Three-Nucleon Forces and Masses of Neutron-Rich Nuclei Jason D. Holt

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1 Three-Nucleon Forces and Masses of Neutron-Rich Nuclei Jason D. Holt

2 Drip Lines and Magic Numbers: The Evolving Nuclear Landscape 3N forces important in light nuclei, nuclear matter What are the limits of nuclear existence? How do magic numbers form and evolve? 82! 126! Undiscovered protons 28! 20! stable nuclei 50! 82! 50! known nuclei 8! 2! 2! 8! 20! 28! neutrons

3 Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei Important in light nuclei, nuclear matter What are the limits of nuclear existence? How do magic numbers form and evolve? Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL (2010) 82! Heaviest oxygen isotope 126! 50! protons 28! 20! 50! 82! 8! 2! 2! 8! 20! 28! neutrons 3N forces essential for medium mass nuclei

protons 28! 20! Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei Important in light nuclei, nuclear matter What are the limits of nuclear existence? How do magic numbers form and evolve?

4 protons 28! 20! Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei Important in light nuclei, nuclear matter What are the limits of nuclear existence? How do magic numbers form and evolve? Holt, Otsuka, Schwek, Suzuki, arxiv: ! 2 + Energy (MeV) ! 82! N=28 magic number in calcium (d) NN + 3N (pfg 9/2 shell) 50! (NN) + 3N(N 2 LO) + 3N(N 2 LO) [MBPT] 126! Mass Number A 8! 2! 2! 8! 20! 28! neutrons 3N forces essential for medium mass nuclei

Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei Important in light nuclei, nuclear matter What are the limits of nuclear existence? How do magic numbers form and evolve?

5 Drip Lines and Magic Numbers: 3N Forces in Medium-Mass Nuclei Important in light nuclei, nuclear matter What are the limits of nuclear existence? How do magic numbers form and evolve? Advances in microscopic theory: Calculate all theoretical inputs consistently with 126! NN+3N 82! protons Can we perform adequate spectroscopic calculations? 50! What are the effects of 3N to higher-order in MBPT? 82! 28! 20! 50! 8! 2! 2! 8! 20! 28! neutrons 3N forces essential for medium mass nuclei

6 Chiral Effective Field Theory: Nuclear Forces LO Nucleons interact via pion exchanges and contact interactions Hierarchy: V NN > V 3N >... NLO Consistent treatment of NN, 3N, electroweak operators N 2 LO N 3 LO Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Meissner,

7 Chiral Effective Field Theory: Nuclear Forces LO Nucleons interact via pion exchanges and contact interactions Hierarchy: V NN > V 3N >... NLO Consistent treatment of NN, 3N, electroweak operators Couplings fit to experiment once N 2 LO N 3 LO Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Meissner,

8 Solving the Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons Interaction and energies of valence space orbitals from original This alone does not reproduce experimental data 0h,1f,2p 0g,1d,2s 0f,1p 0p - sd -valence space Active nucleons occupy valence space 0s Assume filled core

9 Solving the Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons Interaction and energies of valence space orbitals from original This alone does not reproduce experimental data allow explicit breaking of core 0h,1f,2p Hjorth-Jensen, Kuo, Osnes (1995) 0g,1d,2s 0f,1p Strong interactions with core generate effective interaction between valence nucleons 0p - sd -valence space Active nucleons occupy valence space 0s Assume filled core

Solving the Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons Interaction and energies of valence space orbitals from original This alone does

10 Solving the Nuclear Many-Body Problem Nuclei understood as many-body system starting from closed shell, add nucleons Interaction and energies of valence space orbitals from original This alone does not reproduce experimental data allow explicit breaking of core Effective two-body matrix elements Single-particle energies (SPEs) 0h,1f,2p Hjorth-Jensen, Kuo, Osnes (1995) 0g,1d,2s 0f,1p Strong interactions with core generate effective interaction between valence nucleons 0p - sd -valence space Active nucleons occupy valence space 0s Assume filled core

11 Calculation Details Convergence in terms of Harmonic Oscillator basis size NN matrix elements derived from: - Chiral N 3 LO (Machleidt, 500 MeV) using smooth-regulator - 3 rd -order in perturbation theory - 13 major HO shells for intermediate state configurations (converged) Energy (MeV) O (1st) (2nd) (3rd) Major Shells

