Shell evolution in neutron rich nuclei

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Shell evolution in neutron rich nuclei Gustav R. Jansen 1,2 gustav.jansen@utk.edu 1 University of Tennessee, Knoxville 2 Oak Ridge National Laboratory March 18. 2013

Collaborators and acknowledgements Andreas Ekström (UiO, MSU) Christian Forrsen (Chalmers) Gaute Hagen (ORNL) Morten Hjorth-Jensen (UiO, MSU) Gustav R. Jansen (UTK, ORNL) Ruprecht Machleidt (UI) Hai Ah Nam (ORNL) Witold Nazarewicz (UTK, ORNL) Thomas Papenbrock (UTK, ORNL)

Outline Motivation. Shell evolution in oxygen isotopes. Dripline Magic nuclei. 26 O vs. 26 F. Shell evolution in calcium isotopes. The single particle picture. Magic numbers and shell closures. Consistent nuclear forces

Understanding matter

The nuclear shell model Maria Goeppert Mayer, Phys. Rev. 75 (1949). Otto Haxel, J. Hans D. Jensen, and Hans E. Suess, Phys. Rev. 75 (1949). Nobel prize in physics in 1963: Eugene Wigner, Maria Goeppert Mayer, J. Hans D. Jensen. Defined by shell closures at magic numbers, where a spin-orbit force added to a harmonic oscillator potential reproduce the magic numbers.

The nuclear manybody problem Need to solve the Schrödinger equation Ĥ Ψ = (ˆT+ ˆV1 + ˆV 2 + ˆV ) 3... Ψ = E Ψ Two ingredients 1. The nuclear interaction. 2. A method to solve the many body problem.

Nuclear interactions Want an interaction between nucleons, with the pion as force carrier. Not the fundamental degrees of freedom. Hadrons are color neutral, in much the same way as atoms are charge neutral. The interaction between hadrons are residual interactions, much like the van der Waals interactions between molecules. Complicated interaction with manybody components

Chiral effective field theory D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001 (2003) Direct link to QCD. Perturbative expansion in momentum. Chiral symmetry is spontaneously and explicitly broken. The hierarchy of nuclear forces unfolds automatically.

Density dependent chiral threebody force

Finite basis expansion The wavefunction is expanded in Slater determinants Ψ = D c i Φ i. i The number of possible Slater determinants is ( n A), where n is the number of single particle states and A is the number of nucleons.

Curse of dimensionality 10 52 10 48 10 44 4 He 10 Be 16 O 40 Ca 10 40 Slater determinant basis 10 36 10 32 10 28 10 24 10 20 10 16 10 12 10 8 10 4 10 0 0 50 100 150 200 Number of single particle states

The coupled-cluster method

Excited states using EOM-CC Eigenvalues of H = e ˆT ĤeˆT Φ 0 Ĥ Φ 0 ( HˆR) = ωˆr c Properties of H. Non-symmetric (non-hermetian) operator. For CCSD and a twobody hamiltonian - six-body operator. The matrix representation is very sparse. Generally too large to store and diagonalize exactly. ( HˆR) Efficient implementation of C is key.

Open Quantum Systems

The oxygen dripline

Oxygen isotopes from chiral interaction

Evolution of single particle energies S. M. Lenzi, F. Nowacki, A. Poves and K. Sieja, PRC 82 054301 (2010) Main features Shell-model calculation in the sd-shell including 0f 7/2 and 1p 3/2 for neutrons. Inversion of the 0f 7/2 and the 1p 3/2 single particle states in 28 O.

Negative parity states in O25

Partial summary 28 O is not a closed shell nucleus. Contiuum effects crucial for level ordering. New shell consisting of 0d 3/2, 1p 3/2 and 1p 1/2? Next doubly magic oxygen nuclei would be 34 O? Coupled cluster calculation in this region will be very difficult, since we have no good closed shell reference.

The effects of a single proton 26 F probing the proton-neutron interaction Simple structure 24 O plus πd 5/2 and νd 3/2. First approximation J π = 1 + 4 +. Weakly bound S n 0.8 MeV. Adding a proton changes the dripline by 6 neutrons. Not all fluorine isotopes towards the dripline are bound Interplay between proton-neutron interactions, neutron-neutron interactions, threebody interactions and continuum degrees of freedom.

Threebody forces in 26 F A. Lepailleur et al., Phys. Rev. Lett. 110, 082502 (2013) Technical details Chiral interaction at N 3 LO. Identical threebody force as established in the oxygen chain. 17 major harmonic oscillator shells with a Gamow-Hartree-Fock basis for νs 1/2 and νd 3/2 CCSD with triples corrections (Λ-CCSD(T)) for 24 O, with 2PA-EOMCC. 26 F free = B( 25 O) + B( 25 F) B( 24 O) Threebody forces are crucial for correct levelspacing.

