiyama et al. (2011) using the interactions of Ref. Hebeler et al. (2011) in comparison with experiment (black horizontal lines).

Figure 1 shows the computed ground-state energies per nucleon for He, O, Ca, Ni, Zr, and Sn with the single-reference IMSRG Tsukiyama et al. (2011); Hergert et al. (2013). The 1.8/2.0(EM) interaction consistently yields the best agreement with data. Presently, it is unclear what distinguishes this interaction from the other similarly obtained interactions; however this soft interaction puts us in a fortuitous situation to make theoretical predictions (albeit without rigorous uncertainty quantification) for binding energies and spectra in nuclei as heavy as Sn.

Coupled-cluster calculations use a Hartree-Fock basis constructed from a harmonic-oscillator basis of up to 15 major oscillator shells. For VS-IMSRG we use a similar basis, except that the Hartree-Fock reference is constructed with respect to an ensemble state above the Zr core following Ref. Stroberg et al. (2017). All calculations are performed at oscillator frequencies in the range  MeV, which include the minimum in energy for the largest model space we consider. We use the normal-ordered two-body approximation Hagen et al. (2007); Roth et al. (2012); Binder et al. (2014) for the interaction with an additional energy cut on three-body matrix elements . When is increased from 16 to 18, the binding energy of Sn changes by 2% for the hardest interaction 2.0/2.0(PWA), while for the softest interaction, 1.8/2.0(EM), the change is less than 1%.

Method – The coupled-cluster method is an ideal tool to compute doubly magic nuclei and their neighbors Kümmel et al. (1978); Mihaila and Heisenberg (2000); Dean and Hjorth-Jensen (2004); Włoch et al. (2005); Hagen et al. (2008, 2010); Roth et al. (2011); Binder et al. (2013); Hagen et al. (2014). This method computes the similarity transform of the Hamiltonian , obtained by normal ordering the free-space Hamiltonian (1) with respect to the closed-shell Hartree-Fock reference of Sn. The cluster operator includes particle-hole excitations and is truncated at the coupled-cluster singles-doubles (CCSD) level. Usually CCSD accounts for about 90% of the correlation energy (i.e., the energy beyond Hartree Fock) Bartlett and Musiał (2007). For a higher precision of the ground-state energy, we include triples excitations of the cluster operator perturbatively within the -CCSD(T) method Taube and Bartlett (2008). Excited states in Sn are computed with an equation-of-motion (EOM) method including 3-3 corrections via a generalization of the ground state -CCSD(T) approximations to excited states with EOM-CCSD(T)  Watts and Bartlett (1995). The neighboring nuclei Sn are computed as one- and two-particle attached states Gour et al. (2005); Jansen et al. (2011); Jansen (2013) of the Sn similarity transformed Hamiltonian . The two-particle attached states of Sn are truncated at the - level, while the particle-attached states of Sn are computed at the - level with perturbative - corrections included (described below). Further details of the coupled-cluster approach to nuclei are presented in a recent review Hagen et al. (2014).

We briefly describe our new approach to include perturbative - corrections to the particle-attached states of Sn. Generalizing the completely renormalized (CR) EOM-CCSD(T) approximation from quantum chemistry Kowalski and Piecuch (2000); Piecuch et al. (2002) and nuclear physics Kowalski et al. (2004); Włoch et al. (2005); Binder et al. (2013) to particle-attached excited states yields the correction

(2)

Here denotes the state of interest, (and ) are occupied (and unoccupied) orbitals in the Sn reference , and represent the left and right - moments

are - excited states, and is the resolvent

(3)

Here is the - energy corresponding to the states and of Sn. We draw the reader’s attention to the similar structure between the bi-variational expression (2) and second-order perturbation theory. This method is the completely renormalized particle-attached equation-of-motion (CR-PA-EOM). In our results for Sn, we used three different approximations (labeled A,B,C) for the energy denominator in Eq. (3). Approximation A uses in place of the Hartree-Fock single-particle energies, approximation B uses the one-body part of , and approximation C uses both the one- and two-body parts of . Thus, approximation C is the most complete choice for the resolvent and most accurately approximates the full calculation Włoch et al. (2005).

The IMSRG and its VS-IMSRG variant are effective tools for computing doubly magic nuclei and for constructing valence-space interactions from and interactions that can be subsequently diagonalized using shell-model techniques Tsukiyama et al. (2011, 2012); Bogner et al. (2014); Hergert et al. (2016); Stroberg et al. (2016, 2017). These methods also rely on similarity transformations where is the normal-ordered Hamiltonian with respect to the ensemble reference of each target nucleus. For nuclei in the Sn region, the VS-IMSRG yields an anti-Hermitian , truncated at the one- and two-body level, which decouples the major oscillator shell above Zr. The ensuing large-scale eigenvalue problem is solved via the importance-truncated shell model Stumpf et al. (2016).

