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K. P. Drumev

SU(3) realization of the Pairing-plus- Quadrupole Model in One or More Oscillator Shells. K. P. Drumev. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012.

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K. P. Drumev

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  1. SU(3) realization of the Pairing-plus-Quadrupole Model in One or More Oscillator Shells K. P. Drumev Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria

  2. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 Outline • Motivation • SU(3) Realization of Pairing-plus-Quadrupole Model • - full-space results for 20Ne 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 (f5/2,p3/2,p1/2) + g9/2 shell model space - 64Ge and • 68Se • Conclusion 2

  3. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 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 ~109 basis states) • Ab-initio no-core techniques – applicable for light nuclei only • A challenge - not many realistic interactions available in the pf5/2g9/2 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. 3

  4. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 Pairing-plus-Quadrupole Model H = Hpairing + HQQ Elliott`s model SU(3) (β,γ) shape parameters ~ (λ,μ) labels SU(3): Microscopic theory since the SU(3) group generators – Lμ and Qμ (μ=1,2,3) are given in terms of individual nucleon coordinate and momentum variables 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 4

  5. πU νU πN νN A Z X N πU νU πU νU πN νN πN νN πU νU πU νU πN νN πN νN SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 Four Model Spaces Shell U FOUR SPACES (unique spaces explicitly included) Shell N …mixed with … ππ + ν ν + + ππ + νν πν 5

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

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

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

  9. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 Calculations in ds+fp shells hω hω • Scenario 2 f7/2 NOT an intruder level (belongs to the upper shell - fp) • Scenario 1 f7/2 is an intruder level (belongs to the lower shell - ds) 0 0 fp fp f7/2 f7/2 ds ds • Systems: 2, 3 and 4 particles of the same kind in the ds+fp shells 2p+2n in the ds+fp shells • full-space calculation • pairing strength G = 0.05 MeV, 0.2 MeV (mild to medium) • single-particle strength hω = 5, 10, 20 MeV (small to considerable) • quadrupole-quadrupole strength χ = 0 ,…, 0.3 MeV 9

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

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

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

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

  14. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 Beta shape parameter – scenario 1 k = (5/9π)1/2A<r2> <r2> r.m.s. radius A mass number hω J = 0+ J = 0+ J = 0+ J = 1/2+ J = 1/2+ J = 1/2+ G J = 0+ J = 0+ J = 0+ J = 1/2+ J = 1/2+ J = 1/2+ 14

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

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

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

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

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

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

  21. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 Quadrupole collectivity for 3p and 4p Quadr. Coll. = < C2(λ,μ) >/C2,ref(λref,μref) Scenario 1 J = 0+ J = 0+ J = 1/2+ J = 0+ J = 1/2+ J = 1/2+ Scenario 2 J = 0+ J = 0+ J = 0+ J = 1/2+ J = 1/2+ J = 1/2+ 21

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

  23. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 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 ) 23

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

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

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

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

  28. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 Extended (Pseudo-) SU(3) Shell Model (SUMMARY) Microscopic theory since the SU(3) group generators – Lμ and Qμ (μ=1,2,3) are given in terms of individual nucleon coordinate and momentum variables Related to the Bohr-Mottelson model pseudo-ds (ds) upper-fp (f5/2p) ~ ~ g9/2 ~ p1/2 s1/2 pseudospin ~ d5/2 f5/2 ~ d3/2 p3/2 transformation Hext SU(3)= f7/2 SU(3) symmetry broken by the s.p. terms in the Hamiltonian f7/2 +GHpairing – χ/2 HQQ +aKJ2+bJ2 INERT CORE SU(3) symmetry is reasonably good K. P. Drumev – Towards an Extended Microscopic Theory for Upper-fp-Shell Nuclei, Ph.D. Dissertation, Louisiana State University, USA, 2008 28

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

  30. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 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. 30

  31. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012 Future Work • 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 pf5/2g9/2( JUN45?* Honma et al. PRC 80, 064323 (2009) ) or pf5/2gds 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 31

  32. Thank you !

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