Resonances in neutron-rich 24 O

Oxygen isotopes from chiral interaction

Evolution of single particle energies Technical details J. Meng, H. Toki, J. Y. Zeng, S. Q. Zhang and S. -G. Zhou, PRC 65 041302(R) (2002). Relativistic mean-field including continuum effects. Main features Bunching of single-particle energies outside the pf-shell. No shell-gap in 60 Ca - 70 Ca. Large deformations and no shell-closure. Continuum effects responsible for bound 60 Ca - 72 Ca.

Evolution of single particle energies S. M. Lenzi, F. Nowacki, A. Poves and K. Sieja, PRC 82 054301 (2010) Main features Shell-model calculation in the pf-shell including 0g 9/2 and 2d 5/2 for neutrons. Inversion of the 0g 9/2 and the 2d 5/2 single particle states in 60 Ca. Bunching of levels including the 0f 5/2 state indicates no shell-closure.

Binding energies in calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL109 032502 (2012) E (MeV) -320-340 -360-380 -400-420 -440-460 -480-500 -520 NN + 3NF eff Experiment 39 40 41 42 47 48 49 50 51 52 53 54 55 56 59 60 61 62 A Technical details Chiral interaction at N 3 LO. Density dependent three body force with k F = 0.95fm 1, c D = 0.2 and c E = 0.735. N max = 18 and ω = 26 MeV. Mass of 51 Ca and 52 Ca from A. T. Gallant et al., PRL 109, 032506 (2012) Main features Total binding energies agree well with experimental masses. 60 Ca is not magic. Three nucleon force is repulsive.

Binding energies in calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL109 032502 (2012) E (MeV) -320-340 -360-380 -400-420 -440-460 -480-500 -520 NN only NN + 3NF eff Experiment NN only 39 40 41 42 47 48 49 50 51 52 53 54 55 56 59 60 61 62 A Technical details Chiral interaction at N 3 LO. Density dependent three body force with k F = 0.95fm 1, c D = 0.2 and c E = 0.735. N max = 18 and ω = 26 MeV. Mass of 51 Ca and 52 Ca from A. T. Gallant et al., PRL 109, 032506 (2012) Main features Total binding energies agree well with experimental masses. 60 Ca is not magic. Three nucleon force is repulsive.

Shell evolution in neutron rich calcium isotopes. Details J. D. Holt, T. Otsuka, A. Schwenk and T. Suzuki, J Phys G39 085111 (2012).. J π = 2 + systematics in even calcium isotopes. Main features Threebody forces needed to make 48 Ca magic. Different models have 54 Ca magic, semi magic and not magic at all.

J π = 2 + systematics in even calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL109 032502 (2012) 5 E 2 + (MeV) 4 3 2 1 0 NN+3NF eff Exp 42 48 50 52 54 56 A Ca Technical details Chiral interaction at N 3 LO. Density dependent three body force with k F = 0.95fm 1, c D = 0.2 and c E = 0.735. N max = 18 and ω = 26 MeV. Main features Good agreement between theory and experiment. Shell closure in 48 Ca. Sub-shell closure in 52 Ca. Predict weak sub-shell closure in 54 Ca.

J π = 2 + systematics in even calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL109 032502 (2012) 5 E 2 + (MeV) 4 3 2 1 0 NN only NN+3NF eff Exp NN only 42 48 50 52 54 56 A Ca Technical details Chiral interaction at N 3 LO. Density dependent three body force with k F = 0.95fm 1, c D = 0.2 and c E = 0.735. N max = 18 and ω = 26 MeV. Main features Good agreement between theory and experiment. Shell closure in 48 Ca. Sub-shell closure in 52 Ca. Predict weak sub-shell closure in 54 Ca.

Spectra in calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL109 032502 (2012) Energy (MeV) 7 6 5 4 3 2 1 4 + 52 3 Ca + 54 Ca 55 Ca 4 + 1 + 2 + 3 + 4 + 1 + 2 + 2 + 5/2 + 7/2-53 Ca 3/2-5/2-4 + Technical 1 + 2 + 3 + 4 + 3 + 2 + 9/2 + 5/2 + 56 Ca 4 + 2 + details Chiral interaction at N 3 LO. Density dependent three body force with k F = 0.95fm 1, c D = 0.2 and c E = 0.735. N max = 18 and ω = 26 MeV. Continuum included for selected weakly bound and resonant states. 0 NN+3NF eff 0 + Exp 0 + 1/2-1/2 - NN+3NF eff Exp NN+3NF eff 0 + 0 + Exp NN+3NF eff 5/2 - Exp 5/2-0 + 0 + Exp NN+3NF eff Main features Inversion of g 9/2 and d 5/2. 1/2 + groundstate in 61 Ca. Continuum effects are crucial.

Consistent nuclear forces Want to derive consistent forces. All contributions at a given order are evaluated. Currently NNLO. Apply numerical optimization algorithms to find the optimal parameters.

NNLO (POUNDerS) POUNDerS Practical Optimization Using No Derivatives for sums of Squares Part of TAO (http://www.mcs.anl.gov/research/projects/tao/) Finding a better solution POUNDerS is allowed to explore the NNLO parameter space to find the best fit to phaseshifts. Compute χ 2 /datum against available scattering data, to evaluate the quality of the interaction.