12 Calculation Details Convergence in terms of Harmonic Oscillator basis size NN matrix elements derived from: - Chiral N 3 LO (Machleidt, 500 MeV) using smooth-regulator - 3 rd -order in perturbation theory - 13 major HO shells for intermediate state configurations (converged) Energy (MeV) nd order 3 rd order G-matrix 18 O Major Shells Major Shells

13 Calculation Details Convergence in terms of Harmonic Oscillator basis size NN matrix elements derived from: - Chiral N 3 LO (Machleidt, 500 MeV) using smooth-regulator - 3 rd -order in perturbation theory - 13 major HO shells for intermediate state configurations (converged) Monopole 5 ms 7 ms 9 ms 11 ms 13 ms p 1/2 p 1/ p 3/2 p 1/ p 3/2 p 3/ f 5/2 p 1/ f 5/2 p 3/ f 5/2 f 5/ f 7/2 p 1/ f 7/2 p 3/ f 7/2 f 5/ f 7/2 f 7/

14 Limits of Nuclear Existence: Oxygen Anomaly Where is the nuclear dripline? Limits defined as last isotope with positive neutron separation energy - Nucleons drip out of nucleus Neutron dripline experimentally established to Z=8 (Oxygen)

15 Limits of Nuclear Existence: Oxygen Anomaly Where is the nuclear dripline? Limits defined as last isotope with positive neutron separation energy - Nucleons drip out of nucleus Neutron dripline experimentally established to Z=8 (Oxygen) Regular dripline trend except oxygen Adding one proton binds 6 additional neutrons

16 Limits of Nuclear Existence: Oxygen Anomaly Where is the nuclear dripline? Limits defined as last isotope with positive neutron separation energy - Nucleons drip out of nucleus Neutron dripline experimentally established to Z=8 (Oxygen) Prediction with NN forces Microscopic picture: NN-forces too attractive Incorrect prediction of dripline

17 Physics in Oxygen Isotopes Calculate evolution of sd-orbital energies from interactions 16 - O O 28 - O - O 16 - O O 28 - O - O - 16 O Fit to experiment Microscopic NN Theories d 3/2 orbit bound to 28 O Phenomenological Models d 3/2 orbit unbound

18 Physics in Oxygen Isotopes Calculate evolution of sd-orbital energies from interactions 16 - O O 28 - O - O 16 - O O 28 - O - O - 16 O Fit to experiment 0-10 Energy (MeV) sd-shell sdf 7/2 p 3/2 shell USDb Mass Number A Microscopic NN Theories d 3/2 orbit bound to 28 O Dripline at 28 O Phenomenological Models d 3/2 orbit unbound Dripline at 24 O Oxygen anomaly unexplained with NN forces

19 Comparison to Coupled Cluster Many-body method insufficient? Benchmark against ab-initio Coupled Cluster at NN-only level Energy [MeV] Oxygen CC MBPT Experiment (a) CC/MBPT Mass Number A Energy [MeV] Calcium CC MBPT CC/MBPT Mass Number A SPEs: one-particle attached CC energies in 17 O and 41 Ca Small difference in many-body methods Include 3N forces

20 Chiral Effective Field Theory: 3N Forces Only two new couplings at N 2 LO LO NLO N 2 LO c terms given from NN fits: constrained by NN, πn data c D c E fit to properties of light nuclei: Triton binding energy, 4 He radius No new 3N, 4N couplings at N 3 LO N 3 LO

21 3N Forces for Valence-Shell Theories Normal-ordered 3N: contribution to valence neutron interactions Effective two-body Effective one-body 16 O core 16 O core ab V 3N,eff a'b' = αab V 3N αa'b' a V 3N,eff a' = 1 2 α =core Combine with microscopic NN: eliminate empirical adjustments αβa V 3N αβa' αβ =core

22 Calculation Details Convergence in terms of Harmonic Oscillator basis size NN matrix elements derived from: - Chiral N 3 LO (Machleidt, 500 MeV) using smooth-regulator - 3 rd -order in perturbation theory - 13 major HO shells for intermediate state configurations (converged) 3N force contribution to TBMEs and SPEs - Chiral N 2 LO: c D, c E fit to properties of light nuclei with above NN (Λ = 2.0 fm -1 ) - Normal-ordered two-body part included to Third Order in MBPT - Calculated within 5 major HO shells Are 3N forces fit in light nuclei appropriate for medium-mass systems?