Results – Results for Sn are shown in Fig. 2. Panel (a) shows the low-lying states in Sn computed in the EOM-CCSD and EOM-CCSD(T) approximations with the 1.8/2.0(EM) interaction. We also show the phenomenological LSSM results of Ref. Hinke et al. (2012). The relatively large excitation gap of about 4 MeV is consistent with Sn being doubly magic, a finding which is—to our knowledge—qualitatively ubiquitous in all previous theoretical investigations. Panel (b) shows our EOM-CCSD predictions for the in Sn for the 1.8/2.0(EM), 2.0/2.0(EM), 2.2/2.0(EM), and 2.0/2.0(PWA) interactions together with the experimental values for the isotopes Sn Banu et al. (2005); Vaman et al. (2007); Ekström et al. (2008); Guastalla et al. (2013); Bader et al. (2013); Doornenbal et al. (2014). Our computed values are similar in size to that of Sn and consistent with Sn being doubly magic. They also fall within expectations from phenomenological shell-model calculations Faestermann et al. (2013) and from extrapolations of data in light tin isotopes. Panel (c) shows CC results for 1.8/2.0(EM), 2.0/2.0(EM), 2.2/2.0(EM), and 2.0/2.0(PWA) interactions and the VS-IMSRG result for the 1.8/2.0(EM) interaction for the energy of the first state in light even isotopes Sn, and data for Sn. The values computed are consistent with the computed excitation energies of the first , in the sense that, for a given interaction, the larger the energy, the smaller the value. We note that despite the consistency with experiment, the 1.8/2.0(EM) interaction produces radii that are too small, and this would certainly affect the . The systematic trend of known and computed energies in the tin isotopes again suggests that Sn is doubly magic. In Sn, this energy is similar to that of the doubly magic nucleus Sn Björnstad et al. (1980); Jones et al. (2010). The VS-IMSRG result for the state in Sn is about  MeV higher than the EOM-CCSD(T) result, but close to the similarly approximated EOM-CCSD result shown in panel (a). Using the discrepancy between methods as an estimate of the uncertainty in the many-body method, our results for the energy of state in Sn are consistent with the data.

(a)(b)(c)
Figure 2: (Color online) Panel (a) shows low-lying states in Sn computed with the chiral interaction 1.8/2.0(EM) in the EOM-CCSD and EOM-CCSD(T) approximations and compared to LSSM calculations based on phenomenological interactions Faestermann et al. (2013). Panel (b) shows the EOM-CCSD results for the transition strength in Sn with the interactions as indicated, and the experimental data for all other even tin isotopes. Panel (c) shows the energy of the states in even tin isotopes, with coupled-cluster results for Sn [labelled as in panel (b)] and VS-IMSRG results for Sn with interactions as indicated, and data for Sn (black circles).

In Sn, the two lowest states are separated by only 172 keV Seweryniak et al. (2007); Darby et al. (2010). Observation of Te decay in coincidence with the 172 keV line indicates that the dominant decay of Te is to the first excited state in Sn, implying that these states have identical spins Darby et al. (2010). We recall that the lowest two states in the odd isotopes Te and Sn are only about 0.2 MeV apart and lack definite spin assignments. In tin, this near degeneracy between the and states persists up to Sn, and the ground-state spin changes from in Sn Cerizza et al. (2016) and Sn to in Sn. The level ordering in Sn to Sn between the and states is not known. This is reflected in panel (a) of Fig. 3 which compares available data (full and open black points for definite and tentative spins assignments, respectively) with CC and VS-IMSRG predictions for the energy splitting in odd tin isotopes using the interactions 1.8/2.0(EM) and 2.0/2.0(EM). Both interactions yield a small splitting between the and states, but they differ on its precise size and sign. Panel (b) of Fig. 3 plots the calculated energy splitting between the and states versus the neutron separation energy of Sn computed with the CR-PA-EOM using the 1.8/2.0(EM), 2.0/2.0(EM), 2.2/2.0(EM), and 2.0/2.0(PWA) interactions. Also shown are estimated uncertainties due to finite model-space sizes and the employed methods, and a blue (diagonal) band encompassing these uncertainties (see Ref. Hagen et al. (2016a) for details). The horizontal and vertical green lines indicate experimental data. The intersection of the blue diagonal band with the precisely known neutron separation energy yields one estimate of the systematic uncertainty for the energy splitting between the and states in Sn. Clearly, theory is not sufficiently precise to make a definite prediction for the ground-state spin of Sn as the predicted range for the energy splitting can support either or as the ground state. Again, the 1.8/2.0(EM) interaction is closest to data. Panel (c) of Fig. 3 shows the lowest states in Sn, computed with the - particle-attached EOM-CC method, the CR-PA-EOM developed in this work, and the VS-IMSRG for the 1.8/2.0(EM) interaction. We find that for this interaction, the different methods agree on the level ordering, and the energy splitting varies by at most  keV. While the upcoming measurements will yield definite spin assignments Garcia Ruiz et. al (2016), getting theory to a level where such fine details can be unambiguously resolved will require more work.

(a)(c)(b)
Figure 3: (Color online) Panel (a) shows the energy splitting between the lowest and states in light odd-mass tin isotopes. Data are black circles for isotopes with definite (full) and tentative spin assignments (open symbols). Theoretical results are from coupled cluster (full) and VS-IMSRG (open symbols) for the interactions as indicated. Panel (b) shows the correlation between the neutron separation energy and the energy splitting between the lowest and states for the interactions and computational method as labeled (symbols with error bars and an encompassing blue uncertainty band) compared to data (vertical and horizontal green lines). Panel (c) shows the low-lying levels in Sn based on the chiral interaction 1.8/2.0(EM) computed with various coupled-cluster equation-of-motion (EOM) methods, the VS-IMSRG, and compared to data.

