1. Spontaneous symmetry breaking and rotational bands S. Frauendorf Department of Physics University of Notre Dame

3. The collective model x Even-even nuclei, low spin Deformed surface breaks rotational the spherical symmetry band

4. Collective and single particle degrees of freedom On each single particle state (configuration) a rotational band is built (like in molecules).

5. Single particle and collective degrees of freedom become entangled at high spin and low deformation. Limitations: Rotational bands in

6. More microscopic approach: Retains the simple picture of an anisotropic object going round. Mean field theory + concept of spontaneous symmetry breaking for interpretation.

7. Rotating mean field (Cranking model): Start from the Hamiltonian in a rotating frame Mean field approximation: find state |> of (quasi) nucleons moving independently in mean field generated by all nucleons. Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …), Relativistic mean field, Micro-Macro (Strutinsky method) ……. Reaction of the nucleons to the inertial forces must be taken into account

8. Low spin: simple droplet. High spin: clockwork of gyroscopes. Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries Rotational response Mean field theory: Tilted Axis Cranking TAC S. Frauendorf Nuclear Physics A557, 259c (1993) Quantization of single particle motion determines relation J(  ).

9. Spontaneous symmetry breaking Symmetry operation S and Full two-body Hamiltonian H’ Mean field approximation Mean field Hamiltonian h’ and m.f. state h’|>=e’|>. Symmetry restoration Spontaneous symmetry breaking

10. Which symmetries can be broken? Combinations of discrete operations is invariant under Broken by m.f. rotational bands Obeyed by m.f. spin parity sequence broken by m.f. doubling of states

11. Rotational degree of freedom and rotational bands. Deformed charge distribution nucleons on high-j orbits specify orientation

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

13. Rotational bands in

14. No deformation – no bands? E2 radiation - electric rotation M1 radiation - magnetic rotation I-1/2 19 20 21 22 2324 25 26 27 28 10’ Baldsiefen et al. PLB 275, 252 (1992)

15. Magnetic rotor composed of two current loops 2 neutron holes 2 proton particles The nice rotor consists of four high-j orbitals only!

16. Why so regular? repulsive loop-loop interaction JE Shears mechanism Most of the l-l interaction due to a slight quadrupole polarization of the nucleus. Keeps two high-j holes/particles in the blades well aligned. The 4 high-j orbitals contribute incoherently to staggering. Staggering in Multiplets!

17. TAC Long transverse magnetic dipole vectors, strong B(M1) B(M1) decreases with spin: band termination Experimental magnetic moment confirms picture. Experimental B(E2) values and spectroscopic quadrupole moments give the calculated small deformation. First clear experimental evidence: Clark et al. PRL 78, 1868 (1997)

18. Magnetic rotorAntimagnetic rotor Anti-Ferromagnet Ferromagnet 18 19 20 21 22 23 24 18 20 22 24 strong magnetic dipole transitions weak electric quadrupole transitions

19. A. Simons et al. PRL 91, 162501 (2003) Band termination

20. J Degree of orientation (A=180, width of Ordinary rotorMagnetic rotor Many particles 2 particles, 2 holes Terminating bands Deformation:

21. Chirality Chiral or aplanar solutions: The rotational axis is out of all principal planes. 20’

22. Consequence of chirality: Two identical rotational bands.

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

24. Composite chiral bands Demonstration of the symmetry concept: It does not matter how the three components of angular momentum are generated. 23 0.20 29 20 0.22 29 Best candidates

25. S. Zhu et al. Phys. Rev. Lett. 91, 132501 (2003) Composite chiral band in

26. Tunneling between the left- and right-handed configurations causes splitting. chiral regime Rotational frequency Energy difference between chiral sister bands chiral regime chiral regime Chiral sister states:

27. Transition rates - + B(-in)B(-out) Branching B(out)/B(in) sensitive to details. Robust: B(-in)+B(-out)=B(+in)+B(+out)=B(lh)=B(rh) Sensitive to details of the system

28. Rh105 Chiral regime J. Timar et al. Phys Lett. B 598 178 (2004)

29. Odd-odd: 1p1h Even-odd: 2p1h, 1p2h Even-even: 2p-2h Best Chirality

30. 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 Predicted regions of chirality

31. mass-less particle nucleus New type of chirality molecule

32. Reflection asymmetric shapes Two mirror planes Combinations of discrete operations 29’

33. Good simplex Several examples in mass 230 region

34. Parity doubling Only good case.

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

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

37. E3 M2

38. Prolate ground state Tetrahedral isomer at 2 MeV Predicted as best case (so far): Comes down by particle alignment

39. Summary Orientation does not always mean a deformed charge density: Magnetic rotation – axial vector deformation. Nuclei can rotate about a tilted axis: New discrete symmetries. New type of chirality in rotating triaxial nuclei: Time reversal changes left-handed into right handed system. Bands in nuclei with tetrahedral symmetry predicted 34’ Thanks to my collaborators! V. Dimitrov, S. Chmel, F. Doenau, N. Schunck, Y. Zhang, S. Zhu Orientation is generated by the asymmetric distribution quantal orbits near the Fermi surface

41. Microscopic (“finite system”) Rotational levels become observable. Spontaneous symmetry breaking = Appearance of rotational bands. Energy scale of rotational levels in

42. Tiniest external fields generate a superposition of the |JM> that is oriented in space, which is stable. Spontaneous symmetry breaking Macroscopic (“infinite”) system

43. Weinberg’s chair Hamiltonian rotational invariant Why do we see the chair shape?

44. Symmetry broken state: approximation, superposition of |IM> states: calculate electronic state for given position of nuclei. 1 2 3

45. Quadrupole deformation Axial vector deformation J Degree of orientation (width of Orientation is specified by the order parameter Electric quadrupole moment magnetic dipole moment Ordinary “electric” rotorMagnetic rotor

46. Transition rates - + inout Branching sensitive to details. Robust:

47. Nuclear chirality