1. Tidal Waves and Spatial Symmetry Daniel Almehed Stefan Frauendorf Yongquin Gu

2. E I qp. excitations Tidal waves

3. Generation of angular momentum Angular velocity Deformation Rotor increases stays constant Tidal wave Vibrator stays constant increases (Other degrees of freedom) 3

4. The symmetry of the tidal wave determines the spin-parity sequence of the band. Low-spin quadrupole waves: 4

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7. Quadrupole waves: Theoretical method Cranking model: semiclassical treatment of angular momentum Micro-macro method (Nilsson+fixed pairing). Find the equilibrium shape for the rotating mean field. Minimizing at fixed frequency problematic: Minimizing 7

9. Transition from tidal waves to rotation at low I. Energies of “vibrational nuclei” strongly anharmonic, B(E2) more harmonic. Z=48, N=60-66: after neutron alignment, smaller deformation  approach of antimagnetic rotation Z=46, N=56,60 and Z=44, N=62,64 angular velocity nearly constant during neutron alignment – tidal wave with quasiparticle degrees of freedom More B(E2) values to check theory Remarkable reproduction of data by calculations 9

10. High-spin waves Combination of Angular momentum reorientation Triaxial deformation 10

11. tidal C. Thwaites et al. PRC 66, 054309 (2002) 11

12. Line distance: 200 keV L. K. Pattinson et al. PRL 91, 182501 (2003) 12

13. Heart-shaped waves - good simplex W. Reviol et al. Phys. Rev. C74, 044305 (2006) 13

14. Nuclides in mass 230 region 14

15. Boson condensation + - + - + - =3 phonon =2 phonon Two waves with different angular velocity rotational frequency  =33 phonons At condensation 15

16. X. Wang, R.V.F. Janssens, I. Wiedenhoever et al. to be published. Preliminary 16

17. n=0 n=1 n=2 n=3 n=0 n=1 n=2 n=3 harmonic (non-interacting) phonons anharmonic (interacting) phonons 0-2 1-3 Data: J.F.Smith et al.PRL 75, 1050(95) The quadrupole-octupole interaction tends to synchronize the two motions. 17

18. n=0 n=1 n=2 n=3 0 2 18

19. Increasing quadrupole-octupole coupling Locking of 33-octupole vibration to quadrupole rotation not fully reached. Best cases 224 and 226. Example of how heart-shaped waves/rotors show up in real nuclei. “Octupole deformation” is a mixture of the 0- and 2-phonon states. 18

20. Parity doubling Tilted heart-shaped wave <60keV 19

21. z z z + - + - + -  = 20+ 22+ 33  = 20+ 22+ 32 + - + + - - Shovel-shaped tidal waves 20

22. W. Reviol et al. Phys. Rev. C74, 044305 (2006) 21

23. Octupole deformation materializes as condensation of rotationally aligned (j=3) bosons. Octupole phonons not fully locked to the quadrupole tidal wave/rotation. Exotic “shapes” are expected to show up as condensing bosons at the best. Spin-parity sequence (+ selection rules) dictated by symmetry: Evidence for the type of shape. 22

24. Parity doubling E3M3 23 Tetrahedral waves - tip

25. E3 24 Tetrahedral waves - edge

26. Tetrahedral shapes will likely show up as tidal waves as the discussed reflection asymmetric shapes. Both symmetry types are expected to show up. No quadrupole transitions would be a clean signal. However, coupling to quadrupole vibrations is expected. Complex symmetries – not conclusive. 25

27. Condensation of non-rotating vs. rotating octupole phonons + - + - + - j=3 phonon Angular momentum rotational frequency 17  =30 phonons  axial refl. as. Rotor  =33 phonons At condensation

28. Rotating octupole does not completely lock to the rotating quadrupole. + - + - + - 19