Figure 4 shows the convergence of the and states in Sn and Te with the number of particle-hole excitations () in the importance-truncated large-scale shell-model calculations using the 1.8/2.0(EM) and 2.0/2.0(EM) interactions. In both, Sn and Te, we obtain nearly degenerate and states consistent with data.

Figure 4: (Color online) Ground- and first excited states in Sn and Te obtained in VS-IMSRG for the 1.8/2.0(EM) and 2.0/2.0(EM) interactions, with spins (red) and (black).

Conclusions and Outlook – Our computations demonstrated that tin nuclei can be described by and interactions constrained by few-body data. We found that Sn is doubly magic and presented results and predictions for its structure and low-lying collectivity. For an increased precision of excited states in Sn, we developed a method that includes three-particle-two-hole excitations in our coupled-cluster calculations. One interaction reproduced both binding energies and the near degeneracy between the lowest and states in the odd-mass isotopes Sn and Te. This work opens the avenue for reliable calculations of even heavier nuclei based on and interactions.

Acknowledgements.
We thank K. Hebeler for providing us with matrix elements in Jacobi coordinates for the interaction at next-to-next-to-leading order. This work was supported by the Office of Nuclear Physics, U.S. Department of Energy, under Grants DE-FG02-96ER40963, DE-SC0008499 (NUCLEI SciDAC collaboration), the Field Work Proposal ERKBP57 at Oak Ridge National Laboratory (ORNL), by the ERC Grant No. 307986 STRONGINT, the BMBF under Contract No. 05P15RDFN1, the DFG under Grant SFB 1245 DFG, the National Research Council Canada, and NSERC. Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility located at ORNL, which is supported by the Office of Science of the Department of Energy under Contract No. DE-AC05-00OR22725. Computations were also performed at the LICHTENBERG high performance computer of the TU Darmstadt, the Jülich Supercomputing Center (JURECA), the LOEWE-CSC Frankfurt, and at the Max-Planck-Institute for Nuclear Physics.

