1. Structure of neutron rich calcium isotopes from coupled cluster theory Gaute Hagen (ORNL) Collaborators: Andreas Ekström (MSU) Christian Forrsen (Chalmers) Morten Hjorth-Jensen (UiO/CMA) Gustav Jansen (UT/ORNL) Ruprecht Machleidt (UI) Witold Nazarewicz (UT/ORNL) Thomas Papenbrock (UT/ORNL) Jason Sarich (ANL) Stefan Wild (ANL) CREX Workshop Jefferson Laboratory, March 18, 2013

2. Nuclear forces from chiral effective field theory [Weinberg; van Kolck; Epelbaum et al.; Entem & Machleidt; …] [Epelbaum, Hammer, Meissner RMP 81, 1773 (2009)] Low energy constants from fit of NN data, A=3,4 nuclei, or light nuclei. CECE CDCD

3. Chiral interactions from Practical Optimization Using No Derivatives (for Squares) NNLO(POUNDerS) NNLO(EGM 450/500) Nijmegen PWA cD = -0.2, cE=-0.36

4. Coupled-cluster method (in CCSD approximation) Ansatz: Correlations are exponentiated 1p-1h and 2p-2h excitations. Part of np-nh excitations included! Coupled cluster equations Scales gently (polynomial) with increasing problem size o 2 u 4. Truncation is the only approximation. Size extensive (error scales with A)  Most efficient for doubly magic nuclei Alternative view: CCSD generates similarity transformed Hamiltonian with no 1p-1h and no 2p-2h excitations.

5. Light nuclei from NN2LO-POUNDerS Rapid Convergence for ground states of oxygen isotopes with NNLO- POUNDerS. Already with N =12-14 major harmonic oscillator shells results are well converged. Rapid Convergence for ground states of oxygen isotopes with NNLO- POUNDerS. Already with N =12-14 major harmonic oscillator shells results are well converged. NNLO-POUNDerS is a “soft” potential No dramatic overbinding is found for light nuclei NNLO-POUNDerS is a “soft” potential No dramatic overbinding is found for light nuclei

6. Oxygen isotopes from chiral NN forces 16 O 22 O 24 O NNLO(POUNDerS)-130.28-159.76-168.45 NNLO(EGM 450/500)-156.76-208.85-225.65 Experiment-127.62-162.06-168.48

7. Shell model calculations of oxygen from chiral NN forces Shell model calculations done in s-d model space Effective interaction from g-matrix and third order perturbation theory. Folded diagrams to infinite order hw = 14MeV, N = 12 for intermediate excitations. Shell model calculations done in s-d model space Effective interaction from g-matrix and third order perturbation theory. Folded diagrams to infinite order hw = 14MeV, N = 12 for intermediate excitations. 1.NNLO(POUNDerS) gives remarkable agreement with experiment, and the dripline in oxygen is correctly placed at 24 O. 2.Two-body forces alone get the structure to leading order right! 1.NNLO(POUNDerS) gives remarkable agreement with experiment, and the dripline in oxygen is correctly placed at 24 O. 2.Two-body forces alone get the structure to leading order right!

8. Evolution of shell structure in neutron rich Calcium How do shell closures and magic numbers evolve towards the dripline? Is the naïve shell model picture valid at the neutron dripline? How do shell closures and magic numbers evolve towards the dripline? Is the naïve shell model picture valid at the neutron dripline? 3NFs are responsible for shell closure in 48 Ca Different models give conflicting result for shell closure in 54 Ca. J. D. Holt et al, J. Phys. G 39, 085111 (2012) 3NFs are responsible for shell closure in 48 Ca Different models give conflicting result for shell closure in 54 Ca. J. D. Holt et al, J. Phys. G 39, 085111 (2012)

9. Evolution of shell structure in neutron rich Calcium Inversion of shell order in 60 Ca S. M. Lenzi, F. Nowacki, A. Poves, and K. Sieja Phys. Rev. C 82, 054301 (2010) Inversion of d5/2 and g9/2 in 60 Ca. Bunching of levels pointing to no shell-closure. S. M. Lenzi, F. Nowacki, A. Poves, and K. Sieja Phys. Rev. C 82, 054301 (2010) Inversion of d5/2 and g9/2 in 60 Ca. Bunching of levels pointing to no shell-closure.

10. Evolution of shell structure in neutron rich Calcium Relativistic mean-field show no shell gap in 60-70 Ca Bunching of single- particle orbitals large deformations and no shell closure J. Meng et al, Phys. Rev. C 65, 041302(R) (2002) Relativistic mean-field show no shell gap in 60-70 Ca Bunching of single- particle orbitals large deformations and no shell closure J. Meng et al, Phys. Rev. C 65, 041302(R) (2002)

11. How many protons and neutrons can be bound in a nucleus? Skyrme-DFT: 6,900±500 syst Literature: 5,000-12,000 288 ~3,000 Erler et al., Nature 486, 509 (2012) Description of observables and model-based extrapolation Systematic errors (due to incorrect assumptions/poor modeling) Statistical errors (optimization and numerical errors)

12. Including the effects of 3NFs (approximation!) [J.W. Holt, Kaiser, Weise, PRC 79, 054331 (2009); Hebeler & Schwenk, PRC 82, 014314 (2010)] Parameters: For calcium we use k F = 0.95 fm -1, c E = 0.735, c D = -0.2 from binding energy of 40 Ca and 48 Ca (The parameters c D, c E differ from values proposed for light nuclei) 3NFs as in-medium effective two-nucleon forces Integration of Fermi sea of symmetric nuclear matter: k F