Quality of the fit Preliminary T lab (MeV) 0 35 35 125 125 183 183 290 0 290 pp χ 2 23.95 /datum 1.11 1.56 4.35 a 29.26 17.10 14.03 a np χ 2 /datum 0.85 1.17 1.87 6.09 2.95 N 3 LO NNLO-POUNDerS Empirical a C pp -7.8188-7.8174-7.8196(26) -7.8149(29) rpp C 2.795 2.755 2.790(14) 2.769(14) a N pp -17.083-17.825 r N pp 2.876 2.817 a N nn -18.900-18.889-18.95(40) rnn N 2.838 2.797 2.75(11) a np -23.732-23.749-23.740(20) r np 2.725 2.684 2.77(5) B D (MeV) 2.224575 2.224582 2.224575(9) r D (fm) 1.975 1.967 1.97535(85) Q D (fm 2 ) 0.275 0.272 0.2859(3) P D 4.51 4.05

Triton binding energy Preliminary E NCSM (MeV) -6-6.5-7 -7.5-8 POUNDerS-NNLO (NN+NNN) POUNDerS-NNLO (NN) Experiment N3LO (NN) -8.5-9 0 10 20 30 40 Nmax

4 He binding energy Preliminary -20-22 -24 E NCSM (MeV) -26-28 -30-32 -34 POUNDerS-NNLO(NN) POUNDerS-NNLO(NN+NNN) N3LO(NN) Experiment 0 5 10 15 20 Nmax

Oxygen isotopes with NNLO (POUNDerS) Preliminary E (MeV) 100 110 120 130 140 E (MeV) 0-10 -20-30 -40-50 -60-70 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 A 150 160 170 Experiment NNLO (POUNDerS) N 3 LO 15 16 17 18 19 20 21 22 23 24 25 26 27 28 A

Oxygen spectra with NNLO (POUNDerS) Preliminary Energy (MeV) 8 7 6 5 4 3 2 1 0 N 3 LO 3 + 2 + 0 + 2 + 4 + 4 + 2 + 0 + 18 O 4 + 23 4 + O NNLO 0 + 3 + 2 + 2 + Exp 3 + 2 0+ + 4 + 2 + N 3 LO 2 + 3 + 4 + 3 + 22 O NNLO 2 + 3 + 3 + 2 + 0 + 0 + 0 + 0 + Exp 2 + 3 + 2 + 0 + N 3 LO 3/2 + 5/2 + 5/2 + 1/2 + 1/2 + 1/2 + NNLO 3/2 + 3/2 + Exp 5/2 + N 3 LO 1 + 2 + 24 O NNLO 2 + 3 +? 4 + 1 + Exp 1 + 2 + 2 + 0 + 0 + 0 +

J π = 2 + systematics in oxygen isotopes with NNLO (POUNDerS) Preliminary E 2 + (MeV) 5 4 3 2 N 3 LO 3nfeff Exp N 3 LO NNLO (POUNDerS) 1 0 18 22 24 26 A O

J π = 2 + systematics with NNLO (POUNDerS) Preliminary 5 E 2 + (MeV) 4 3 2 NN+3NF eff Exp NN only NNLO (POUNDerS) 1 0 N2LO (POUNDerS) 42 48 50 52 54 56 A Ca

Calcium isotopes with NNLO (POUNDerS) Preliminary -320-340 E (MeV) -360-380 -400-420 -440-460 -480-500 -520 NN only NN + 3NF eff Experiment NN only 39 40 41 42 47 48 49 50 51 52 53 54 55 56 59 60 61 62 A

Calcium isotopes with NNLO (POUNDerS) Preliminary -320-340 E (MeV) -360-380 -400-420 -440-460 -480-500 -520 NNLO (POUNDerS) NNLO (POUNDerS) NN + 3NF eff Experiment N3LO NN only 39 40 41 42 47 48 49 50 51 52 53 54 55 56 59 60 61 62 A

Calcium isotopes with NNLO (POUNDerS) Preliminary -320-340 E (MeV) -360-380 -400-420 -440-460 -480-500 -520 NNLO (Shifted) NN + 3NF eff Experiment N3LO NN only 39 40 41 42 47 48 49 50 51 52 53 54 55 56 59 60 61 62 A

Calcium isotopes with NNLO (POUNDerS) Preliminary -320-340 E (MeV) -360-380 -400-420 -440-460 -480-500 -520 NNLO (Shifted) NN + 3NF eff Experiment N3LO NN only 39 40 41 42 47 48 49 50 51 52 53 54 55 56 59 60 61 62 A

Questions? Gustav R. Jansen gustav.jansen@utk.edu This work was partly supported by the Office of Nuclear Physics, U.S. Department of Energy (Oak Ridge National Laboratory), under Contracts No. DE-FG02-96ER40963 (University of Tennessee) and No.DE-SC0008499 (NUCLEI SciDAC-3 Collaboration). An award of 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 in the Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract DE-AC05-00OR22725 and used computational resources of the National Center for Computational Sciences, the National Institute for Computational Sciences, and the Notur project in Norway.

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