1. Chirality: From Symmetry to Dynamics S. Frauendorf Department of Physics University of Notre Dame USA Stockholm, April 20, 2015

3. Chirality “ I call a physical object, chiral, and say it has chirality, if its image, generated by space inversion, cannot brought to coincide with itself by a rotation. ” Lord Kelvin 11/37

4. The rotational axis is out of all principal planes. Possible for nuclei – not for classical mechanics

5. Consequence of strong chirality: Two identical rotational bands. 15

6. Chirality “ I call a physical object, chiral, and say it has chirality, if its image, generated by space inversion or time reversal, cannot brought to coincide with itself by a rotation. ” 11/37

7. H H H H H H H H C C N N F F Rotational frequency: 100meV + - 100meV 6000 GHz Chirality of molecules COOH rightleft 11

8. TAC Mean field calculations 134 Pr  =350keV + Shell correction Contour lines 18keV Dimitrov, Frauendorf, Doenau PRL 84 (2000) 5732 ε=0.175

9. Frauendorf, Meng, Nucl. Phys. A617, 131 (1997) Triaxial rotor + high-j particle and hole

10. Static vs. dynamic chiral rotation Chiral vibration Chiral rotation Chiral transition Soft chiral rotation

11. Dynamics Multi Quasiparticle+Triaxial Rotor Model (MQPTR) Two Quasiparticle+IBA core (IBFFM) Two Quasiparticle+BH core (Prochniak) Random Phase Approximation Adiabatic Collective Hamiltonian (Zhang) Triaxial Projected Shell Model (TPSM)

12. Multi-Quasiparticle+Triaxial Rotor Model Generic example: Odd-Odd Triaxial nuclei

13. TQPTR calculation Chiral vibration transverse Chiral vibration longitudinal Transient phenomenon

14. Chiral geometry in the MQPTR

15. Frozen Alignment Approximation (FAA): One dimensional - very well suited for analysis. Semiclassics sphereellipsoid intersection orbit

16. chiral Vibration transverse chiral rotation FAZ: 1D problem on The angular momentum sphere

17. Chiral vibrator (transverse): Chiral rotor Vibrational frequency decreases with I Constant staggering parameter S

18. B. Qi et al. Systematic study of TQPTR with hydro MoI, β=0.22, various γ

19. Angles between the angular momentum vectors See Chris Starosta, Dimiter Tonev

20. Probability distribution of angular momentum in cylinder projection S. F. + Bin Qi to be published

21. -π/2-ππ/2π0 0 π Band B I=9 Band A -π/2-ππ/2π0 0 A B Transverse Chiral Vibrator TCV j l, j s transverse to m-axis s l l mm

22. -π/2-ππ/2π0 0 π Band B I=12 Band A -π/2-ππ/2π0 0 A B TCV

23. -π/2-ππ/2π0 0 π Band B I=15 Band A -π/2-ππ/2π0 0 A B Chiral Rotor CR

24. -π/2-ππ/2π0 0 π Band B I=18 Band A -π/2-ππ/2π0 0 A B Longitudinal Chiral Vibrator LCV j l, j s longitudinal to m-axis

25. -π/2-ππ/2π0 0 π Band B I=21 Band A -π/2-ππ/2π0 0 A B LCV

26. -π/2-ππ/2π0 0 π Band B I=24 Band A -π/2-ππ/2π0 0 A B Longitudinal Wobbler LW

27. Evidence for chiral vibration (CV) and rotation (CR)

28. chiral regime TCVCR LCV T transverse L longitudinal E 2 -E 1 and ω “Degeneracy”: left-right tunneling slow compared rotation

29. chiral regime CVTCR CVL T transverse L longitudinal E 2 -E 1 and ω 0 Constant “staggering parameter”

30. CR: Same in-band BE2, weak inter-band, in- and inter- may be exchanged CV: sensitive to symmetry of geometry symmetric: weak in-band, strong but different inter-band asymmetric: strong in-band, weak inter-band Increasing with I

31. CR: Same in-band BE2, BM1, weak inter-band, in- and inter- may be exchanged CV: Strong but different in-band, weak inter-band

34. CR: pronounced staggering for symmetric geometry, weak for asymmetric See Koike, Starosta,Hamamoto PRL 93 (2004) 172502 for a special scenario

35. CR: No B(M1 I->I), B(E2 I->I) CV: Weak (compared to in-band)

36. Direct consequence of the left-right symmetry. Accidentally close states with with different qp configurations will radiate. Bin Qi +SF, to be published

37. Examples

38. Comparison with TQPTR calculations ε=0.175, γ=25 o, J s /J l =1.52, J i /J l =3.55 (different from hydro J s /J l =1, J i /J l =4) Comparison with IBFFM calculations Fit to 134 Ce core, shallow minimum at γ=25 o See Dimiter Tonev.

