Structure of even-even nuclei using a mapped collective hamiltonian

1. Structure of even-even nuclei using a mapped collective hamiltonian and the D1S Gogny interaction J.-P. Delaroche, M. Girod, H. Goutte, S. Hilaire, S. Péru, N. Pillet (CEA Bruyères-le-Châtel, France) J. Libert (IPN Orsay, France) G. F. Bertsch (INT, Seattle, USA)

2. Introduction Reminder of formalism Ground state properties Yrast spectrum Non yrast spectrum Summary Outline

3. Motivations Methodology Calculations for ~1700 nuclei (dripline to dripline) (10<Z<110, N<200) Benchmarking Predictions for future studies (SPIRAL 2, FAIR, RIA, ...) Introduction Computing time : over 25 years of CPU time if calculation were performed on a single processor.

4. Formalism More details in: J. Libert et al., PRC60, 054301 (1999) 1) Hartree-Fock-Bogoliubov equations with constraints with Constraints on and and Self-consistent symmetries and parity

5. Formalism CHFB equations solved by expanding sp states onto triaxial harmonic oscillator basis • Number of major shells: N0= 6 - 16 • Linear constraints used throughout • (q0, q2)  Bohr coordinates (β, γ) • CHFB equations solved on a grid 0 < β < 0.9 0 < γ < π/3 Δβ = 0.05 Δγ = 10°

6. CHFB -> GCM -> GOA -> 5DCH No free parameters beyond those in the Gogny D1S force.

7. Formalism 2) Collective Hamiltonian in 5 quadrupole collective coordinates Jk(a0, a2): moment of inertia Bmn(a0, a2): collective mass (vibration) D(a0, a2): metric ZPEpot neglected

8. Formalism Approximations Bmn(a0, a2): cranking (Inglis-Belyaev) Notations Correlation energy not fullfilled for ~80 nuclei at (near) double-closed-shells.

9. Formalism 3) Limitations of present CHFB+5DCH theory • Adiabatic approximation  low spins only • Quasiparticle degrees of freedom ignored • No coupling to other collective modes

10. Shape coexistence E.Clément et al. PRC75, 054313 (2007)

11. Transitional nucleus

12. GS static and dynamic deformations

13. Frequency distributions of β and γ deformations

14. Rigidity parameters

15. Charge radii

16. Charge radii for Sr isotopes

17. Correlation energy and residuals

18. 2-nucleon separation energies

19. Energy weighted sum rules

20. First 2+ level collective properties G.F. Bertsch et al., PRL 99, 032502 (2007)

21. Frequency distributions of the R42 ratio R42= E(4+1) / E(2+1) Exp. Th.

22. R42 ratio versus deformation properties δβ/<β> <β>

23. R42 frequency distribution R42: comparison Th. / Exp.

24. R62 versus R42 : comparison Th. / Exp.

25. Probability distribution of K components for 22+ states

26. P(K=2)frequency distribution for 22+ and 23+ states

27. 5DCH Systematics for 2+γlevels

28. Comparison Th. / Exp. for 22+ energies γ vibration

29. Comparison Th. / Exp. for 02+ energies Cranking masses too small !!!

30. Exp. and Th. for R02 versus R42 R02= E(0+2)/E(2+1)

31. Energy distribution of 02+ levels

32. Model criteria for the occurrence of β-vibration Relationship between quadrupole Transition operator for 21+ 02+, 23+ 21+, 23+ 01+ transitions (Bohr and Mottelson, Eq. 4-219) Crossover matrix elements Form the ratio of |M20|, |M02|, |M22| to their total.  Conditions for the existence of β vibration should be quite common.

33. Chart of the nuclei in the vicinity of the center of the triangle

34. E0 transition strengths versus neutron number

35. Ratio of transition strengths for excited K=0 over ground state bands versus neutron number : indicator for shape coexistence

36. ~1700 nuclei have been studied between drip-lines in the present microscopic model Yrast band properties: well described especially for well deformed nuclei 22+ levels: energy well described most of these levels are 2γ+vibrations 02+ levels: energy high (cranking masses) off-band E2 transition: β-vibration? E0 transition strength: high →CHFB+5DCH questionable for the 02+ excitations. Extension required to include coupling to quasiparticle and pairing vibration modes. How with GOA? Better collective masses: Thouless-Valatin, QRPA Next to come: γ band properties Summary and outlook