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

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? Bohr and Mottelson: The nuclear surface

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

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

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

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

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

10 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

11 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

12 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 The nucleus: A clockwork of gyroscopes Quantization of single particle motion Adds elements that specify the orientation. rigid irrotational

13 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

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

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

19 No deformation – no bands? E2 radiation - electric rotation M1 radiation - magnetic rotation I-1/ ’ Baldsiefen et al. PLB 275, 252 (1992)

20 No deformation-no bands M1 bands

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

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

23 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)

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

25 Why are magnetic bands so regular? Staggering in Multiplets! Keeps two high-j holes/particles in the blades well aligned. 2) Somewhat away from the magic numbers Dominates the long range interaction due to a slight quadrupole polarization of the nucleus. Staggers only weakly. The 4 high-j orbitals contribute incoherently to staggering. 1)There are two particles and two holes

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

27

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

29 Magnetic rotorAntimagnetic rotor Anti-Ferromagnet Ferromagnet strong magnetic dipole transitions weak electric quadrupole transitions

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

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 Composite chiral bands Demonstration of the symmetry concept: It does not matter how the three components of angular momentum are generated Best candidates

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

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

38 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:

39 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

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

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

42 Chiral sister bands Representative nucleus observed predicted observed Predicted regions of chirality

43 mass-less particle nucleus New type of chirality molecule

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

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)

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

49 E3 M2

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

51 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. A host of different discrete symmetries for non-reflection symmetric nuclei. 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

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

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

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

55 Symmetry broken state: approximation, superposition of |IM> states: calculate electronic state for given position of nuclei

56 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

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

58

59 rigid irrotational Pair correlations

60 Imax>20 Normal persistent currents 10’