39. TQPTR: good energies, sharp crossing, problems IBFFM: good transitions, problems Conclusion: soft dynamical chirality

40. Best case so far: S. Zhu et al. PRL 91, 132501 (2003) 15 short medium long Chiral vibrator ->Chiral rotor

41. PLB 675 (2009) 175–180 B. Qi, S.Q. Zhang, J. Meng, S.Y. Wang, S. Frauendorf 3QPTR calculations Mukhopadhyay, Almehed et al. PRL 99, 172501 (2007) life times, TAC+RP calculations instability planar chiral n=0 n=1 S. Zhu et al. PRL 91, 132501 (2003) energies 17

43. -π/2-ππ/2π0 0 π Band B I=29/2 Band A -π/2-ππ/2π0 0 TCV

44. -π/2-ππ/2π0 0 π Band B I=39/2 Band A -π/2-ππ/2π0 0 CR

45. suppressed I->I transitions predicted Bin Qi + SF to be published

46. Transverse Wobbling Transverse Chiral Vibration Transverse Wobbling and Chirality come together axis of max MoI: m transverse W: j perp. m longitudinal W: j par. m m m s s l l

47. Frozen alignment approximation l m s 9 S. F. and F. Doenau PRC C 89, 014322 (2014)

48. et al. 1 phonon -> 0 phonon transitions See Costel Petrache. Signatures for TW: -Wobbling energy decreases with I -Strong collective E2 transitions between 1 and 0 phonon states

49. 163 Lu – a transverse wobbler QTPR calculation S. F. and F. Doenau PRC C 89, 014322 (2014)

50. 7 Transverse Wobbling and Chiral vibrations come together: 135 Pr TW and 134 Pr, 135 Nd TCV 163 Lu TW and ??? TAC calculation

51. Problem with the QPTR model Z-dependence of the TW and TCV vibrational frequencies not reproduced. TPSM OK. Pauli Principle core-qp? J. Peng et al. 133 La 131 Cs 135 Pr exp TPSM

52. Tilted Axis Cranking combined with Random Phase Approximation Almehed, Doenau, Frauendorf, PRC 83, 054308 (2011) Cranking combined with Random Phase Approximation works well for Transverse Wobblers. Y. R. Shimizu and M. Matsuzaki, Nucl. Phys. A 588, 559 (1996)

53. Almehed, Doenau, and Frauendorf PRC 83, 054308 (2011) TAC+RPA in Odd-odd nuclei <20keV TCV LCV CVL LCV

54. PLB 675 (2009) 175–180 B. Qi, S.Q. Zhang, J. Meng, S.Y. Wang, S. Frauendorf 3QPTR calculations Mukhopadhyay, Almehed et al. PRL 99, 172501 (2007) life times, TAC+RP calculations instability planar chiral n=0 n=1 S. Zhu et al. PRL 91, 132501 (2003) energies 17

55. RPA limited to CV Near onset of CR gives wrong results. The orientation degrees of freedom are well decoupled from the shape degrees In CV mode Adiabatic treatment of the orientation degrees of freedom seperately seems promising.

56. Triaxial Projected Shell Model TPSM Microscopic (no phenomenological core) Non-adiabatic Large amplitude Fixed shape 128 Cs o-o A≈100, SF o-e A≈135 TW

57. generates triaxial mean field and the two – quasiparticle configurations which are projected on good angular momentum The state is the superposition The coefficients are obtained by solving

58. 128 Cs ε=0.22, ε’=0.14, corresponds to γ=30 o chosen experiment MoI OK, bit to much staggering

59. Impressive agreement, Signature splitting somewhat overestimated experiment

60. Interpretation: “band diagram” Sequences of good K not revealing chirality

62. More complex scenarios: Combination of high-j legs with “spectators” MχD

63. More complex scenarios: No three legs associated with high-j qp’s +R Two-quasineutron excitation h 11/2 (g 7/2 -d 5/2 ) TAC+RPA

64. Challenges I->I transitions: measure/establish limits Identify chiral sisters with longer legs (2qp+2qh) Neutron-rich region around 114 Ru Find chiral bands at high spin – “strongly deformed triaxial nuclei” around A=163 Interpretation of TPSM (chiral sisters vs. qp. configurations ) Sharp vs. avoided crossings between chiral sister bands

65. Low energy dynamics is dominated orientation degrees of freedom The adiabatic approximation good when RPA energy is ’small’ The simplest Hamiltonian that can describe our tilted system is a forth order rotor Hamiltonian. where the forth order terms mimic the tilted mean field solution. This Hamiltonian has the energy surface at a constant I which we used to fit the c i parameters to the energy surface calculated with the full TAC Hamiltonian. 4th order Rotor Hamiltonian

66. Potential energy surface 134-Pr  =0.4

67. 134-Pr Energy

68. change in  or  can give a factor of 2 in B(E2) Transition Quadrupole moment

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

70. I=14 I=10I=12 J1J1 J2J2 J3J3 134 Pr Acquires a Berry phase? 19

71. Chirality Expected for high-j particles + high-j holes + triaxial core Rotating mean field becomes chiral, i.e. it breaks above a critical spin, but weakly. Chiral vibration is the precursor of this symmetry breaking: slow motion of angular momentum through the two chiral sectors Decreasing phonon energy, B(E2), and B(M1) well described by TAC+RPA and TPR for 135 Nd Odd-odd nuclides not well understood Some cross sharply 200keV TAC+RPA: Orientation mode well decoupled from deformation modes. In contrast, core+particle+hole models give substantial coupling 20