References

  • Lewitowicz et al. (1994) M. Lewitowicz, R. Anne, G. Auger, D. Bazin, C. Borcea, V. Borrel, J. M. Corre, T. Dörfler, A. Fomichov, R. Grzywacz, D. Guillemaud-Mueller, R. Hue, M. Huyse, Z. Janas, H. Keller, S. Lukyanov, A. C. Mueller, Yu. Penionzhkevich, M. Pfützner, F. Pougheon, K. Rykaczewski, M. G. Saint-Laurent, K. Schmidt, W. D. Schmidt-Ott, O. Sorlin, J. Szerypo, O. Tarasov, J. Wauters,  and J. Zylicz, “Identification of the doubly-magic nucleus Sn in the reaction Sn Ni at 63 MeV/nucleon,” Phys. Lett. B 332, 20 – 24 (1994).
  • Hinke et al. (2012) C. B. Hinke, M. Böhmer, P. Boutachkov, T. Faestermann, H. Geissel, J. Gerl, R. Gernhäuser, M. Gorska, A. Gottardo, H. Grawe, J. L. Grebosz, R. Krücken, N. Kurz, Z. Liu, L. Maier, F. Nowacki, S. Pietri, Zs. Podolyák, K. Sieja, K. Steiger, K. Straub, H. Weick, H.-J. Wollersheim, P. J. Woods, N. Al-Dahan, N. Alkhomashi, A. Atac, A. Blazhev, N. F. Braun, I. T. Celikovic, T. Davinson, I. Dillmann, C. Domingo-Pardo, P. C. Doornenbal, G. de France, G. F. Farrelly, F. Farinon, N. Goel, T. C. Habermann, R. Hoischen, R. Janik, M. Karny, A. Kaskas, I. M. Kojouharov, Th. Kröll, Y. Litvinov, S. Myalski, F. Nebel, S. Nishimura, C. Nociforo, J. Nyberg, A. R. Parikh, A. Prochazka, P. H. Regan, C. Rigollet, H. Schaffner, C. Scheidenberger, S. Schwertel, P.-A. Söderström, S. J. Steer, A. Stolz,  and P. Strmen, “Superallowed Gamow-Teller decay of the doubly magic nucleus Sn,” Nature 486, 341–345 (2012).
  • Erler et al. (2012) J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. M. Perhac,  and M. Stoitsov, “The limits of the nuclear landscape,” Nature 486, 509 – 512 (2012).
  • Liddick et al. (2006) S. N. Liddick, R. Grzywacz, C. Mazzocchi, R. D. Page, K. P. Rykaczewski, J. C. Batchelder, C. R. Bingham, I. G. Darby, G. Drafta, C. Goodin, C. J. Gross, J. H. Hamilton, A. A. Hecht, J. K. Hwang, S. Ilyushkin, D. T. Joss, A. Korgul, W. Królas, K. Lagergren, K. Li, M. N. Tantawy, J. Thomson,  and J. A. Winger, “Discovery of and : Superallowed decay near doubly magic ,” Phys. Rev. Lett. 97, 082501 (2006).
  • Seweryniak et al. (2006) D. Seweryniak, K. Starosta, C. N. Davids, S. Gros, A. A. Hecht, N. Hoteling, T. L. Khoo, K. Lagergren, G. Lotay, D. Peterson, A. Robinson, C. Vaman, W. B. Walters, P. J. Woods,  and S. Zhu, “ decay of ,” Phys. Rev. C 73, 061301 (2006).
  • Faestermann et al. (2013) T. Faestermann, M. Górska,  and H. Grawe, “The structure of Sn and neighbouring nuclei,” Prog. Part. Nucl. Phys. 69, 85 – 130 (2013).
  • Seweryniak et al. (2007) D. Seweryniak, M. P. Carpenter, S. Gros, A. A. Hecht, N. Hoteling, R. V. F. Janssens, T. L. Khoo, T. Lauritsen, C. J. Lister, G. Lotay, D. Peterson, A. P. Robinson, W. B. Walters, X. Wang, P. J. Woods,  and S. Zhu, “Single-neutron states in ,” Phys. Rev. Lett. 99, 022504 (2007).
  • Darby et al. (2010) I. G. Darby, R. K. Grzywacz, J. C. Batchelder, C. R. Bingham, L. Cartegni, C. J. Gross, M. Hjorth-Jensen, D. T. Joss, S. N. Liddick, W. Nazarewicz, S. Padgett, R. D. Page, T. Papenbrock, M. M. Rajabali, J. Rotureau,  and K. P. Rykaczewski, “Orbital dependent nucleonic pairing in the lightest known isotopes of tin,” Phys. Rev. Lett. 105, 162502 (2010).
  • Banu et al. (2005) A. Banu, J. Gerl, C. Fahlander, M. Górska, H. Grawe, T. R. Saito, H.-J. Wollersheim, E. Caurier, T. Engeland, A. Gniady, M. Hjorth-Jensen, F. Nowacki, T. Beck, F. Becker, P. Bednarczyk, M. A. Bentley, A. Bürger, F. Cristancho, G. de Angelis, Zs. Dombrádi, P. Doornenbal, H. Geissel, J. Grebosz, G. Hammond, M. Hellström, J. Jolie, I. Kojouharov, N. Kurz, R. Lozeva, S. Mandal, N. Mărginean, S. Muralithar, J. Nyberg, J. Pochodzalla, W. Prokopowicz, P. Reiter, D. Rudolph, C. Rusu, N. Saito, H. Schaffner, D. Sohler, H. Weick, C. Wheldon,  and M. Winkler, “ studied with intermediate-energy coulomb excitation,” Phys. Rev. C 72, 061305 (2005).
  • Vaman et al. (2007) C. Vaman, C. Andreoiu, D. Bazin, A. Becerril, B. A. Brown, C. M. Campbell, A. Chester, J. M. Cook, D. C. Dinca, A. Gade, D. Galaviz, T. Glasmacher, M. Hjorth-Jensen, M. Horoi, D. Miller, V. Moeller, W. F. Mueller, A. Schiller, K. Starosta, A. Stolz, J. R. Terry, A. Volya, V. Zelevinsky,  and H. Zwahlen, “ Shell Gap near from Intermediate-Energy Coulomb Excitations in Even-Mass Isotopes,” Phys. Rev. Lett. 99, 162501 (2007).
