1. Makito Oi Senshu University, Tokyo

2. Study of “rotation” Apparent retrograde motion of Saturn in 2011 (Apr-July) … observation by MO Eudoxus Hipparcus 3D 1D

3. Nuclear rotation Static rotation Uniform rotation One-dimensional rotation “Yrast” states (1D) R coll

4. Collectivity: many-body Physics Microscopic and Macroscopic viewpoints Degrees of freedom: collectivity

5. Quadrupole deformation prolate triaxial oblate sphere NDSD 0.0 0.20.6

6. Wobbling motion as a 3D rotation Triaxial rotor High spin (J >> 1) Quasi-1D rotation (Jx ~ J) Bohr and Mottelson (1975)

7. Non-abelian algebra: QM flucutation Classical: J 2 Quantum: J 2 + J

8. Definition of K quantum number K : quantisaion axis | J K >

9. QM flucutation (II) Example: state |J K=J >

10. Nuclear chiral rotation Valence neutron (hole) Valence proton (particle) Triaxial rotor (collective) Three Spin Vectors PRL84 (2000), Dimitrov, Frauendorf, Donau

11. Angular momentum projection Norm overlap kernels NPA 80 (1966), Onishi and Yoshida

12. The Onishi formula Square root of determinant NPA 80 (1966), Onishi and Yoshida

13. Nodal lines A collection of singular points PLB 606 (2005), M.Oi and Tajima

14. A Pfaffian formula of Robledo Pfaffian: polynomials Robledo, PRC 79 (2009)

15. Another Pfaffian formula for odd-A systems Chirality needs a particle and a hole M. Oi and Mizusaki, PLB 707 (2012)

16. A Pfaffian formula of Oi and Mizusaki Several unfamiliar bipartite matrices… M. Oi and Mizusaki, PLB 707 (2012)

17. Meaning of X -1 Antisymmetrised density matrix (with a bipartite form) Mizusaki and M.OI, arXiv:1204.6531(2012)

18. Extended Pfaffian formula of Mizusaki and Oi Mizusaki and M.OI, arXiv:1204.6531(2012)

19. “Lewis Carroll” formula for Pfaffian Avez and Bender, PRC 85 (2012) Mizusaki and M.OI, arXiv:1204.6531(2012)

20. Summary Nuclear excited states through angular momentum Structural transitions to exotic shape and 3D rotation Quantization of classical / semi-classical 3D models New formulae were discovered Ready for numerical analysis!

23. University of Tokyo LB ２０４５： NaI scintillator