Shell evolution and pairing in calcium isotopes with two- and three-body forces

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1 Shell evolution and pairing in calcium isotopes with two- and three-body forces Javier Menéndez Institut für Kernphysik, TU Darmstadt ExtreMe Matter Institute (EMMI) with Jason D. Holt, Achim Schwenk and Johannes Simonis Computational and Theoretical Advances for Exotic Isotopes in the Medium Mass Region INT Seattle, 8 April

2 Outline Ca isotopes: ground states Ca ground states: pairing, shell evolution Ca excited states: shell evolution, spectra, transitions : oxygen and calcium isotopes

3 The nuclear many-body approach Big variety of nuclei in the nuclear chart, A... Exact (ab initio) calculations only possible in the lightest nuclei Poses a hard many-body problem: design approximate methods valid in different regions In the rest of the nuclear chart, pick only the (more) relevant degrees of freedom for observable to study: idea of the Interacting Shell Model

4 Medium-mass nuclei: standard view Standard studies of medium-mass nuclei (A 8) are performed with theoretical approaches based on phenomenology: Shell Model calculations use fitted interactions or modified G-matrices Energy Density Functional interactions use Skyrme, Gogny or Relativistic fitted interactions Interactions are made to reproduce experiment for stable nuclei When extrapolated to exotic (neutron rich) regions results differ Need to guide (ideally avoid) fits! Why microscopic interactions have to be modified? Need to include N forces explicitly!

5 Towards microscopic medium-mass nuclei calculation Microscopic calculation of medium-mass nuclei including N forces Use Chiral Effective Field Theory (chiral EFT) interactions, includes naturally and N forces. Perform a renormalization group evolution to V lowk interaction to enhance convergence of the MBPT calculation Apply Many-Body Perturbation Theory (MBPT) to rd order to obtain interactions to be used in Shell Model (SM) calculations Full diagonalizations using codes ANTOINE and NATHAN Caurier et al. RMP77 47(5) N forces are naturally included Shown necessary to reproduce light nuclei spectra All the parameters that appear in the SM hamiltonian calculated from the input of the microscopic interaction (no fits!)

6 Chiral EFT +N Forces Systematic expansion: state-of-the-art chiral EFT forces forces included up to N LO N forces included up to N LO fitted to: scattering data π-n scattering data N fitted to: H Binding Energy 4 He radius Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Meißner...

7 Normal-ordered N Forces Approximate treatment of N forces: normal-ordered B: valence, core particle (effective) Two-body Matrix Elements (TBME) normal-ordered B: valence, core particles (effective) Single particle energies (SPE) residual B: Estimated to be suppressed by N valence /N core

8 Ca isotopes: Ground-state energies Energy (MeV) N (emp) +N (MBPT) Ca calculations with respect to 4 Ca core ω =.48 MeV, appropriate for Ca radii Normal-ordered N force contributions to TBMEs and SPEs N forces provide crucial repulsion (similar to O case) +N calculations nice agreement with experimental data

9 Ca isotopes: Ground-state energies Energy (MeV) N (emp) +N (MBPT) Results particularly sensitive to SPEs, especially more neutron-rich systems, consider two sets: MBPT (calculated from +N forces) Empirical (from GXPF interaction) to have an estimate of this uncertainty Calculation performed in pf g 9/ valence space Flat behaviour towards 6 Ca does not allow clear prediction of the dripline: enlarge valence space include continuum degrees of freedom

10 Nuclear Pairing Gaps Pairing correlations are important in nuclei Odd-even mass staggering Moments of Inertia... Theoretical pairing gaps compared to experiment via the three point mass formula: N = ( )N BE (N ) + BE (N + ) BE (N ) Holt, JM, Schwenk, arxiv:4.44

11 MBPT convergence (MeV) n () (a) st order (b) + nd-order ladders (c) + rd-order ladders N AME TITAN AME+ Succesive orders pp, hh ladder diagrams build up pairing gaps At third order pp, hh ladders results seem to be converged Third order ladders results still away from experiment Incorrect even-odd staggering (too attractive mean-field)

12 Pairing, N forces and EDF (MeV) n () (a) rd-order ladders (b) EDF Results At pp-hh level N forces reduce the pairing gaps in -5 kev first observed in EDF: Lesinski et al. JPhysG9 58 () EDF N slightly closer to experiment, and more staggering No indication of shell closures

13 Nuclear Pairing Gaps: vs +N (MeV) n () (a) rd-order ladders (pf) (b) Full rd-order (pf) (c) Full rd (pfg 9/ ) +N (emp) +N (MBPT) Full third order MBPT improves agreement with experiment Core-polarization effects significantly enhance pairing gaps Good agreement with experimental trends

14 Pairing gaps and seniority (MeV) n () (a) rd-order ladders (pf) (b) rd-order ladders (pf) (c) Full rd order (pfg 9/ ) s= s=s max (d) Full rd order (pfg 9/ ) Perform seniority truncation (limited number broken J = pairs) Lowest order (no broken pairs) already good approximation apart from mid-shell nuclei broken pairs needed in mid-shell, especially when including g 9/ orbital

