2. NPSC-2003Gabriela Popa Microscopic interpretation of the excited K  = 0 +, 2 + bands of deformed nuclei Gabriela Popa Rochester Institute of Technology Collaborators: J. P. Draayer Louisiana State University J. G. Hirsch UNAM, Mexico A.Georgieva Bulgarian Academy of Science

3. NPSC-2003Gabriela Popa Outline Introduction Introduction What we know about the nucleus Characteristic energy spectra Theoretical Model Configuration space System Hamiltonian Results Conclusion and future work

4. Nuclear Physics Summer School June 15-27, 2003Gabriela Popa Chart of the nuclei N (number of neutrons) Z (protons)

5. NPSC-2003Gabriela Popa Nuclear vibrations  -vibration 3 1 2 3  -vibration 1 2

6. 0, 2, 4 0 0 2 4 6 8 6 8 2 0 2 4 6 2 3 4 5 6 spherical deformed E = n   E = I(I+1)/(2  ) Schematic level schemes of spherical and deformed nuclei

7. NPSC-2003Gabriela Popa Experimental energy levels

8. Nuclear Physics Summer School June 15-27, 2003Gabriela Popa Experimental Experimental energy spectra of 162 Dy

9. NPSC-2003Gabriela Popa Single particle energy levels 1s 1p 2s 1g N = 3 N = 2 N = 1 N = 0 N = 4 1d l·l l·s d5/2 d3/2 s1/2 1f 2p 3s 2d 1h N = 5 s1/2 p1/2 p3/2 f7/2 f5/2 p5/2 f9/2 p1/2 d3/2 d5/2 h11/2 s1/2 g7/2

10. NPSC-2003Gabriela Popa Valence space: U(   )  U(  ) total normal unique   =32  n  =20  u  =12  =44  n =30  u =14 Particle distributions:  protons: neutrons:  total 152 Nd 10 6 4 10 6 4 (12,0) (18,0) (30, 0)  156 Sm 12 6 6 12 6 6 (12,0) (18,0) (30, 0)  160 Gd 14 8 6 14 8 6 (10,4) (18,4) (28, 4)  164 Dy 16 10 6 16 10 6 (10,4) (20,4) (30, 4)  168 Er 18 10 8 18 10 8 (10,4) (20,4) (30, 4)  172 Yb 20 12 8 20 12 8 (36,0) (12,0) (24, 0)  176 Hf 22 14 8 22 14 8 (8,30) (0,12) (8, 18)  Particle distribution

11. NPSC-2003Gabriela Popa Wave Function |    =  C  i |  i  |  i  = |{   ;  }  (,  )KL S; JM    (  = , ) = n  [f  ] ( ,   ), S  ( ,   )  (,  ) =  (,  ) 

12. NPSC-2003Gabriela Popa Coupling proton and neutron irreps to total (coupled) SU(3):           +      +      +   -   +   +     +     +   -     +   -    +   -     m,l   +  -2m+l   +  +m-2l) k with the multiplicity denoted by k = k(m,l) Direct Product Coupling

13. NPSC-2003Gabriela Popa (10,4)(18,4)(28,8)(29,6)(30,4)(31,2)(32,0)(26,9)(27,7) (10,4)(20,0)(30,4) (10,4)(16,5)(26,9)(27,7) (10,4)(17,3)(27,7) (12,0)(18,4)(30,4) (12,0)(20,0)(32,0) (8,5)(18,4)(26,9)(27,7) (9,3)(18,4)(27,7) (7,7)(18,4)(25,11)(26,9)(27,7) (7,7)(20,0)(27,7) (4,10)(18,4)(22,14) ( ,   )(,  )(,  ) 21 st SU(3) irreps corresponding to the highest C 2 values were used in 160 Gd

14. NPSC-2003Gabriela Popa Tricks Invariants  Invariants Rot(3) SU(3) Tr(Q 2 )  C 2 Tr(Q 3 )  C 3    ~    +  +  +  +   =  tan   + 

15. NPSC-2003Gabriela Popa SU(3) conserving Hamiltonian: H =  Q   Q + a L 2 + b K J 2 + a sym C 2 + a 3 C 3 this Hamiltonian becomes... + one-body and two-body terms + H s.p.  H s.p.   G  H P  G H P Rewriting H = H sp  +  H sp  +  G  H P  +  G H P +  Q Q + a L 2 + b K J 2 + a sym C 2 + a 3 C 3. System Hamiltonian

16. NPSC-2003Gabriela Popa Parameters of the Pseudo-SU(3) Hamiltonian Fit to experiment(fine tuning): a, b, a sym and a 3   = 0.0637   = 0.60  = 0.0637  = 0.60 From systematics:  = 35/A 5/3 MeV G  = 21/A MeV G  = 19/A MeV h  = 41/A 1/3 MeV