  • Ekström et al. (2008) A. Ekström, J. Cederkäll, C. Fahlander, M. Hjorth-Jensen, F. Ames, P. A. Butler, T. Davinson, J. Eberth, F. Fincke, A. Görgen, M. Górska, D. Habs, A. M. Hurst, M. Huyse, O. Ivanov, J. Iwanicki, O. Kester, U. Köster, B. A. Marsh, J. Mierzejewski, P. Reiter, H. Scheit, D. Schwalm, S. Siem, G. Sletten, I. Stefanescu, G. M. Tveten, J. Van de Walle, P. Van Duppen, D. Voulot, N. Warr, D. Weisshaar, F. Wenander,  and M. Zielińska, “ transition strengths in and ,” Phys. Rev. Lett. 101, 012502 (2008).
  • Guastalla et al. (2013) G. Guastalla, D. D. DiJulio, M. Górska, J. Cederkäll, P. Boutachkov, P. Golubev, S. Pietri, H. Grawe, F. Nowacki, K. Sieja, A. Algora, F. Ameil, T. Arici, A. Atac, M. A. Bentley, A. Blazhev, D. Bloor, S. Brambilla, N. Braun, F. Camera, Zs. Dombrádi, C. Domingo Pardo, A. Estrade, F. Farinon, J. Gerl, N. Goel, J. Grȩbosz, T. Habermann, R. Hoischen, K. Jansson, J. Jolie, A. Jungclaus, I. Kojouharov, R. Knoebel, R. Kumar, J. Kurcewicz, N. Kurz, N. Lalović, E. Merchan, K. Moschner, F. Naqvi, B. S. Nara Singh, J. Nyberg, C. Nociforo, A. Obertelli, M. Pfützner, N. Pietralla, Z. Podolyák, A. Prochazka, D. Ralet, P. Reiter, D. Rudolph, H. Schaffner, F. Schirru, L. Scruton, D. Sohler, T. Swaleh, J. Taprogge, Zs. Vajta, R. Wadsworth, N. Warr, H. Weick, A. Wendt, O. Wieland, J. S. Winfield,  and H. J. Wollersheim, ‘‘Coulomb excitation of and the strength of the shell closure,” Phys. Rev. Lett. 110, 172501 (2013).
  • Bader et al. (2013) V. M. Bader, A. Gade, D. Weisshaar, B. A. Brown, T. Baugher, D. Bazin, J. S. Berryman, A. Ekström, M. Hjorth-Jensen, S. R. Stroberg, W. B. Walters, K. Wimmer,  and R. Winkler, ‘‘Quadrupole collectivity in neutron-deficient Sn nuclei: Sn and the role of proton excitations,” Phys. Rev. C 88, 051301 (2013).
  • Doornenbal et al. (2014) P. Doornenbal, S. Takeuchi, N. Aoi, M. Matsushita, A. Obertelli, D. Steppenbeck, H. Wang, L. Audirac, H. Baba, P. Bednarczyk, S. Boissinot, M. Ciemala, A. Corsi, T. Furumoto, T. Isobe, A. Jungclaus, V. Lapoux, J. Lee, K. Matsui, T. Motobayashi, D. Nishimura, S. Ota, E. C. Pollacco, H. Sakurai, C. Santamaria, Y. Shiga, D. Sohler,  and R. Taniuchi, “Intermediate-energy Coulomb excitation of : Moderate strength decrease approaching ,” Phys. Rev. C 90, 061302 (2014).
  • Lipoglavšek et al. (2002) M. Lipoglavšek, C. Baktash, J. Blomqvist, M. P. Carpenter, D. J. Dean, T. Engeland, C. Fahlander, M. Hjorth-Jensen, R. V. F. Janssens, A. Likar, J. Nyberg, E. Osnes, S. D. Paul, A. Piechaczek, D. C. Radford, D. Rudolph, D. Seweryniak, D. G. Sarantites, M. Vencelj,  and C.-H. Yu, “Breakup of the doubly magic Sn core,” Phys. Rev. C 66, 011302 (2002).
  • Blazhev et al. (2004) A. Blazhev, M. Górska, H. Grawe, J. Nyberg, M. Palacz, E. Caurier, O. Dorvaux, A. Gadea, F. Nowacki, C. Andreoiu, G. de Angelis, D. Balabanski, Ch. Beck, B. Cederwall, D. Curien, J. Döring, J. Ekman, C. Fahlander, K. Lagergren, J. Ljungvall, M. Moszyński, L.-O. Norlin, C. Plettner, D. Rudolph, D. Sohler, K. M. Spohr, O. Thelen, M. Weiszflog, M. Wisell, M. Wolińska,  and W. Wolski, ‘‘Observation of a core-excited isomer in ,” Phys. Rev. C 69, 064304 (2004).
  • Boutachkov et al. (2011) P. Boutachkov, M. Górska, H. Grawe, A. Blazhev, N. Braun, T. S. Brock, Z. Liu, B. S. Nara Singh, R. Wadsworth, S. Pietri, C. Domingo-Pardo, I. Kojouharov, L. Cáceres, T. Engert, F. Farinon, J. Gerl, N. Goel, J. Grbosz, R. Hoischen, N. Kurz, C. Nociforo, A. Prochazka, H. Schaffner, S. J. Steer, H. Weick, H.-J. Wollersheim, T. Faestermann, Zs. Podolyák, D. Rudolph, A. Ataç, L. Bettermann, K. Eppinger, F. Finke, K. Geibel, A. Gottardo, C. Hinke, G. Ilie, H. Iwasaki, J. Jolie, R. Krücken, E. Merchán, J. Nyberg, M. Pfützner, P. H. Regan, P. Reiter, S. Rinta-Antila, C. Scholl, P.-A. Söderström, N. Warr, P. J. Woods, F. Nowacki,  and K. Sieja, ‘‘High-spin isomers in Ag: Excitations across the and , closed shells,” Phys. Rev. C 84, 044311 (2011).
