1. Symmetries of the Cranked Mean Field S. Frauendorf Department of Physics University of Notre Dame USA IKH, Forschungszentrum Rossendorf, Dresden Germany

2. In collaboration with A.Afanasjev, UND, USA B.V. Dimitrov, ISU, USA F. Doenau, FZR, Germany J. Dudek, CRNS, France J. Meng, PKU, China N. Schunck, US, GB Y.-ye Zhang, UTK, USA S. Zhu, ANL, USA

3. Rotating mean field: Tilted Axis Cranking model Seek a mean field state |> carrying finite angular momentum, where |> is a Slater determinant (HFB vacuum state) Use the variational principle with the auxiliary condition The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity  about the z axis. TAC: The principal axes of the density distribution need not coincide with the rotational axis (z).

4. Variational principle : Hartree-Fock effective interaction Density functionals (Skyrme, Gogny, …) Relativistic mean field Micro-Macro (Strutinsky method) ……. (Pairing+QQ) X S. Frauendorf Nuclear Physics A557, 259c (1993)

5. Spontaneous symmetry breaking Symmetry operation S

6. Which symmetries Combinations of discrete operations leave invariant? Broken by m.f. rotational bands Obeyed by m.f. spin parity sequence

7. Common bands Principal Axis Cranking PAC solutions TAC or planar tilted solutions Many cases of strongly broken symmetry, i.e. no signature splitting

8. Chirality Chiral or aplanar solutions: The rotational axis is out of all principal planes.

9. Consequence of chirality: Two identical rotational bands.

10. band 2 band 1 134 Pr  h 11/2 h 11/2

11. The prototype of a chiral rotor Frauendorf, Meng, Nucl. Phys. A617, 131 (1997 )

12. There is substantial tunneling between the left- and right-handed configurations chiral regime Rotational frequency Energy difference Between the chiral sisters chiral regime

13. Chiral sister bands Representative nucleus observed13 0.21 14 13 0.21 40 13 0.21 14 predicted 45 0.32 26 observed13 0.18 26 31/37

14. Composite chiral bands Demonstration of the symmetry concept: It does not matter how the three components of angular momentum are generated. observed 23 0.20 29 observed 20 0.22 29 Is it possible to couple 3 quasiparticles to a chiral configuration?

15. Reflection asymmetric shapes Two mirror planes Combinations of discrete operations

16. Good simplex Several examples in mass 230 region Other regions?Substantial tunneling

17. Parity doubling Only good case. Must be better studied! Substantial tunneling

18. Tetrahedral shapes J. Dudek et al. PRL 88 (2002) 252502

19. Which orientation has the rotational axis? minimum maximum Classical no preference

20. E3

21. Prolate ground state Tetrahedral isomer at 2 MeV

22. Isospatial analogy Which symmetries leave invariant? Broken by m.f. isorotational bands Proton-neutron pairing: symmetries of the pair-field Analogy between angular momentum J and isospin T Broken by m.f. Pair-rotational bands

23. Isovector pair field breaks isorotational invariance. Isoscalar pair field keeps isorotational invariance.

24. The isovector scenario Calculate without np-pair field. Add isorotational energy. preferred axis

25. The isovector scenario works well (see poster 161).

26. Isorotational energy gives the Wigner term in the binding energies Structure of rotational bands inreproduced For the lowest states in odd-odd nuclei with No evidence for the presence of an isoscalar pair field See poster 161

27. Isoscalar pairing at high spin? Isoscalar pairs carry finite angular momentum total angular momentum A. L. Goodman Phys. Rev. C 63, 044325 (2001) Predicted by Which evidence?

28. Adding nn pairs to the condensate does not change the structure. Pair rotational bands are an evidence for the presence of a pair field. Ordinary nn pair field

29. which symmetries leave invariant? Either even or odd A belong to the band. Even and odd N belong to the band. Both signatures belong to the band. total angular momentum If an isoscalar pair field is present,

30. Pair rotational bands for an isoscalar neutron-proton pair field Even-even, even IOdd-odd, odd I Not enough data yet.

31. Summary Symmetries of the mean field are very useful to characterize nuclear rotational bands. Nuclei can rotate about a tilted axis: New discrete symmetries manifest by the spin and parity sequence in the rotational band: -New type of chirality in nuclei: Time reversal changes left-handed into right handed system. -Spin-parity sequence for reflection asymmetric (tetrahedral) shapes The presence of an isovector pair field and isospin conservation explain the binding energies and rotational spectra of N=Z nuclei.

32. Out of any plane: parity doubling + chiral doubling

35. Banana shapes Z=70, N=86,88 J. Dudek, priv. comm.

36. Doublex quantum number

37. Restrictions due to the States with good N, Z –parity are in general no eigenstates of If they are (T=0) the symmetry restricts the possible configurations, if not (T=1/2) the symmetry does not lead to anything new.

39. Rotational bands in 11’2347 PAC TAC