13. Calcium isotopes from chiral interactions Main Features: 1.Total binding energies agree well with experimental masses. 2.Masses for 40-52 Ca are converged in 19 major shells. 3. 60 Ca is not magic 4. 61-62 Ca are located right at threshold. See also: Meng et al PRC 65, 041302 (2002), Lenzi et al PRC 82, 054301 (2010) and Erler et al, Nature 486, 509 (2012) Main Features: 1.Total binding energies agree well with experimental masses. 2.Masses for 40-52 Ca are converged in 19 major shells. 3. 60 Ca is not magic 4. 61-62 Ca are located right at threshold. See also: Meng et al PRC 65, 041302 (2002), Lenzi et al PRC 82, 054301 (2010) and Erler et al, Nature 486, 509 (2012) G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, T. Papenbrock, Phys. Rev. Lett. 109, 032502 (2012). k F =0.95fm -1, c D =-0.2, c E =0.735 N max = 18, hw = 26MeV A peninsula of weak stability?

14. Is 54 Ca a magic nucleus? (Is N=34 a magic number?) 48 Ca 52 Ca 54 Ca 2+2+ 4+4+ 4 + /2 + 2+2+ 4+4+ 2+2+ 4+4+ 3.584.201.172.193.951.801.894.462.36 3.834.501.172.56????? CC Exp Main Features: 1.Good agreement between theory and experiment. 2.Shell closure in 48 Ca due to effects of 3NFs 3.Predict weak (sub-)shell closure in 54 Ca. Main Features: 1.Good agreement between theory and experiment. 2.Shell closure in 48 Ca due to effects of 3NFs 3.Predict weak (sub-)shell closure in 54 Ca. G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, T. Papenbrock, Phys. Rev. Lett. 109, 032502 (2012).

15. Spectra and shell evolution in Calcium isotopes 1.Inversion of the 9/2+ and 5/2+ resonant states in 53,55,61 Ca 2. We find the ground state of 61 Ca to be ½+ located right at threshold. 3.A harmonic oscillator basis gives the naïve shell model ordering of states. 1.Inversion of the 9/2+ and 5/2+ resonant states in 53,55,61 Ca 2. We find the ground state of 61 Ca to be ½+ located right at threshold. 3.A harmonic oscillator basis gives the naïve shell model ordering of states. New penning trap measurement of masses of 51,52 Ca A. T. Gallant et al Phys. Rev. Lett. 109, 032506 (2012) New penning trap measurement of masses of 51,52 Ca A. T. Gallant et al Phys. Rev. Lett. 109, 032506 (2012) Continuum coupling is crucial!

16. Calcium isotopes from NNLO-POUNDerS 1MeV per nucleon overall shift. Basic features such as separtation eneriges and spectra well reproduced at the NN level. 1MeV per nucleon overall shift. Basic features such as separtation eneriges and spectra well reproduced at the NN level.

17. Calcium isotopes from NNLO-POUNDerS

18. Neutron matter from NNLO-POUNDerS Neutron matter from the NNLO-POUNDerS (red line) and the N3LO NN potentials (blue dashed line). Coupled-cluster calculations (particle- particle ladders and exact Pauli operator) The NNLO-POUNDerS results are agreement with results from I. Tews, T. Krüger, K. Hebeler, and A. Schwenk PRL110, 032504 (2013). Neutron matter from the NNLO-POUNDerS (red line) and the N3LO NN potentials (blue dashed line). Coupled-cluster calculations (particle- particle ladders and exact Pauli operator) The NNLO-POUNDerS results are agreement with results from I. Tews, T. Krüger, K. Hebeler, and A. Schwenk PRL110, 032504 (2013).

19. Modelspace: N =14 major harmonic oscillator for the NN interaction while E 3max = 12 for three- nucleon force. Binding energies of Calcium isotopes are not converged => need E 3max ~ 20. Modelspace: N =14 major harmonic oscillator for the NN interaction while E 3max = 12 for three- nucleon force. Binding energies of Calcium isotopes are not converged => need E 3max ~ 20. Point radii from NNLO-POUNDerS 16 O 22 O 24 O 48 Ca NN-only2.292.552.692.97 NN+3NF2.572.853.013.49 Exp2.54(2)2.75(15)3.19(13) Neutron skin (NN+3NF) = 0.16fm Neutron skin (NN-only) = 0.17fm Preliminary

20. Summary/perspectives 1.Effects of 3NF and continuum give good agreement with experiment for binding energy and spectra in oxygen and calcium 2.Predict weak sub-shell closure in 54 Ca. 3.Level ordering in the gds shell in neutron calcium is reversed compared to naïve shell model 4.A proper optimized NNLO chiral interaction gets leading order structure right from light nuclei to neutron matter 1.Effects of 3NF and continuum give good agreement with experiment for binding energy and spectra in oxygen and calcium 2.Predict weak sub-shell closure in 54 Ca. 3.Level ordering in the gds shell in neutron calcium is reversed compared to naïve shell model 4.A proper optimized NNLO chiral interaction gets leading order structure right from light nuclei to neutron matter 1.Compute covariance matrix of optimized LECs and study correlations between LECs 2.Quantify how uncertainties propagate from optimized interaction to observables in medium mass nuclei 3.Compute radii and neutron skin in calcium-48 with NN+3NFs and quantified uncertainties 1.Compute covariance matrix of optimized LECs and study correlations between LECs 2.Quantify how uncertainties propagate from optimized interaction to observables in medium mass nuclei 3.Compute radii and neutron skin in calcium-48 with NN+3NFs and quantified uncertainties