  • Brock et al. (2010) T. S. Brock, B. S. Nara Singh, P. Boutachkov, N. Braun, A. Blazhev, Z. Liu, R. Wadsworth, M. Górska, H. Grawe, S. Pietri, C. Domingo-Pardo, D. Rudolph, S. J. Steer, A. Ataç, L. Bettermann, L. Cáceres, T. Engert, K. Eppinger, T. Faestermann, F. Farinon, F. Finke, K. Geibel, J. Gerl, R. Gernhäuser, N. Goel, A. Gottardo, J. Grebosz, C. Hinke, R. Hoischen, G. Ilie, H. Iwasaki, J. Jolie, A. Kaşkaş, I. Kojuharov, R. Krücken, N. Kurz, E. Merchán, C. Nociforo, J. Nyberg, M. Pfützner, A. Prochazka, Zs. Podolyák, P. H. Regan, P. Reiter, S. Rinta-Antila, H. Schaffner, C. Scholl, P.-A. Söderström, N. Warr, H. Weick, H.-J. Wollersheim,  and P. J. Woods (RISING Collaboration), ‘‘Observation of a new high-spin isomer in Pd,” Phys. Rev. C 82, 061309 (2010).
  • Caurier et al. (2010) E. Caurier, F. Nowacki, A. Poves,  and K. Sieja, “Collectivity in the light xenon isotopes: A shell model study,” Phys. Rev. C 82, 064304 (2010).
  • Hjorth-Jensen et al. (1995) M. Hjorth-Jensen, T. T. S. Kuo,  and E. Osnes, ‘‘Realistic effective interactions for nuclear systems,” Phys. Rep. 261, 125 – 270 (1995).
  • Kolck (1999) U. Van Kolck, “Effective field theory of nuclear forces,” Prog. Part. Nucl. Phys. 43, 337 – 418 (1999).
  • Epelbaum et al. (2009) E. Epelbaum, H.-W. Hammer,  and U.-G. Meißner, “Modern theory of nuclear forces,” Rev. Mod. Phys. 81, 1773–1825 (2009).
  • Machleidt and Entem (2011) R. Machleidt and D. R. Entem, “Chiral effective field theory and nuclear forces,” Phys. Rep. 503, 1 – 75 (2011).
  • Navrátil et al. (2009) P. Navrátil, S. Quaglioni, I. Stetcu,  and B. R. Barrett, “Recent developments in no-core shell-model calculations,” J. Phys. G 36, 083101 (2009).
  • Barrett et al. (2013) B. R. Barrett, P. Navrátil,  and J. P. Vary, “Ab initio no core shell model,” Prog. Part. Nucl. Phys. 69, 131 – 181 (2013).
  • Hagen et al. (2014) G. Hagen, T. Papenbrock, M. Hjorth-Jensen,  and D. J. Dean, “Coupled-cluster computations of atomic nuclei,” Rep. Prog. Phys. 77, 096302 (2014).
  • Hergert et al. (2016) H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk,  and K. Tsukiyama, “The in-medium similarity renormalization group: A novel ab initio method for nuclei,” Phys. Rep. 621, 165 – 222 (2016).
  • Pieper and Wiringa (2001) S. C. Pieper and R. B. Wiringa, “Quantum Monte Carlo calculations of light nuclei,” Ann. Rev. Nucl. Part. Sci. 51, 53–90 (2001).
  • Mihaila and Heisenberg (2000) B. Mihaila and J. H. Heisenberg, “Microscopic Calculation of the Inclusive Electron Scattering Structure Function in O,” Phys. Rev. Lett. 84, 1403–1406 (2000).
  • Dean and Hjorth-Jensen (2004) D. J. Dean and M. Hjorth-Jensen, “Coupled-cluster approach to nuclear physics,” Phys. Rev. C 69, 054320 (2004).
  • Dickhoff and Barbieri (2004) W. H. Dickhoff and C. Barbieri, “Self-consistent green’s function method for nuclei and nuclear matter,” Prog. Part. Nucl. Phys. 52, 377 – 496 (2004).
  • Hagen et al. (2008) G. Hagen, T. Papenbrock, D. J. Dean,  and M. Hjorth-Jensen, “Medium-mass nuclei from chiral nucleon-nucleon interactions,” Phys. Rev. Lett. 101, 092502 (2008).
  • Hagen et al. (2010) G. Hagen, T. Papenbrock, D. J. Dean,  and M. Hjorth-Jensen, ‘‘Ab initio coupled-cluster approach to nuclear structure with modern nucleon-nucleon interactions,” Phys. Rev. C 82, 034330 (2010).
  • Tsukiyama et al. (2011) K. Tsukiyama, S. K. Bogner,  and A. Schwenk, “In-Medium Similarity Renormalization Group For Nuclei,” Phys. Rev. Lett. 106, 222502 (2011).
  • Roth et al. (2012) R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer,  and P. Navrátil, “Medium-Mass Nuclei with Normal-Ordered Chiral Interactions,” Phys. Rev. Lett. 109, 052501 (2012).
  • Hergert et al. (2013) H. Hergert, S. K. Bogner, S. Binder, A. Calci, J. Langhammer, R. Roth,  and A. Schwenk, “In-medium similarity renormalization group with chiral two- plus three-nucleon interactions,” Phys. Rev. C 87, 034307 (2013).
  • Somà et al. (2014) V. Somà, A. Cipollone, C. Barbieri, P. Navrátil,  and T. Duguet, “Chiral two- and three-nucleon forces along medium-mass isotope chains,” Phys. Rev. C 89, 061301 (2014).
  • Binder et al. (2013) S. Binder, P. Piecuch, A. Calci, J. Langhammer, P. Navrátil,  and R. Roth, “Extension of coupled-cluster theory with a noniterative treatment of connected triply excited clusters to three-body hamiltonians,” Phys. Rev. C 88, 054319 (2013).
  • Lähde et al. (2014) T. A. Lähde, E. Epelbaum, H. Krebs, D. Lee, U.-G. Meißner,  and G. Rupak, “Lattice effective field theory for medium-mass nuclei,” Phys. Lett. B 732, 110 – 115 (2014).
