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Structure of neutron rich calcium isotopes from coupled cluster theory

Explore the evolution of shell structure in neutron-rich calcium isotopes from chiral interactions. Study shell closures, magic numbers, and shell order inversions towards the dripline. Utilize coupled cluster theory coupled with practical optimization methods.

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Structure of neutron rich calcium isotopes from coupled cluster theory

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  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) RuprechtMachleidt (UI) WitoldNazarewicz (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; Epelbaumet al.; Entem & Machleidt; …] [Epelbaum, Hammer, Meissner RMP 81, 1773 (2009)] CD CE Low energy constants from fit of NN data, A=3,4 nuclei, or light nuclei.

  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) • Scales gently (polynomial) with increasing problem size o2u4 . • Truncation is the only approximation. • Size extensive (error scales with A) •  Most efficient for doubly magic nuclei Ansatz: Correlations are exponentiated 1p-1h and 2p-2h excitations. Part of np-nh excitations included! Coupled cluster equations 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. • NNLO-POUNDerSis a “soft” potential • No dramatic overbinding is found for light nuclei

  6. Oxygen isotopes from chiral NN forces

  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. NNLO(POUNDerS) gives remarkable agreement with experiment, and the dripline in oxygen is correctly placed at 24O. 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? • 3NFs are responsible for shell closure in 48Ca • Different models give conflicting result for shell closure in 54Ca. • J. D. Holt et al, J. Phys. G 39, 085111 (2012)

  9. Evolution of shell structure in neutron rich CalciumInversion of shell order in 60Ca • S. M. Lenzi, F. Nowacki, A. Poves, • and K. Sieja Phys. Rev. C 82, 054301 (2010) • Inversion of d5/2 and g9/2 in 60Ca. • 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 60-70Ca • 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±500syst Literature: 5,000-12,000 288 ~3,000 Description of observables and model-based extrapolation • Systematic errors (due to incorrect assumptions/poor modeling) • Statistical errors (optimization and numerical errors) Erler et al., Nature 486, 509 (2012)

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

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

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

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

  16. Calcium isotopes from NNLO-POUNDerS 1MeV per nucleon overall shift. Basic features such as separtationeneriges 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).

  19. Point radii from NNLO-POUNDerS Preliminary • Modelspace: N =14 major harmonic oscillator for the NN interaction while E3max= 12 for three-nucleon force. • Binding energies of Calcium isotopes are not converged => need E3max~ 20. Neutron skin (NN+3NF) = 0.16fm Neutron skin (NN-only) = 0.17fm

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

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