23 Oxygen Anomaly First calculations using NN+3N Probe limits of nuclear existence with 3N forces F dripline 3N repulsion amplified with N: vital for neutron-rich nuclei? d 3/2 unbound at 24 O with 3N forces Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL (2010)

24 Oxygen Anomaly First calculations using NN+3N Probe limits of nuclear existence with 3N forces F dripline 3N repulsion amplified with N: vital for neutron-rich nuclei? d 3/2 unbound at 24 O with 3N forces Isotopes unbound beyond 24 O First microscopic explanation of oxygen anomaly Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL (2010)

25 One-Body 3N: Single Particle Energies NN-only microscopic SPEs yield poor results rely on empirical adjustments sd-shell: SPEs much too bound, unreasonable splitting Orbit Exp USDb T+V NN d 5/ s 1/ d 3/

26 One-Body 3N: Single Particle Energies NN-only microscopic SPEs yield poor results rely on empirical adjustments sd-shell: SPEs much too bound, unreasonable splitting 3N forces: additional repulsion reasonable values! Orbit USDb T+V NN +V 3N d 5/ s 1/ d 3/

5/2-3.93-3.78-3.95-3.46 s 1/2-3.21-2.42-3.16-2.20 d 3/2 2.11 1.45 1.65 1.

27 One-Body 3N: Single Particle Energies Effects of correlations beyond one major oscillator shell: 16 O 16 O Orbit USDb T+V NN +V 3N SDPF-M T+V NN +V 3N d 5/ s 1/ d 3/ f 7/ p 3/ Fully microscopic framework and extended valence space

28 Energy (MeV) Binding Energies of Oxygen Isotopes Interaction and self-consistent SPEs from NN+3N (3 rd -order) Empirical SPEs for NN-only sd-shell (2nd) sd-shell (3rd) sdf 7/2 p 3/2 -shell (b) NN Mass Number A NN+3N: more modest trend past 24 O USDb sd-shell sdf 7/2 p 3/2 -shell (c) NN + 3N Mass Number A Holt, Menendez, Schwenk, arxiv: sd-shell results underbound; improved in sdf 7/2 p 3/2 Continuum: ~300keV more binding beyond 24 O (from CC)

29 Impact on Spectra: 23 O Neutron-rich oxygen spectra with NN+3N (3 rd order) 5/2 +, 3/2 + indicate position of d 5/2 and d 3/2 orbits 5 23 O 4 3/2 + (3/2 + ) Energy (MeV) S 3/2 + n 5/2 + (5/2 + ) 3/2 + 3/2 + 5/2 + 5/ /2 + 1/2 + 1/2 + 1/2 + 1/ NN sd 5/2 + NN sdf 7/2 p 3/2 NN+3N sd NN+3N sdf 7/2 p 3/2 Expt. Holt, Menendez, Schwenk, arxiv:

30 Impact on Spectra: 23 O Neutron-rich oxygen spectra with NN+3N (3 rd order) 5/2 +, 3/2 + indicate position of d 5/2 and d 3/2 orbits Energy (MeV) /2 + 3/2 + 5/ O 3/2 + 5/2 + (3/2 + ) S 3/2 + n 5/2 + (5/2 + ) sd-shell NN-only Wrong ground state! 5/2 + too low 3/2 + bound Microscopic NN+3N Improvements in extended valence space 0 1/2 + 1/2 + 1/2 + 1/2 + 1/ NN sd 5/2 + NN sdf 7/2 p 3/2 NN+3N sd NN+3N sdf 7/2 p 3/2 Expt. Holt, Menendez, Schwenk, arxiv:

31 Impact on Spectra: 22 O Neutron-rich oxygen spectra with NN+3N 22 O: N=14 new magic number not reproduced with NN Energy (MeV) O S n (4 + ) (2 + ) (0 + ) NN-only 2 + too low Spectrum too compressed Microscopic NN+3N Extended space essential Reproduces N=14 magic number 1 0 NN sd 0 + NN sdf 7/2 p 3/ NN+3N sd NN+3N sdf 7/2 p 3/2 Expt. Holt, Menendez, Schwenk, arxiv: Contributions from 3N and extended valence orbitals important

