Spontaneous symmetry breaking and rotational bands

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

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

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

7. Reaction of the nucleons to the inertial forces must be taken into account 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) …….

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

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

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

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

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. E2 radiation - electric rotation I-1/2 23 24 25 22 26 27 21 M1 radiation - magnetic rotation 28 20 19 No deformation – no bands? 10’ Baldsiefen et al. PLB 275, 252 (1992)

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

16. repulsive loop-loop interaction E J Shears mechanism Staggering in Multiplets! Why so regular? 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.

17. First clear experimental evidence: Clark et al. PRL 78 , 1868 (1997) 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.

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

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

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

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

22. Consequence of chirality: Two identical rotational bands.

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

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

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

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

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

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

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

30. 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 Predicted regions of chirality Chiral sister bands Representative nucleus

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

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

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

37. E3 M2 E3 M2

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

39. Summary 34’ Orientation is generated by the asymmetric distribution quantal orbits near the Fermi surface 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 Thanks to my collaborators! V. Dimitrov, S. Chmel, F. Doenau, N. Schunck, Y. Zhang, S. Zhu

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

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” rotor Magnetic rotor

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

47. Nuclear chirality