Three-nucleon forces and shell structure of neutron-rich Ca isotopes
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1 Three-nucleon forces and shell structure of neutron-rich Ca isotopes Javier Menéndez Institut für Kernphysik (TU Darmstadt) and ExtreMe Matter Institute (EMMI) NUSTAR Week 3, Helsinki, 9 October 13
2 Outline Theoretical Approach: NN+3N forces in Shell Model Neutron rich Ca isotopes: Shell evolution N = 3 shell closure Proton rich N=8 and N= isotopes / 1
3 Outline Theoretical Approach: NN+3N forces in Shell Model Neutron rich Ca isotopes: Shell evolution N = 3 shell closure Proton rich N=8 and N= isotopes
4 Nuclear Structure approach Big variety of nuclei in the nuclear chart, A...3 Systematic ab initio calculations only possible in the lightest nuclei Poses a hard many-body problem: design approximate methods suited for different regions Interacting Shell Model: Solve the problem choosing the (more) relevant degrees of freedom Use realistic nucleon-nucleon (NN) and three-nucleon (3N) interactions 3 / 1
5 The Interacting Shell Model Chose as basis states that of the 3D Harmonic Oscillator To keep the problem feasible, the configuration space is separated into Outer orbits: orbits that are always empty Valence space: the space in which we explicitly solve the problem Inner core: orbits that are always filled ( (p + 1)(p + )ν Dim N ) ( (p + 1)(p + )π Z ) 4 / 1
6 Many Body Perturbation Theory Better convergence through V lowk transformation Many-body Perturbation Theory up to third order to build an effective Shell Model interaction in a valence space H Ψ = E Ψ H eff Ψ eff = E Ψ eff Single Particle Energies (SPEs) Two-Body Matrix Elements (TBMEs) Full diagonalizations using codes ANTOINE and NATHAN Caurier et al. RMP77 47(5) and compare to experiment 5 / 1
7 3N forces: Oxygen dripline Z 8 H 1934 He 1961 Li 1966 Be 1973 stability line B 1984 O 197 N 1985 C 1986 F 1999 Ne Na 8 8 3N forces provide repulsion missing in NN-only forces 3N forces crucial also for reliable description of spectra 6 / 1 Energy (MeV) stable isotopes unstable isotopes Mg 7 Si 7 Al 7 unstable fluorine isotopes unstable oxygen isotopes neutron halo nuclei N sd-shell (nd) sd-shell (3rd) sdf 7/ p 3/ -shell Mass Number A Chiral NN+3N forces to describe O dripline at 4 O Otsuka et al. PRL (1) (b) NN USDb sd-shell sdf 7/ p 3/ -shell (c) NN + 3N Mass Number A
8 Forces and Currents in Chiral EFT Chiral EFT: low energy approach to QCD for nuclear structure energies Approximate chiral symmetry of QCD: pions pseudo-goldstone bosons Short-range couplings are fitted to experiment once Systematic expansion: nuclear forces and electroweak currents N force 3N force 4N force LO NN forces up to N 3 LO 3N forces up to N LO NLO N LO 3 N LO Weinberg, van Kolck, Savage, Epelbaum, Kaiser, Meißner... 7 / 1 NN fitted to: NN scattering data 3N fitted to: 3 H Binding Energy 4 He radius
9 3N Forces Treatment of 3N forces: normal-ordered B: valence, 1 core particle (effective) Two-body Matrix Elements (TBME) normal-ordered 1B: 1 valence, core particles (effective) Single particle energies (SPE) residual 3B: Estimated to be suppressed by N valence /N core 8 / 1
10 Residual 3N Forces Neutron Number (N) (c) 3-body interaction 8 1 Neutron Nu (d) 3-body intera more neutro In the most neutron-rich oxygen isotopes, 3N forces between 3 valence neutrons can give a relevant contribution 16 O core Energy (MeV) R3B-LAND (this work) MoNA/NSCL (8, 1) NN+3N + residual 3N NN+3N 3/ + residual 3N Residual 3N contributions are repulsive They are small compared to normal-ordered 3N force, but increase with N Very good agreement with resonances in 5 O and 6 O 4 O 5 O 6 O Caesar, Simonis et al PRC (13) 9 / 1