32 Shell Formation/Evolution in Calcium Isotopes Important in light nuclei, nuclear matter What are the limits of nuclear existence? How do magic numbers form and evolve? 82! 126! protons 8! 2! 2! 8! 28! 20! 20! 28! neutrons 50! 82! 50! N=28 magic number in calcium not predicted with NN-only theories Zuker, Poves

Calcium Isotope Physics: Magic Numbers 40 Ca Single-Particle Energy (MeV) 0-5 -10-15 (a) Phenomenological Forces (b) NN-only Theory f 5/2 f 5/2 p 1/2 p 3/2 f 7/2 KB3G GXPF1 40 44 48 52 56 60 Mass

33 Calcium Isotope Physics: Magic Numbers 40 Ca Single-Particle Energy (MeV) (a) Phenomenological Forces (b) NN-only Theory f 5/2 f 5/2 p 1/2 p 3/2 f 7/2 KB3G GXPF Mass Number A p 1/2 p 3/2 f 7/2 G Mass Number A GXPF1: Honma, Otsuka, Brown, Mizusaki (2004) KB3G: Poves, Sanchez-Solano, Caurier, Nowacki (2001) 2 + Energy (MeV) (a) Phenomenological Forces Experiment GXPF1 KB3G (b) NN-only Theory G G [SPE_KB3G] [SPE_KB3G] [fp+g 9/2 ] Phenomenological Forces Large gap at 48 Ca Discrepancy at N=34 Microscopic NN Theory Small gap at 48 Ca N=28: first standard magic number not reproduced in microscopic NN theories

34 Single-Particle Energy (MeV) Evolution of Shell Structure SPE evolution with 3N forces in pf and pfg 9/2 spaces: (a) Phenomenological Forces (b) NN-only Theory f 5/2 f 5/2 p 1/2 p 3/2 f 7/2 KB3G GXPF Mass Number A p 1/2 p 3/2 f 7/2 f 7/2 G Mass Number A (c) NN + 3N G + 3N( ) + 3N(N 2 LO) Mass Number A NN+3N pf-shell: Trend across: improved binding energies Increased gap at 48 Ca: enhanced closed-shell features Holt, Otsuka, Schwenk, Suzuki arxiv: Include g 9/2 orbit, calculated SPEs Different behavior of ESPEs (not observable, model dependent) + 3N(N 2 LO) [MBPT] Mass Number A Small gap can give large 2 + energy: due to many-body correlations (physics beyond monopole part of Hamiltonian) f 5/2 p 1/2 p 3/2 (d) NN + 3N (pfg 9/2 shell) p 1/2 f 5/2 p 3/2 f 7/2 g 9/2 Duguet, Hagen, arxiv:

35 N=28 Magic Number in Calcium First excited 2 + energies in calcium isotopes with NN+3N 2 + Energy (MeV) (c) NN + 3N G + 3N( ) + 3N( ) + 3N(N 2 LO) + 3N(N 2 LO) [SPE_KB3G] (d) NN + 3N (pfg 9/2 shell) (NN) + 3N(N 2 LO) + 3N(N 2 LO) [MBPT] Mass Number A Mass Number A Holt, Otsuka, Schwenk, Suzuki arxiv: pf-shell: robust but modest improvement in 2 + energies, below experiment pfg 9/2 -shell: reproduce experimental 2 + in 48 Ca Both 3N and extended space essential

36 2 + Energy (MeV) Evolution of Magic Numbers: N=34 First excited 2 + energies in calcium isotopes with NN+3N (c) NN + 3N G + 3N( ) + 3N( ) + 3N(N 2 LO) + 3N(N 2 LO) [SPE_KB3G] (d) NN + 3N (pfg 9/2 shell) (NN) + 3N(N 2 LO) + 3N(N 2 LO) [MBPT] Mass Number A Mass Number A pf-shell: Very pronounced closed-shell properties pfg 9/2 -shell: More modest, similar to 52 Ca Holt, Otsuka, Schwenk, Suzuki arxiv:

