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Microscopic interpretation of the excited K  = 0 + , 2 + bands of deformed nuclei

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 Georgieva Bulgarian Academy of Science. Outline.  Introduction

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Microscopic interpretation of the excited K  = 0 + , 2 + bands of deformed nuclei

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  1. 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 • Georgieva • Bulgarian Academy of Science Gabriela Popa

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

  3. Chart of the nuclei Z (protons) N (number of neutrons) Gabriela Popa

  4. 3 3 Nuclear vibrations 2 2 1 1 -vibration -vibration Gabriela Popa

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

  6. Experimental energy levels Gabriela Popa

  7. Experimental energy spectra of 162Dy Gabriela Popa

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

  9. Particle distribution Valence space: U(Wp) U(Wn) totalnormal unique Wp =32 Wnp =20 Wup =12 Wn =44 Wnn =30 Wun=14 • Particle distributions: (l,m) • protons: neutrons: p n total • 152Nd 10 6 4 10 6 4(12,0)(18,0) (30, 0) • 156Sm 12 6 612 6 6(12,0)(18,0) (30, 0) • 160Gd 14 8 614 8 6(10,4)(18,4) (28, 4) • 164Dy 16 10 616 10 6(10,4)(20,4) (30, 4) • 168Er 18 10 818 10 8(10,4)(20,4) (30, 4) • 172Yb 20 12 820 12 8(36,0)(12,0) (24, 0) • 176Hf 22 14 822 14 8(8,30)(0,12) (8, 18) Gabriela Popa

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

  11. Direct Product Coupling Coupling proton and neutron irreps to total (coupled) SU(3): (lp, mp)  (ln, mn)  (lp+ln , mp+mn) + (lp+ln- 2,mp +mn+ 1) + (lp+ln+1,mp+mn- 2) + (lp+ln- 1,mp+mn- 1)2 + ...  S m,l (lp+ln-2m+l,mp+mn+m-2l)k with the multiplicity denoted by k = k(m,l) Gabriela Popa

  12. 21st SU(3) irreps corresponding to the highestC2 values were used in 160Gd (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) (,)(,) (,) Gabriela Popa

  13. Tricks Invariants Invariants Rot(3) SU(3) Tr(Q2)  C2 Tr(Q3)  C3 b2~ l2+ lm + m + 3(l + m+1) g =tan-1 [3 m / (2 l + m+3)] Gabriela Popa

  14. System Hamiltonian SU(3) conserving Hamiltonian: H = c Q•Q + aL2 + b KJ2 + asym C2 + a3C3 + one-body and two-body terms + Hs.p.p + Hs.p.n+Gp HPp +Gn HPn Rewriting this Hamiltonian becomes ... H = Hspp +Hspn+Gp HPp +Gn HPn+ cQ•Q + aL2 + b KJ2 + asymC2 + a3 C3 Gabriela Popa .

  15. Parameters of the Pseudo-SU(3) Hamiltonian From systematics:  = 35/A 5/3 MeV Gp = 21/A MeV Gn = 19/A MeV h = 41/A 1/3 MeV p= 0.0637 p = 0.60 n= 0.0637 n= 0.60 Fit to experiment(fine tuning): a, b, asym and a3 Gabriela Popa

  16. Energy spectrum for 160Gd Exp. Th. Exp. Th. Exp. Th. 2.5 + 8 2 + 160 8 + Gd 6 + + 7 4 1.5 + 6 + 2 + 5 + 0 + 4 p + + = K 0 1 3 + 2 + 8 2 p + = 2 K + 0.5 6 + 4 + 2 + 0 0 p + = K 0 Gabriela Popa

  17. Direct Product Coupling Coupling proton and neutron irreps to total (coupled) SU(3): (lp, mp)  (ln, mn)  (lp+ln , mp+mn) + (lp+ln- 2, mp +mn+ 1) + (lp+ln+1, mp+mn- 2) + (lp+ln- 1, mp+mn- 1)2 + ... m,l  (lp+ln-2m+l,mp+mn+m-2l)k Scissors Twist Scissors + Twist … Orientation of the p-n system is quantized with the multiplicity denoted by k = k(m,l) Gabriela Popa

  18. M1 Transition Strengths [m2N]in the Pure Symmetry Limit of the Pseudo SU(3) Model Nucleus B(M1) mode 160 ,162 Dy (10,4) (18,4) (28,8) ( 29, 6) 0.56 t ( 26, 9) 1.77 s 1 (27 ; 7) 1.82 s+t 2 (27 ; 7) 0.083 t+s 164 Dy (10,4) (20,4) (30,8) (31,6) 0.56 t (28,9) 1.83 s (29,7) 1.88 s+t (29,7) 0.09 t+s ()() () ()1+ Gabriela Popa

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

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

  21. Total B(M1) strength (N2) Nucleus Calculated Experiment Pure S U (3) Theory 160 Dy 2.48 4.24 2.32 162 Dy 3.29 4.24 2.29 164 Dy 5.63 4.36 3.05 B(M1)[N2] Gabriela Popa

  22. First excited K=0+ and K=2+ states Gabriela Popa

  23. Gabriela Popa

  24. Gabriela Popa

  25. Gabriela Popa

  26. Gabriela Popa

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

  28. Conclusions 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 • B(E2) transitions within the g.s. band well • reproduced • 1+ energies fall in the correct energy range • fragmentation in the B(M1) transition probabilities correctly predicted Gabriela Popa

  29. Conclusions • 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. • A proper description of collective properties of the first excited K = 0+andK=2+states must take into account the mixing of different SU(3)-irreps, which is driven by the Hamiltonian. Gabriela Popa

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

  31. Thank you! Gabriela Popa

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