11 Outline Theoretical Approach: NN+3N forces in Shell Model Neutron rich Ca isotopes: Shell evolution N = 3 shell closure Proton rich N=8 and N= isotopes
12 Ca isotopes: Masses Ca isotopes: explore nuclear shell evolution N =, 8, 3?, 34? Energy (MeV) NN NN+3N (emp) NN+3N (MBPT) Mass Number A Ca with respect to 4 Ca core 3N forces repulsive contribution, chiral NN-only forces too attractive Flat behaviour towards 6 Ca does not allow clear prediction of the dripline Results sensitive to SPEs, especially more neutron-rich systems, MBPT (calculated from NN+3N forces) Empirical (from GXPF1 interaction) Estimate of the uncertainty 1 / 1
13 Two-Neutron separation energies Ca isotopes (on top of 4 Ca core) Compare S n = [B(N, Z ) B(N, Z )] with experiment S n (MeV) AME3 TITAN NN+3N (MBPT) Neutron Number N 11 / 1 Precision measurements with TITAN changed AME MeV in 5 Ca More flat behaviour in 5 Ca 5 Ca 3N forces needed in theoretical calculation pfg 9/ valence space Gallant et al. PRL (1)
14 Two-Neutron separation energies Ca isotopes (on top of 4 Ca core) Compare S n = [B(N, Z ) B(N, Z )] with experiment S n (MeV) S n (theo) S n (exp) (MeV) AME3 TITAN NN+3N (MBPT) NN Neutron Number N 11 / 1 Precision measurements with TITAN changed AME MeV in 5 Ca More flat behaviour in 5 Ca 5 Ca 3N forces needed in theoretical calculation pfg 9/ valence space Gallant et al. PRL (1)
15 54 Ca and N = 3 shell closure Ca isotopes (on top of 4 Ca core) Compare S n = [B(N, Z ) B(N, Z )] with experiment S n (MeV) S n (theo) S n (exp) (MeV) ISOLDE NN+3N (MBPT) NN Neutron Number N 1 / 1 Very recently 53,54 Ca measured at ISOLDE Excellent agreement between calculation and experiment S n evolution: 5 Ca 54 Ca as 48 Ca 5 Ca: N = 3 shell closure Wienholtz et al. Nature (13)
16 Two-neutron separation energies Compare S n = [B(N, Z ) B(N, Z )] with experiment S n also calculated by other approaches: S n (MeV) S n (theo) S n (exp) (MeV) ISOLDE NN+3N (MBPT) CC (Hagen et al.) KB3G GXPF1A Neutron Number N 13 / 1 Phenomenological interactions also reproduce quite well experiment (input about masses/gaps into the interactions) Differ markedly beyond 54 Ca Coupled-Cluster calculations Hagen et al. PRL19 35 (1) very good agreement for even isotopes Wienholtz et al. Nature (13)
17 Two-neutron separation energies Compare S n = [B(N, Z ) B(N, Z )] with experiment S n also calculated by other approaches: S n (MeV) S n (theo) S n (exp) (MeV) ISOLDE UNEDF1 UNEDF SV-min SLY4 HFB Neutron Number N 13 / 1 Modern Energy Density Functionals reproduce correctly overall trends and neutron/proton driplines have more dificulties in describing shell closures Erler et al. Nature486 59(1) Wienholtz et al. Nature (13)
18 Shell closures and 1 energies 1 energies characterise shell closures of the neutron rich calcium isotopes Holt, JM, Schwenk, JPG (13) Energy (MeV) NN NN+3N [emp] NN+3N [MBPT] Mass Number A Correct closure at N = 8 when 3N forces are included Holt et al. JPG (1) 3N forces enhance closure at N = 3 3N forces reduce strong closure at N = 34 (1.7-. MeV) Measured at.1 MeV Steppenbeck et al. Nature, in press 14 / 1
19 48 Ca spectrum Energy (MeV) Challenge: Doubly-closed nucleus 48 Ca NN pf NN pfg 9/ 48 Ca NN+3N NN+3N pf pfg 9/ (+) Expt. GXPF1 KB3G Spectra too compressed with NN forces only or pf space 1 state only appropriate energy in pfg 9/ NN+3N calculation 1 state too low (1st excited state) especially compared to phenomenological interactions Importance of 3N forces Importance of including g 9/ orbit 15 / 1
