Symmetries of the Cranked Mean Field

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

2. In collaboration with • Afanasjev, UND, USA • 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 w 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. invariant? leave Broken by m.f. rotational bands Combinations of discrete operations spin parity sequence Obeyed by m.f. Which symmetries

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 134Pr ph11/2 nh11/2

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

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

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

14. 20 0.22 29 observed 23 0.20 29 observed Composite chiral bands Demonstration of the symmetry concept: It does not matter how the three components of angular momentum are generated. 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. minimum maximum Which orientation has the rotational axis? Classical no preference

20. E3 E3

21. Prolate ground state Tetrahedral isomer at 2 MeV

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

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

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

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

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

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

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

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

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

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

36. Doublex quantum number

37. 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. Restrictions due to the

39. TAC PAC Rotational bands in 1 1’ 2 3 4 7