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

2. HCl Microwave absorption spectrum Moment of inertia of the dumbbell

3. Indistinguishable Particles.. 2 Upper particlesLower particles Restriction of orientation

4. Nuclei are different Nucleons are not on fixed positions Most particles are identical All particles have the same mass. What is rotating? The nuclear mean field

5. 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. S. Frauendorf Nuclear Physics A557, 259c (1993)

6. Variational principle : Hartree-Fock effective interaction Density functionals (Skyrme, Gogny, …) Relativistic mean field Micro-Macro (Strutinsky method) ……. (Pairing+QQ) X NEW: The principal axes of the density distribution need not coincide with the rotational axis (z).

7. The nucleus is not a simple piece of matter, but more like a clockwork of gyroscopes. Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. Tilted rotation

8. Spontaneous symmetry breaking Symmetry operation S

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

10. Rotational degree of freedom and rotational bands. Microscopic approach to the Unified Model. Deformed charge distribution

12. Moments of inertia The moment of inertia are determined by the quantal orbits of the nucleons and the pair correlations. A complicated relationship, but the cranking model provides accurate values.

13. No deformation – no bands?

14. E2 radiation - electric rotation M1 radiation - magnetic rotation

17. Rotor composed of current loops, which specify the orientation. Orientation specified by the magnetic dipole moment. Magnetic rotation. Axial vector deformation.

18. Shears mechanism Most of interaction is due to polarization of the core. TAC calculations describe the phenomenon. Residual interaction between high-j orbitals may play an important role.

19. TAC Long transverse magnetic dipole vectors, strong B(M1) B(M1) decreases with spin.

21. Antimagnetic rotation Magnetic rotorAntimagnetic rotor Anti-Ferromagnet Ferromagnet

22. A. Simons et al. PRL, in press

23. Magnetic rotation is manifest by regular rotational bands in nuclei with near spherical charge distribution. J Quadrupole deformation Axial vector deformation Orientation is specified by the order parameter Electric quadrupole moment magnetic dipole moment 23/42

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

25. Principal Axis Cranking PAC solutions Tilted Axis Cranking TAC or planar tilted solutions Chiral or aplanar solutions Doubling of states

26. Rotational bands in 11’2347 PAC TAC

28. Consequence of chirality: Two identical rotational bands.

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

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

31. Chiral sister bands Representative nucleus observed13 0.21 14 13 0.21 40 13 0.21 14 predicted 45 0.32 26 observed 23 0.20 29 observed13 0.18 26 31/37

32. Chirality of molecules mirror The two enantiomers of 2-iodubutene

33. mirror Chirality of mass-less particles z

34. New type of chirality Chirality Changed invariant Molecules Massless particles space inversion time reversal Nuclei time reversal space inversion

35. Combinations of discrete operations

36. Good simplex Several examples in mass 230 region Other regions?

37. Parity doubling Only good case. Must be better studied!

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

39. Combinations of discrete operations

40. E3

41. Parity doubling E3M3

42. Summary Symmetries of the mean field are very useful to characterize nuclear rotational bands. Orientation does not always mean a deformed charge density: Magnetic rotation. Nuclei can rotate about a tilted axis: New discrete symmetries. New type of chirality: Time reversal changes left-handed into right handed system. Paradigm for non-nuclear fermionic systems.