20 B(M1) Transition in 48 Ca 5 (a) pf-shell B(M1) [µ N ] Experiment GXPF1 G G + 3N( ) NN NN + 3N B(M1) strength in 48 Ca too fragmented in pf space Phenomenological calculations reproduce experimental concentration B(M1) [µ N ] (b) pfg 9/ -shell NN NN + 3N (emp) NN + 3N (MBPT) In the extended pfg 9/ space NN forces also fragmented strength NN+3N calculation in pfg 9/ very good agreement with experiment Excitation Energy (MeV) 16 / 1
21 B(E) Transition Strengths Isotope Transition KB3G GXPF1A MBPT EXP. 46 Ca ± ±.6 46 Ca ±.9 47 Ca 3/ 7/ ±. 48 Ca ± Ca 7/ 3/ ±.1 5 Ca ±. B(E)s in reasonable agreement with experiment (order of magnitude) Similar quality as phenomenological interactions (very close to KB3G) 46 Ca: sd degrees of freedom? 17 / 1
22 Outline Theoretical Approach: NN+3N forces in Shell Model Neutron rich Ca isotopes: Shell evolution N = 3 shell closure Proton rich N=8 and N= isotopes
23 Proton dripline at N = 8 Ground-State Energy (MeV) NN NN+3N NN+3N (sdf 7/ p 3/ ) AME11 IMME N= Mass Number A Holt, JM, Schwenk PRL11 5 (13) Theory complements/improves mass extrapolations and isomeric mass-multiplet formula (IMME) E(A, T, T z ) = E(A, T, T z ) + b(a, T )T z NN forces oberbind 3N forces essential to describe masses and the predict the proton dripline Proton dripline not certain predicted either in Mg or Si: S p = -.1 (Theory) / +.1 (IMME) Measurement needed! Calculations in standard and extended spaces 18 / 1
24 Spectra of N = 8 isotones Excitation Energy (MeV) Ne Na Mg 1/ (1/ ) 4 + (4 + ) 3/ 5/+ + (3/ (5/ ) ) Exp Exp Exp 1 Al Si 1/ 7/ 3/ 5/ 3/ 1/ 5/ Including NN+3N forces good agreement with known spectra Prediction of,4 + doublet close to previously unpublished 4 + state in Mg (I. Mukha) Prediction of 1 Al and Si spectra Holt, JM, Schwenk PRL11 5 (13) In Si calculations point to a sub-shell closure (analogous to O) More experimental information greatly appreciated! 19 / 1
25 Masses and spectra of N = isotones Ground-State Energy (MeV) N= NN NN+3N NN+3N (pfg 9/ ) Exp and AME11 extrapolation IMME Mass Number A Excitation Energy (MeV) Ti 6 + Exp ( ) 43 V 44 Cr 45 Mn 46 Fe 47 Co 48 Ni 9/ 11/ 3/ 3/ 5/ 7/ / 3/+ + 1/ 3/ / 7/ 5/ 7/ 3/ 1/ 7/ 3/ Holt, JM, Schwenk PRL11 5 (13) 4 + Dripline robustly predicted at 46 Fe Good description of 48 Ni: S p = -1. (Th) vs -1.8(6) (Exp) Pomorski (1) / 1
26 Summary and Outlook Shell Model calculation based on chiral EFT (NN+3N forces) and MBPT gives good agreement with experimental masses, two-neutron separation energies, pairing gaps and excitation spectra for oxygen, calcium isotopes and proton-rich N=8, isotones: Oxygen dripline, unbound 5,6 O reproduced with residual 3N forces Predicted neutron rich Ca S n s with NN+3N forces agree with recent measurements of 51,5 Ca (TRIUMF) and 53,54 Ca (ISOLTRAP) Shell structure: prominent closure at N = 3 Ca spectroscopy: spectra, electromagnetic strengths Dripline and spectra of proton-rich N = 8, isotones predicted Outlook: Heavier isotope and isotone chains: include T= (pn) TBME Explore uncertainties in the theoretical calculation 1 / 1
27 Collaborators J. D. Holt, A. Schwenk, J. Simonis R 3 B Collaboration (C. Caesar, T. Aumann...) TITAN Collaboration (A. Gallant, J. Dilling...) ISOLTRAP Collaboration (F. Wienholtz, K. Blaum...) 1 / 1
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