New symmetries of rotating nuclei

1. New symmetries of rotating nuclei S. Frauendorf Department of Physics University of Notre Dame

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

3. . Upper particles Lower particles . Indistinguishable Particles 2 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? Bohr and Mottelson: The nuclear surface

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

6. Collective and single particle degrees of freedom Like rotational and electronic motion in molecule: The rotational motion is Much slower than the single particle motion. On each single particle state (configuration) a rotational band is built

7. Limitation: Single particle and collective degrees of freedom become entangled at high spin and low deformation. Rotational bands in Limitation: Shapes and moments of inertia are parameters.

8. ideal : “irrotational flow” viscous: “rotational flow” None is true: complicated flow containing quantal vortices. Microscopic description needed: Rotating mean field

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

10. 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) …….

11. 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

12. The nucleus: A clockwork of gyroscopes rigid irrotational Quantization of single particle motion Adds elements that specify the orientation. Controls the rotational response. High spin: clockwork of gyroscopes Low spin: simple droplet (gyroscopes paired off) Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries

13. 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

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

15. Microscopic foundation of collective model Experimental transition quadrupole moments (deformation) reproduced. Experimental moments of inertia reproduced. Different from rigid rotation (molecules) or irrotational

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

17. Rotational bands in

18. x Even-even nuclei, low spin 20’ The collective model Deformed surface breaks the spherical symmetry No deformation – no rotational bands!

19. 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)

20. M1bands No deformation-no bands

21. 2 proton particles 2 neutron holes Magnetic rotor composed of two current loops This nice rotor consists of four high-j orbitals only!

22. Generation of angular momentum by the Shears mechanism repulsive loop-loop interaction Short-range nucleon-nucleon interaction Long-range phonon exchange E J

23. 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.

24. Short-range nucleon-nucleon interaction (J. Schiffer) V even (odd) odd (even) J Dominates at closed shell.

25. There are two particles • and two holes The 4 high-j orbitals contribute incoherently to staggering. 2) Somewhat away from the magic numbers Dominates the long range interaction due to a slight quadrupole polarization of the nucleus. Staggers only weakly. Keeps two high-j holes/particles in the blades well aligned. Why are magnetic bands so regular? Staggering in Multiplets!

26. Conditions for shears bands Small deformation High-j particles are combined with high-j holes Some polarizability of the core

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

29. 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

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

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

32. Consequence of chirality: Two identical rotational bands.

33. Nuclear chirality

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

35. 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

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

37. Chiral vibration Chiral rotation Left-right tunneling Left-right communcation

38. 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

39. 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)

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

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

42. 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

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

44. 29’ Reflection asymmetric shapes Two mirror planes Combinations of discrete operations

45. Good simplex Several examples in mass 230 region

46. Parity doubling Only good case.

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

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

49. E3 M2 E3 M2

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