17. NPSC-2003Gabriela Popa 0 0.5 1 1.5 2 2.5 K   0 + K   + K   0 2 + 160 Gd 0 + 6 + 0 + 2 + 2 + 2 + 4 + 4 + 8 + 6 + 3 + 8 + 7 + 6 + 4 + 5 + 8 + Exp. Th. Energy spectrum for 160 Gd

18. NPSC-2003Gabriela Popa Coupling proton and neutron irreps to total (coupled) SU(3):           +      +      +   -   +   +     +     +   -     +   -    +   -     m,l   +  -2m+l   +  +m-2l) k Twist Scissors Scissors + Twist … Orientation of the  - system is quantized with the multiplicity denoted by k = k(m,l) Direct Product Coupling

19. NPSC-2003Gabriela Popa NucleusB(M1)mode 160,162 Dy(10,4)(18,4)(28,8)(29,6)0.56t (26,9)1.77s (27;7) 1 1.82s+t (27;7) 2 0.083t+s 164 Dy(10,4)(20,4)(30,8)(31,6)0.56t (28,9)1.83s (29,7)1.88s+t (29,7)0.09t+s (    ) (  ) (  )(  ) 1+ M1 Transition Strengths [  2 N ] in the Pure Symmetry Limit of the Pseudo SU(3) Model

20. NPSC-2003Gabriela Popa -0.5 0 0.5 1 1.5 2 2.5 164 Dy e) K = 2 K = 0 2 + 4 + 6 + 0 + 2 + 4 + 6 + 0 + 2 + 4 + 6 + 0 + 2 + 4 + 0 + 2 + 4 + 0 + 2 + 4 + 6 + 0 + 2 + 4 + 6 + 3 + 5 + 2 + 4 + 6 + 3 + 5 + g.s. c) Energy [MeV] Exp. Th. Exp. Th. Energy levels of 164 Dy … and M1 Strengths

21. NPSC-2003Gabriela Popa 0 0.5 1 1.5 2 2.5 3 168 Er K = 0 K = 2 0 + 2 + 8 + 6 + 4 + 0 + 2 + 8 + 6 + 4 + 0 + 2 + 8 + 6 + 4 + 0 + 2 + 4 + 2 + 8 + 6 + 4 + 7 + 3 + 5 + 2 + 8 + 6 + 4 + 7 + 3 + 5 + 0 + 2 + 6 + 6 + 4 + 0 + 2 + 6 + 4 + g.s. Energy [MeV] Exp. Th. a) Energy Levels of 168 Er … and M1 Strengths

22. NPSC-2003Gabriela Popa Nucleus Experiment Calculated PureSU(3)Theory 160 Dy2.484.242.32 162 Dy3.294.242.29 164 Dy5.634.363.05 B(M1)[  N 2 ] Total B(M1) strength (  N 2 )

23. NPSC-2003Gabriela Popa First excited K=0 + and K=2 + states

24. NPSC-2003Gabriela Popa

25. NPSC-2003Gabriela Popa

26. NPSC-2003Gabriela Popa

27. NPSC-2003Gabriela Popa

28. NPSC-2003Gabriela Popa 0 20 40 60 80 100 0 20 40 60 80 100 0 gs 20 2 0 20 2 a = (36,0)(12,0)x(24,0) b = (28,10)(12,0)x(16,10) c = (20,20)(4,10)x(16,10) SU(3) content in 172 Yb

29. NPSC-2003Gabriela Popa Conclusions B(E2) transitions within the g.s. band well reproduced ground state, , first and second excited K = 0 + bands well described by a few representations calculated results in good agreement with the low- energy spectra 1 + energies fall in the correct energy range fragmentation in the B(M1) transition probabilities correctly predicted

30. NPSC-2003Gabriela Popa Conclusions A proper description of collective properties of the first excited K  = 0 + and K=2 + states must take into account the mixing of different SU(3)-irreps, which is driven by the Hamiltonian. A microscopic interpretation of the relative position of the collective band, as well as that of the levels within the band, follows from an evaluation of the primary SU(3) content of the collective states. The latter are closely linked to nuclear deformation. If the leading configuration is triaxial (nonzero  ), the ground and  bands belong to the same SU(3) irrep; if the leading SU(3) configuration is axial (  =0), the K =0 and  bands come from the same SU(3) irrep.

31. NPSC-2003Gabriela Popa Future work In some nuclei total strength of the M1 distribution is larger than the experimental value Investigate the energy spectra in super-heavy nuclei Investigate the M1 transitions in light nuclei There are new experiments that determine the inter- band B(E2) transitions Improve the model to calculate these transitions Consider the abnormal parity levels Consider the states with J = 1 in the configuration space

32. NPSC-2003Gabriela Popa