  • Binder et al. (2014) S. Binder, J. Langhammer, A. Calci,  and R. Roth, “Ab initio path to heavy nuclei,” Phys. Lett. B 736, 119 – 123 (2014).
  • Hebeler et al. (2011) K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga,  and A. Schwenk, “Improved nuclear matter calculations from chiral low-momentum interactions,” Phys. Rev. C 83, 031301 (2011).
  • Ekström et al. (2013) A. Ekström, G. Baardsen, C. Forssén, G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, W. Nazarewicz, T. Papenbrock, J. Sarich,  and S. M. Wild, “Optimized chiral nucleon-nucleon interaction at next-to-next-to-leading order,” Phys. Rev. Lett. 110, 192502 (2013).
  • Ekström et al. (2015) A. Ekström, G. R. Jansen, K. A. Wendt, G. Hagen, T. Papenbrock, B. D. Carlsson, C. Forssén, M. Hjorth-Jensen, P. Navrátil,  and W. Nazarewicz, “Accurate nuclear radii and binding energies from a chiral interaction,” Phys. Rev. C 91, 051301 (2015).
  • Ekström et al. (2017) A. Ekström, G. Hagen, T. D. Morris, T. Papenbrock,  and P. D. Schwartz, “Delta isobars and nuclear saturation,” ArXiv e-prints  (2017), arXiv:1707.09028 [nucl-th] .
  • Hagen et al. (2016a) G. Hagen, A. Ekström, C. Forssén, G. R. Jansen, W. Nazarewicz, T. Papenbrock, K. A. Wendt, S. Bacca, N. Barnea, B. Carlsson, C. Drischler, K. Hebeler, M. Hjorth-Jensen, M. Miorelli, G. Orlandini, A. Schwenk,  and J. Simonis, “Neutron and weak-charge distributions of the Ca nucleus,” Nat. Phys. 12, 186 (2016a).
  • Hagen et al. (2016b) G. Hagen, M. Hjorth-Jensen, G. R. Jansen,  and T. Papenbrock, “Emergent properties of nuclei from ab initio coupled-cluster calculations,” Phys. Scr. 91, 063006 (2016b).
  • Hagen et al. (2016) G. Hagen, G. R. Jansen,  and T. Papenbrock, “Structure of from first-principles computations,” Phys. Rev. Lett. 117, 172501 (2016).
  • Simonis et al. (2016) J. Simonis, K. Hebeler, J. D. Holt, J. Menendez,  and A. Schwenk, “Exploring sd-shell nuclei from two- and three-nucleon interactions with realistic saturation properties,” Phys. Rev. C 93, 011302 (2016).
  • Garcia Ruiz et al. (2016) R. F. Garcia Ruiz, M. L. Bissell, K. Blaum, A. Ekström, N. Frömmgen, G. Hagen, M. Hammen, K. Hebeler, J. D. Holt, G. R. Jansen, M. Kowalska, K. Kreim, W. Nazarewicz, R. Neugart, G. Neyens, W. Nörtershäuser, T. Papenbrock, J. Papuga, A. Schwenk, J. Simonis, K. A. Wendt,  and D. T. Yordanov, “Unexpectedly large charge radii of neutron-rich calcium isotopes,” Nature Physics  (2016), 10.1038/nphys36451602.07906 .
  • Simonis et al. (2017) J. Simonis, S. R. Stroberg, K. Hebeler, J. D. Holt,  and A. Schwenk, “Saturation with chiral interactions and consequences for finite nuclei,” Phys. Rev. C 96, 014303 (2017).
  • Bartlett and Musiał (2007) R. J. Bartlett and M. Musiał, “Coupled-cluster theory in quantum chemistry,” Rev. Mod. Phys. 79, 291–352 (2007).
  • Stroberg et al. (2017) S. R. Stroberg, A. Calci, H. Hergert, J. D. Holt, S. K. Bogner, R. Roth,  and A. Schwenk, “Nucleus-dependent valence-space approach to nuclear structure,” Phys. Rev. Lett. 118, 032502 (2017).
  • Stumpf et al. (2016) C. Stumpf, J. Braun,  and R. Roth, “Importance-truncated large-scale shell model,” Phys. Rev. C 93, 021301 (2016).
  • Bogner et al. (2007) S. K. Bogner, R. J. Furnstahl,  and R. J. Perry, “Similarity renormalization group for nucleon-nucleon interactions,” Phys. Rev. C 75, 061001 (2007).
  • Entem and Machleidt (2003) D. R. Entem and R. Machleidt, “Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory,” Phys. Rev. C 68, 041001 (2003).
  • van Kolck (1994) U. van Kolck, “Few-nucleon forces from chiral Lagrangians,” Phys. Rev. C 49, 2932–2941 (1994).
  • Epelbaum et al. (2002) E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner,  and H. Witała, “Three-nucleon forces from chiral effective field theory,” Phys. Rev. C 66, 064001 (2002).
  • Hebeler et al. (2015) K. Hebeler, H. Krebs, E. Epelbaum, J. Golak,  and R. Skibiński, “Efficient calculation of chiral three-nucleon forces up to for ab initio studies,” Phys. Rev. C 91, 044001 (2015).
  • Nogga et al. (2004) A. Nogga, S. K. Bogner,  and A. Schwenk, “Low-momentum interaction in few-nucleon systems,” Phys. Rev. C 70, 061002 (2004).