15 () n Ca isotopes: ground states and Shell Evolution () n can also tell us about shell evolution The experimental trend is very well reproduced by +N forces (MeV) n () TITAN+ AME Theoretical results systematically.5 MeV higher than experiment Shell closure at N = 8, sub-shell closure N = (moderate) and no apparent N = 4 closure Neutron Number N A. T. Gallant et al. PRL9 56 () forces fail to reproduce N = 8 or N = predict closed N = 4 Phenomenological interactions also describe data very well

16 () n Ca isotopes: ground states and Shell Evolution () n can also tell us about shell evolution The experimental trend is very well reproduced by +N forces (MeV) n () TITAN+ AME Theoretical results systematically.5 MeV higher than experiment Shell closure at N = 8, sub-shell closure N = (moderate) and no apparent N = 4 closure Neutron Number N A. T. Gallant et al. PRL9 56 () forces fail to reproduce N = 8 or N = predict closed N = 4 Phenomenological interactions also describe data very well

17 Two-neutron separation energies Compare S n theoretical calculations with experimental results S n = [B(N, Z ) B(N, Z )] S n (MeV) TITAN +N (emp) +N (MBPT) Neutron Number N A. T. Gallant et al. PRL9 56 () S n too flat: No N = 8 or N = closures Points to N = 4 closure When +N forces are included Very good agreement with experiment +N point to N = 8 or N = closures and no N = 4 5,54 Ca masses very recently measured at ISOLDE Very good agreement with +N Nature, in print ()

18 Shell closures and + energies + energies characterise shell closures of the neutron rich calcium isotopes + Energy (MeV) N [emp] +N [MBPT] Correct closure at N = 8 when N forces are included N forces enhance closure at N = N forces reduce strong closure at N = 4 (.7-. MeV)

19 Shell closures and + energies Similar results from Coupled-Cluster E + (MeV) 5 4 +NF eff Exp + Energy (MeV) N [emp] +N [MBPT] A Ca (d) + N (pfg 9/ shell) Hagen et al. PRL9 5 () 54 Ca sensitive to N interaction: Difference st/rd MBPT N forces Holt et al. JPG9 85() V low k () V low k + N(N LO) V low k + N(N LO) [MBPT] Shell evolution and Mass pairing Number in Ca A isotopes with +N forces

20 Effective SPEs and shell evolution Calculate effective SPEs (monopoles) Single-Particle Energy (MeV) -5 - g 9/ f 5/ p / p / f 7/ Single-Particle Energy (MeV) -5 - g 9/ f 5/ p / p / f 7/ +N Closed-shells N = and N = 4 with and +N forces guided by ESPEs Neither ESPEs shows closed N = 8 Different behaviour as phenomenological interactions

21 Effective SPEs and shell evolution Calculate effective SPEs (monopoles) Single-Particle Energy (MeV) (a) Phenomenological Forces f 5/ p / p / f 7/ KBG GXPF Single-Particle Energy (MeV) -5 - g 9/ f 5/ p / p / f 7/ +N Closed-shells N = and N = 4 with and +N forces guided by ESPEs Neither ESPEs shows closed N = 8 Different behaviour as phenomenological interactions

22 48 Ca spectra Ca isotopes: ground states Energy (MeV) Challenge: Doubly-closed nucleus 48 Ca pf + pfg 9/ 48 Ca N +N pf pfg 9/ (+) Expt. GXPF KBG Spectra too compressed with forces only or pf space + state only appropriate energy in pfg 9/ +N calculation + state too low (st excited state) especially compared to phenomenological interactions Importance of N forces Importance of including g 9/ orbit

23 B(M) Transition in 48 Ca B(M) [µ N ] 5 4 (a) pf-shell Experiment GXPF G G + N( ) + N B(M) strength in 48 Ca too fragmented in pf space Phenomenological calculations reproduce experimental concentration B(M) [µ N ] 5 4 (b) pfg 9/ -shell + N (emp) + N (MBPT) In the extended pfg 9/ space forces also fragmented strength +N calculation in pfg 9/ very good agreement with experiment Excitation Energy (MeV)

24 49 Ca, 5 Ca neutron rich spectra Energy (MeV) pf 49 Ca / - / - 9/ - 9/ - - 7/ 9/ - 7/ - - / (, ) 5/ 7/ - - 5/ 5/ 5/ (9/ + ) 7/ - 9/ - 7/ - - / 7/ / 9/ - - 7/ 5/ / - - / - / / - / - / - 7/ - 7/ - / - 7/ - - / - / 7/ - pfg 9/ +N +N pf pfg 9/ Expt. GXPF KBG Energy (MeV) Ca Spectrum compressed unless pfg 9/ +N Correct (/) energy, (but too low (5/) ), possibility to assign experimental spins pf / - / - / - / - / 5/-- 7/ - 7/ - / - 7/ - / - / - pfg 9/ 9/ - 7/ - / - / - +N +N pf pfg 9/ Similar quality comparable to phenomenological interactions (?) (?) (?) (?) (?) ( ) (?) 7/ - 9/ - 7/ - / - Expt. GXPF KBG /- 7/ /-- 9/ - / -

