1. K. P. Drumev Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria

2. Motivation SU(3) Realization of Pairing-plus-Quadrupole Model - full-space results for 20 Ne in the ds shell - full-space results for 2, 3 and 4 particles in the ds+fp shell - full-space results with 2 protons and 2 neutrons in the ds+fp shell Extended (pseudo-) SU(3) shell model - application to upper-fp (f 5/2,p 3/2,p 1/2 ) + g 9/2 shell model space - 64 Ge and 68 Se Conclusion Outline SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 2

3. Motivation for the study of N ~ Z systems Interesting area for research (P+QQ compete, nucleosynthesis – rp- process nuclei, interesting N ~ Z effects – isoscalar pairing) Full-space microscopic calculations in two (upper-fp+gds) shells – beyond current capabilities (max ~10 9 basis states) Ab-initio no-core techniques – applicable for light nuclei only A challenge - not many realistic interactions available in the pf 5/2 g 9/2 model space (none in the fp-gds space?) Add the pair scattering and the isoscalar pairing part in the interaction. Classification of states in SO(8) pn-pairing model – not fully resolved. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 3

4. A. Bohr, B. R. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936 S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1959) No. 11 L. S. Kisslinger and R. A. Sorensen, Mat. Fys. Medd. Dan. Vid. Selsk. 32 (1960) No. 9 K. Kumar and M. Baranger, Nucl. Phys. 62 (1965) 113 Bahri, J. Escher, J. P. Draayer, Nucl. Phys. A592 (1995) 171 (SU(3) basis in 1 shell only ) M. Hasegawa, K. Kaneko, T. Mizusaki, J. Zhang - tens of articles published in 1998 – 2011 period H = H pairing + H QQ SU(3) ( β,γ ) shape parameters ~ ( λ,μ ) labels Elliott`s model Pairing-plus-Quadrupole Model SU(3): Microscopic theory since the SU(3) group generators – L μ and Q μ ( μ=1,2,3 ) are given in terms of individual nucleon coordinate and momentum variables SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 4

5. Four Model Spaces FOUR SPACES (unique spaces explicitly included) …mixed with … πUπUπUπU πNπNπNπN νUνUνUνU νNνNνNνN πUπUπUπU πNπNπNπN νUνUνUνU νNνNνNνN πUπUπUπU πNπNπNπN νUνUνUνU νNνNνNνN πUπUπUπU πNπNπNπN νUνUνUνU νNνNνNνN πUπUπUπU πNπNπNπN νUνUνUνU νNνNνNνN ν ν ν ν + + + Shell U π ππ ππ ππ π π ππ ππ ππ π π νπ νπ νπ ν π νπ νπ νπ ν π π + ν ν X AZAZN Shell N SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 5

6. Quadrupole-Quadrupole Model ≡ Extended SU(3) Shell Model Inter-shell (N and U) coupling of irreps Well-defined particle number and total angular momentum U(2 Ω ) {U( Ω ) SU(3) } x SU(2) Basis States Eigenstates: j j SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 6

7. Extended Pairing-plus-Quadrupole Hamiltonian H = ππ pair-scattering ππ and νν pairing πν pairing πν pair-scattering νν pair-scattering mixes configurations with different distributions of particles over the shells mixes configurations with a specific distribution of particles over the shells SU(3) symmetry preserving interaction NEW TERMS in the model! single-particle energies SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 7

8. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 8 Results for 20Ne in the ds shell ( C. E. Vargas, J. G. Hirsch and J. P. Draayer, Nucl. Phys A690, 409 (2001)

9. Calculations in ds+fp shells Scenario 1 Scenario 1 f 7/2 is an intruder level (belongs to the lower shell - ds) Systems: 2, 3 and 4 particles of the same kind in the ds+fp shells 2p+2n in the ds+fp shells Scenario 2 Scenario 2 f 7/2 NOT an intruder level (belongs to the upper shell - fp) full-space calculation pairing strength G = 0.05 MeV, 0.2 MeV (mild to medium) single-particle strength h ω = 5, 10, 20 MeV (small to considerable) quadrupole-quadrupole strength χ = 0,…, 0.3 MeV ds hωhω 0 f 7/2 fp ds hωhω 0 f 7/2 fp SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 9

10. Pairing and Pair-Scattering Operator ds shell ds and fp shells fp shell S+S-S+S- For pairing η = η ’ For pair scattering η ≠ η ’ strength ≡ P SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 10

11. Results: Pure Pairing Spectrum high highdegeneracy Potential to describe complicated structures SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 11

12. ∩ scenario 1 Results: Wave Function Contents – scenario 1 G hωhω 4p in ds+fp shell [ N N, N U ] ( λ, μ ) [ 4, 0 ] (4,2), (0,4), (3,1), … [ 2, 2 ] (10,0), (8,1), (6,2), … [ 0, 4 ] (8,2), (7,1), (4,4), … SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 12