  • Hagen et al. (2007) G. Hagen, T. Papenbrock, D. J. Dean, A. Schwenk, A. Nogga, M. Włoch,  and P. Piecuch, “Coupled-cluster theory for three-body Hamiltonians,” Phys. Rev. C 76, 034302 (2007).
  • Kümmel et al. (1978) H. Kümmel, K. H. Lührmann,  and J. G. Zabolitzky, “Many-fermion theory in exp S- (or coupled cluster) form,” Phys. Rep. 36, 1 – 63 (1978).
  • Włoch et al. (2005) M. Włoch, D. J. Dean, J. R. Gour, M. Hjorth-Jensen, K. Kowalski, T. Papenbrock,  and P. Piecuch, “Ab-Initio coupled-cluster study of ,” Phys. Rev. Lett. 94, 212501 (2005).
  • Roth et al. (2011) R. Roth, J. Langhammer, A. Calci, S. Binder,  and P. Navrátil, “Similarity-Transformed Chiral Interactions for the Ab Initio Description of and ,” Phys. Rev. Lett. 107, 072501 (2011).
  • Taube and Bartlett (2008) A. G. Taube and R. J. Bartlett, “Improving upon CCSD(T): Lambda CCSD(T). I. Potential energy surfaces,” J. Chem. Phys. 128, 044110 (2008).
  • Watts and Bartlett (1995) J. D. Watts and R. J. Bartlett, “Economical triple excitation equation-of-motion coupled-cluster methods for excitation energies,” Chem. Phys. Lett. 233, 81 – 87 (1995).
  • Gour et al. (2005) J. R. Gour, P. Piecuch,  and M. Włoch, “Active-space equation-of-motion coupled-cluster methods for excited states of radicals and other open-shell systems: EA-EOMCCSDt and IP-EOMCCSDt,” J. Chem. Phys. 123, 134113 (2005).
  • Jansen et al. (2011) G. R. Jansen, M. Hjorth-Jensen, G. Hagen,  and T. Papenbrock, “Toward open-shell nuclei with coupled-cluster theory,” Phys. Rev. C 83, 054306 (2011).
  • Jansen (2013) G. R. Jansen, “Spherical coupled-cluster theory for open-shell nuclei,” Phys. Rev. C 88, 024305 (2013).
  • Kowalski and Piecuch (2000) K. Kowalski and P. Piecuch, “The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches,” J. Chem. Phys. 113, 18–35 (2000).
  • Piecuch et al. (2002) P. Piecuch, K. Kowalski, I. S. O. Pimienta,  and M. J. Mcguire, “Recent advances in electronic structure theory: Method of moments of coupled-cluster equations and renormalized coupled-cluster approaches,” Int. Rev. Phys. Chem. 21, 527–655 (2002).
  • Kowalski et al. (2004) K. Kowalski, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock,  and P. Piecuch, “Coupled cluster calculations of ground and excited states of nuclei,” Phys. Rev. Lett. 92, 132501 (2004).
  • Tsukiyama et al. (2012) K. Tsukiyama, S. K. Bogner,  and A. Schwenk, “In-medium similarity renormalization group for open-shell nuclei,” Phys. Rev. C 85, 061304 (2012).
  • Bogner et al. (2014) S. K. Bogner, H. Hergert, J. D. Holt, A. Schwenk, S. Binder, A. Calci, J. Langhammer,  and R. Roth, “Nonperturbative shell-model interactions from the in-medium similarity renormalization group,” Phys. Rev. Lett. 113, 142501 (2014).
  • Stroberg et al. (2016) S. R. Stroberg, H. Hergert, J. D. Holt, S. K. Bogner,  and A. Schwenk, ‘‘Ground and excited states of doubly open-shell nuclei from ab initio valence-space hamiltonians,” Phys. Rev. C 93, 051301 (2016).
  • Björnstad et al. (1980) T. Björnstad, L.-E. De Geer, G.T. Ewan, P.G. Hansen, B. Jonson, K. Kawade, A. Kerek, W.-D. Lauppe, H. Lawin, S. Mattsson,  and K. Sistemich, “Structure of the levels in the doubly magic nucleus Sn,” Phys. Lett. B 91, 35 – 37 (1980).
  • Jones et al. (2010) K. L. Jones, A. S. Adekola, D. W. Bardayan, J. C. Blackmon, K. Y. Chae, K. A. Chipps, J. A. Cizewski, L. Erikson, C. Harlin, R. Hatarik, R. Kapler, R. L. Kozub, J. F. Liang, R. Livesay, Z. Ma, B. H. Moazen, C. D. Nesaraja, F. M. Nunes, S. D. Pain, N. P. Patterson, D. Shapira, J. F. Shriner, M. S. Smith, T. P. Swan,  and J. S. Thomas, Nature 465, 454–457 (2010).
  • Cerizza et al. (2016) G. Cerizza, A. Ayres, K. L. Jones, R. Grzywacz, A. Bey, C. Bingham, L. Cartegni, D. Miller, S. Padgett, T. Baugher, D. Bazin, J. S. Berryman, A. Gade, S. McDaniel, A. Ratkiewicz, A. Shore, S. R. Stroberg, D. Weisshaar, K. Wimmer, R. Winkler, S. D. Pain, K. Y. Chae, J. A. Cizewski, M. E. Howard,  and J. A. Tostevin, “Structure of studied through single-neutron knockout reactions,” Phys. Rev. C 93, 021601 (2016).
  • Garcia Ruiz et. al (2016) R. F. Garcia Ruiz et. al, IS613: Laser Spectroscopy of neutron-deficient Sn isotopes, Proposal INTC-P-456 CERN-INTC-2016-006 (CERN, 2016).
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