25 Spectroscopic factors Have a look at the spectroscopic factors (Shell Model calculation) 49 Ca gs 48 Ca SF Ji+ p / f 7/ + gs sum sum GXPF (Sum Rule) (.96) (.) (.) (.4) (.4) (.5) (.7) (.) (.69) (7.89) pf +N (SR) (.85) (.4) (.) (.) (.) (.5) (.59) (.4) (.5) (7.4) pfg 9/ +N MBPT spe s (SR) (.9) (.) (.9) (.4) (.8) (.) (.) (.7) (.) (6.) 5 Ca gs 49 Ca SF Ji+ p / f 7/ gs 7 GXPF (SR) (.8) (7.9) pf +N (SR) (.95) (7.) pfg 9/ +N MBPT spe s (SR) (.8) (6.9) Occupancy of the f 7/ orbital lower than phenomenological models (and naively expected) Extended g 9/ orbital takes missing occupancy

26 45 Ca, 47 Ca light isotopes spectra Energy (MeV) 4 Test our interaction with light Ca isotopes Results in pf pfg 9/ spaces and based on +N interactions compared to standard phenomenological interactions pf 9/ - / - 9/ - / - 7/ - 9/ - / - pfg 9/ 9/ - 9/ - / - / - 7/ - 9/ - / - 5/ /- - 9/ - / - 4 Ca 5/ 5/- - 5/- 7/ - 7/ - 7/ - 9/ - 9/ - / - / 7/- - / - / - / - 9/ 9/ - - / - 7/ - 7/ - +N +N Expt. GXPF KBG pf pfg 9/ /- Energy (MeV) 4 pf 45 Ca 7/ - 9/ - / - / - 5,7/ - -? 9/ - 9/ - / / - / - / - -? / / - / - 9/ - - /-? (/ - ) - / / - / 7/ - 7/ - pfg 9/ 7/ - - / - 5/ 7/ - 7/ - 7/ - / - 9/ - / - 9/ - / - 7/ - 7/ - +N +N Expt. GXPF KBG pf pfg 9/ Agreement with experiment similar to phenomenological interactions sd degrees of freedom might still play a role for some excited states

27 B(E) Transition Strengths Isotope Transition KBG GXPFA MBPT EXP. 46 Ca ± ±.6 46 Ca ±.9 47 Ca / 7/ ±. 48 Ca ± Ca 7/ / ±. 5 Ca ±. B(E)s in reasonable agreement with experiment (order of magnitude) Similar quality as phenomenological interactions (very close to KBG) 46 Ca: sd degrees of freedom?

28 5 Ca, 54 Ca neutron rich spectra: Energy (MeV) / - 5 Ca? / - / - / - +N Expt. GXPF KBG pfg 9/ 7/ - 7/ - 7/ - / - 9/ - 7/ - / - Energy (MeV) Ca + + +N Expt. GXPF KBG pfg 9/ Phenomenological interactions different results in neutron rich nuclei MBPT: prediction, more controlled Explore sensitivity to theoretical uncertainties (difference with respect to N st order results)

29 Residual N Forces In the most neutron-rich isotopes, N forces between valence neutrons (suppressed by N valence /N core ) can give a relevant contribution Sin + N (Δ) Neutron Number (N) (c) -body interaction Exact treatment of residual N forces would demand V N diagonalization Expected small correction compared to other N contributions Evaluated st order perturbation theory: 6 O core Ψ V N Ψ expected more important in O isotopes

30 in O isotopes E N,res (MeV) O Energy (MeV) N USDb AME Neutron Number N Neutron Number N small and repulsive correction Increase with the number of neutrons Negligible up to 4 O, up to MeV in 8 O Important in the dripline region: flat energy behaviour

31 Residual N Forces around O dripline Energy (MeV) RB-LAND (this work) MoNA/NSCL (8, ) +N + residual N +N / + + residual N + + Compare to experimental energies of unbound 5 O and 6 O (relative to 4 O) Without residual N forces O dripline predicted at 6 O by kev (N forces to rd order in MBPT) drive dripline to 4 O in agreement with experiment O O O Caesar, Simonis et al, arxiv:9.56 Unbound 5 O and 6 O also in very good agreement with MSU and GSI measurements

32 in Ca isotopes E N,res (MeV).4... Ca Neutron Number N Energy (MeV) N (emp) +N (MBPT) repulsive Smaller that in the O case: vs 8 core-neutrons Negligible for the discussion of S n, n or spectra

33 Summary and Outlook Microscopic calculation based on chiral EFT (+N forces) and MBPT gives good agreement with experimental two-neutron separation energies, pairing gaps and excitation spectra for calcium isotopes: Experimental trends in S n s and () n s nicely reproduced S n s and () n together with + energies establish shell closures: N = 8 appears, N = /4 enhanced/reduced by N forces Outlook: Predicted spectra for Ca neutron rich isotopes included and found relevant in O isotopes Explore uncertainties in the theoretical calculation Lee-Suzuki transformation from pf g 9/ to pf space