13. scenario 2 Results: Wave Function Contents – scenario 2 G hωhω 4p in ds+fp shell SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 13

14. scenario 1 Beta shape parameter – scenario 1 G hωhω k = (5/9 π ) 1/2 A k = (5/9 π ) 1/2 A r.m.s. radius A mass number J = 1/2 + J = 0 + J = 1/2 + J = 0 + SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 14

15. scenario 2 Beta shape parameter – scenario 2 G hωhω J = 0 + J = 1/2 + SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 15

16. scenario 2 Beta shape parameter – scenario 2 G hωhω J = 0 + J = 1/2 + SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 16

17. scenario 1 Gamma shape parameter – scenario 1 G hωhω J = 0 + J = 1/2 + SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 17

18. scenario 2 Gamma shape parameter – scenario 2 G hωhω J = 0 + J = 1/2 + SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 18

19. scenario 2 Gamma shape parameter – scenario 2 G hωhω J = 0 + J = 1/2 + J = 0 + J = 1/2 + SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 19

20. Uncertainty of the beta shape parameter for 3p and 4p Scenario 1 Scenario 2 SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 20

21. Quadrupole collectivity for 3p and 4p Scenario 1 Scenario 2 J = 1/2 + J = 0 + Quadr. Coll. = /C 2,ref ( λ ref,μ ref ) SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 21

22. Results: Pure Pairing Spectrum for proton-neutron systems: 2p+2n Isovector (T=1) pairing Total pairing (T=0 + T=1) Total pairing (T=0 + T=1) SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 22

23. Beta shape parameter Scenario 1 Scenario 2 ( ( K. P. Drumev, A. I. Georgieva and J. P. Draayer, J. Phys: Conf. Ser., 356, 012015 (2012) - hw = 0 case only ) SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 23

24. Gamma shape parameter Scenario 1 Scenario 2 SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 24

25. Wave function: Effects of G πν ≠ G ππ (and G νν ) Scenario 1 Scenario 2 SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 25

26. Beta: Effects of G πν ≠ G ππ (and G νν ) Scenario 1 Scenario 2 SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 26

27. Gamma: Effects of G πν ≠ G ππ (and G νν ) Scenario 1 Scenario 2 SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 27

28. Extended (Pseudo-) SU(3) Shell Model (SUMMARY) K. P. Drumev – Towards an Extended Microscopic Theory for Upper-fp-Shell Nuclei, Ph.D. Dissertation, Louisiana State University, USA, 2008 Microscopic theory since the SU(3) group generators – L μ and Q μ ( μ=1,2,3 ) are given in terms of individual nucleon coordinate and momentum variables Related to the Bohr-Mottelson model upper-fp (f 5/2 p) (f 5/2 p) pseudo-ds(ds) SU(3) symmetry broken by the s.p. terms in the Hamiltonian f 7/2 f 5/2 SU(3) symmetry is reasonably good p 1/2 p 3/2 d 3/2 d 5/2 f 7/2 g 9/2 INERT CORE s 1/2 pseudospin transformation ~ ~ ~ ~ H ext SU(3) = +GH pairing – χ/2 H QQ +aK J 2 +bJ 2 SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 28

29. [Interactions provided by P. Van Isacker, see e.g.: E. Caurier, F. Nowacki, A. Poves, & J. Retamosa, Phys. Rev. Lett. 77, 1954 (1996)] up to 50-60% dominance of the leading irreps ! Pseudo-SU(3) Symmetry in 64 Ge and 68 Se SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 29

30. Conclusions Calculations for the systems 2p(n), 3p(n), 4p(n) and 2p+2n were perfomed Effects of the quadrupole, pairing and the single-particle terms of the Hamiltonian were studied, two scenarios for the position of the intruder level were considered Results suggest that the two scenarios lead to a very distinct behavior of the wave functions, shape parameters and the quadrupole collectivity for the ground states of all the systems While the pairing interaction mostly softens the effects, the strength of the s.p. energies drives the main (rapid) changes in the behavior of the systems. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 30

31. Application of the extended SU(3) model to additional upper-fp + gds shell nuclei (Br and Kr isotopes of particular interest) (Challenges: need other realistic interactions in the pf 5/2 g 9/2 ( JUN45?* Honma et al. PRC 80, 064323 (2009) ) or pf 5/2 gds model space, huge model spaces in full-space calculations) Application of the theory to heavier deformed (rare-earth / actinide) nuclei - Origin and multiplicity of 0 + states - B(E2) & B(M1) transition strengths, clusterization effects - Double beta decay - Study of nuclear reactions Role of truncations [e.g., (  ) & S] in the symmetry-adapted basis Search for new and improved interactions (parameter optimization) Evolution of key parameters from the theory of effective interactions Future Work SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 31