37 Energy (MeV) Calcium Ground State Energies and Dripline Ground state energies using NN+3N NN-only: overbinds beyond ~ 46 Ca (c) NN + 3N (d) NN + 3N (pfg 9/2 shell) (NN) G + 3N( ) + 3N( ) + 3N(N 2 LO) Mass Number A (NN) + 3N(N 2 LO) + 3N(N 2 LO) [MBPT] Mass Number A Holt, Otsuka, Schwenk, Suzuki arxiv: pf-shell: 3N forces correct binding energies; good experimental agreement pfg 9/2 -shell: calculate to 70 Ca; modest overbinding beyond 52 Ca Predict heaviest calcium isotope ~ Ca; 70 Ca not unreasonable

38 Experimental Connection: Mass of 52 Ca S 2n energies for exotic calcium isotopes: S 2n (MeV) AME2003 NN-only NN+3N NN-only poor experimental agreement NN+3N Improvement for lighter calcium, wrong behavior past 50 Ca

39 Experimental Connection: Mass of 52 Ca New mass measurements of 51,52Ca at TITAN: Penning trap experiment S 2n (MeV) A. Gallant et al., arxiv: AME2003 TITAN NN-only NN+3N NN-only poor experimental agreement NN+3N Improved agreement with new experimental trend TITAN Measurement 52 Ca mass 1.75MeV more bound than AME2003 value!

40 Experimental Connection: Mass of 52 Ca New mass measurements of 51,52Ca at TITAN: Penning trap experiment S 2n (MeV) (MeV) n (3) A. Gallant et al., arxiv: AME2003 TITAN NN+3N (MBPT) NN+3N (emp) TITAN+ AME2003 NN-only poor experimental agreement NN+3N Improved agreement with new experimental trend TITAN Measurement 52 Ca mass 1.75MeV more bound than AME2003 value! NN+3N(3 rd order) Remarkable experimental agreement Reduced uncertainty from SPEs Neutron Number N Good reproduction of pairing gaps Cont. in talk of J. Menendez!

41 Conclusion Exciting era for nuclear structure experimentally and theoretically Exploring frontiers of nuclear structure theory of medium-mass nuclei with 3N forces Discovered robust and general repulsive 3N mechanism for T=1 neutron-rich nuclei 3N forces now to third order in MBPT

42 Exciting era for nuclear structure experimentally and theoretically Exploring frontiers of nuclear structure theory of medium-mass nuclei with 3N forces Discovered robust and general repulsive 3N mechanism for T=1 neutron-rich nuclei 3N forces now to third order in MBPT Oxygen isotopes: first fully-microscopic results with NN+3N, extended spaces Cures NN-only failings: dripline, shell evolution, spectra in oxygen isotopes Calcium isotopes in pf- and pfg 9/2 -shells: Conclusion First microscopic prediction of N=28 magic number in 48 Ca Shell evolution towards the dripline: modest N=34 closure, quenching of N=40 Dripline near 60 Ca

43 Exciting era for nuclear structure experimentally and theoretically Exploring frontiers of nuclear structure theory of medium-mass nuclei with 3N forces Discovered robust and general repulsive 3N mechanism for T=1 neutron-rich nuclei 3N forces now to third order in MBPT Oxygen isotopes: first fully-microscopic results with NN+3N, extended spaces Cures NN-only failings: dripline, shell evolution, spectra in oxygen isotopes Calcium isotopes in pf- and pfg 9/2 -shells: Conclusion First microscopic prediction of N=28 magic number in 48 Ca Shell evolution towards the dripline: modest N=34 closure, quenching of N=40 Dripline near 60 Ca Clearly improvable upgrade path Much more in next talk!

44 Acknowledgments Collaborators J. Menendez, A. Schwenk T. Otsuka T. Suzuki (Nihon U.) G. Hagen, T. Papenbrock Computing support Travel support

Three-Nucleon Forces and the Structure of Exotic Nuclei Jason D. Holt

Three-Nucleon Forces and the Structure of Exotic Nuclei Jason D. Holt Based on T. Otsuka, T. Suzuki, JDH, A. Schwenk, Y. Akaishi, PRL (11) JDH, J. Menendez, A. Schwenk, arxiv:118.268 JDH, T. Otsuka, A.