Shell evolution in neutron rich nuclei
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1 Shell evolution in neutron rich nuclei Gustav R. Jansen 1,2 1 University of Tennessee, Knoxville 2 Oak Ridge National Laboratory March
2 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)
3 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
4 Understanding matter
5 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.
6 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.
7 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
8 Chiral effective field theory D. R. Entem and R. Machleidt, Phys. Rev. C 68, (2003) Direct link to QCD. Perturbative expansion in momentum. Chiral symmetry is spontaneously and explicitly broken. The hierarchy of nuclear forces unfolds automatically.
9 Density dependent chiral threebody force
10 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.
11 Curse of dimensionality He 10 Be 16 O 40 Ca Slater determinant basis Number of single particle states
12 The coupled-cluster method
13 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.
14 Open Quantum Systems
15 The oxygen dripline
16 Oxygen isotopes from chiral interaction
17 Evolution of single particle energies S. M. Lenzi, F. Nowacki, A. Poves and K. Sieja, PRC (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.
18 Negative parity states in O25
19 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.
20 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 = 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.
21 Threebody forces in 26 F A. Lepailleur et al., Phys. Rev. Lett. 110, (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.
22 Resonances in neutron-rich 24 O
23 Oxygen isotopes from chiral interaction
24 Evolution of single particle energies Technical details J. Meng, H. Toki, J. Y. Zeng, S. Q. Zhang and S. -G. Zhou, PRC (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.
25 Evolution of single particle energies S. M. Lenzi, F. Nowacki, A. Poves and K. Sieja, PRC (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.
26 Binding energies in calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL (2012) E (MeV) NN + 3NF eff Experiment 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 = N max = 18 and ω = 26 MeV. Mass of 51 Ca and 52 Ca from A. T. Gallant et al., PRL 109, (2012) Main features Total binding energies agree well with experimental masses. 60 Ca is not magic. Three nucleon force is repulsive.
27 Binding energies in calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL (2012) E (MeV) NN only NN + 3NF eff Experiment NN only 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 = N max = 18 and ω = 26 MeV. Mass of 51 Ca and 52 Ca from A. T. Gallant et al., PRL 109, (2012) Main features Total binding energies agree well with experimental masses. 60 Ca is not magic. Three nucleon force is repulsive.
28 Shell evolution in neutron rich calcium isotopes. Details J. D. Holt, T. Otsuka, A. Schwenk and T. Suzuki, J Phys G (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.
29 J π = 2 + systematics in even calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL (2012) 5 E 2 + (MeV) NN+3NF eff Exp 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 = 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.
30 J π = 2 + systematics in even calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL (2012) 5 E 2 + (MeV) NN only NN+3NF eff Exp NN only 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 = 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.
31 Spectra in calcium isotopes G. Hagen, M. Hjorth-Jensen, GRJ, R. Machleidt, and T. Papenbrock, PRL (2012) Energy (MeV) Ca + 54 Ca 55 Ca /2 + 7/2-53 Ca 3/2-5/2-4 + Technical /2 + 5/ Ca 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 = 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 Exp NN+3NF eff 5/2 - Exp 5/ 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.
32 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.
33 NNLO (POUNDerS) POUNDerS Practical Optimization Using No Derivatives for sums of Squares Part of 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.
34 Quality of the fit Preliminary T lab (MeV) pp χ /datum a a np χ 2 /datum N 3 LO NNLO-POUNDerS Empirical a C pp (26) (29) rpp C (14) 2.769(14) a N pp r N pp a N nn (40) rnn N (11) a np (20) r np (5) B D (MeV) (9) r D (fm) (85) Q D (fm 2 ) (3) P D
35 Triton binding energy Preliminary E NCSM (MeV) POUNDerS-NNLO (NN+NNN) POUNDerS-NNLO (NN) Experiment N3LO (NN) Nmax
36 4 He binding energy Preliminary E NCSM (MeV) POUNDerS-NNLO(NN) POUNDerS-NNLO(NN+NNN) N3LO(NN) Experiment Nmax
37 Oxygen isotopes with NNLO (POUNDerS) Preliminary E (MeV) E (MeV) A Experiment NNLO (POUNDerS) N 3 LO A
38 Oxygen spectra with NNLO (POUNDerS) Preliminary Energy (MeV) N 3 LO O O NNLO Exp N 3 LO O NNLO Exp 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 O NNLO ? Exp
39 J π = 2 + systematics in oxygen isotopes with NNLO (POUNDerS) Preliminary E 2 + (MeV) N 3 LO 3nfeff Exp N 3 LO NNLO (POUNDerS) A O
40 J π = 2 + systematics with NNLO (POUNDerS) Preliminary 5 E 2 + (MeV) NN+3NF eff Exp NN only NNLO (POUNDerS) 1 0 N2LO (POUNDerS) A Ca
41 Calcium isotopes with NNLO (POUNDerS) Preliminary E (MeV) NN only NN + 3NF eff Experiment NN only A
42 Calcium isotopes with NNLO (POUNDerS) Preliminary E (MeV) NNLO (POUNDerS) NNLO (POUNDerS) NN + 3NF eff Experiment N3LO NN only A
43 Calcium isotopes with NNLO (POUNDerS) Preliminary E (MeV) NNLO (Shifted) NN + 3NF eff Experiment N3LO NN only A
44 Calcium isotopes with NNLO (POUNDerS) Preliminary E (MeV) NNLO (Shifted) NN + 3NF eff Experiment N3LO NN only A
45 Questions? Gustav R. Jansen 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-SC (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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