1. ShuangQuan Zhang (sqzhang@pku.edu.cn) School of Physics, Peking University Static chirality and chiral vibration of atomic nucleus in particle rotor model 17th Nuclear Physics Workshop “Marie & Pierre Curie” in Kazimierz 2010-09-24 Collaborators: B. Qi, S.Y. Wang, J. Meng, S.G. Frauendof

2. Content 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Introduction——Chirality in atomic nucleus Theory——Particle Rotor Model Results –Quantitative description of chiral bands by PRM ( 126,128 Cs, 135 Nd, 106 Rh, 103,105 Rh) –Chiral geometry from PRM (Static chirality; chiral vibration) –An analysis of chiral doublet states with an orientation operator Summary

3. Chirality in Nature Chirality exists commonly in nature. Left- Right- 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

4. Chirality in Atomic Nucleus The rotation of triaxial nuclei can present chiral geometry. There are three perpendicular angular momenta: Collective triaxial rotor R ， Particle-like valence proton j p ， Hole-like valence neutron j n  the total angular momentum J is aplanar. Frauendorf, Meng, Nucl. Phys. A 617,131(1997 ) 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

5. Chiral doublet bands Expected exp. signal ： Two near degenerate  I =1 bands, called chiral doublet bands S.Frauendorf and J.Meng, Nucl. Phys. A617, 131(1997) Intrinsic frame Lab. frame: restoration of symmetry breaking +1 +1 I+4 I+3 I+2 I+1 I 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

6. Claimed chiral nuclei Candidate chiral doublet bands have been claimed in many odd- odd and odd-A nuclei with different configurations in A~80, 100,130,190 mass regions. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

7. Theoretical tools for nuclear chirality Tilted axis cranking – Single-j model Frauendorf and Meng NPA(1997); – Hybird Woods-Saxon and Nilsson model Dimitrov et al PRL(2000) – Skyrme Hartree-Fock model Olbratowski et al PRL(2004), PRC(2006) – Relativistic mean field (RMF) theory Madokoro et al PRC(2000); Peng et al PRC (2008) – TAC+RPA (135Nd) S. Mukhopadhyay et al PRL2007; Particle Core Coupling  Triaxial Particle Rotor Model Frauendorf and Meng NPA(1997); Peng et al PRC(2003); Koike et al PRL(2004), SQZ et.al PRC(2007); Lawrie et al PRC (2008); Qi et al PLB(2009) – Core-quasiparticle coupling model, which follows the KKDF method Starosta et al PRC(2002); Koike et al PRC(2003) – Interacting Boson Fermion Fermion Model (IBFFM) S. Brant et al PRC (2004), PRC (2008), Tonev et al PRL(2006) – Pair Truncated Shell Model K. Higashiyama et al, PRC(2005)  In this talk, the particle rotor model is adopted.

8. Particle Rotor Model The model Hamiltonian: the collective part, the intrinsic part,  We have extended such model for triaxial nuclei with 2-qp and many particle configuration based on single-j model. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

9. Observation in 128 Cs  h 11/2 1  h 11/2 -1

10. Observation in 126 Cs 2010-09-24 17th Nuclear Physics Workshop in Kazimierz  h 11/2 1  h 11/2 -1

11. Observation in 126 Cs 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Electromagnetic properties in Cs isotopes S.Y. Wang et al. PRC 74, 017302 (2006)  h 11/2 1  h 11/2 -1

12. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz PRM description of 126,128 Cs S.Y. Wang, SQZ, B. Qi, J. Meng. PRC75, 024309 (2007)  h 11/2 1  h 11/2 -1

13. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz PRM description of 126,128 Cs Data From: E. Grodner, J. Srebrny et al.  h 11/2 1  h 11/2 -1

14. Observation in 106 Rh 2010-09-24 17th Nuclear Physics Workshop in Kazimierz  g 9/2 -1  h 11/2 1

15. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz PRM description of 106 Rh S.Y. Wang, SQZ, B. Qi, J. Peng, J.M. Yao, J. Meng. PRC77, 034314 (2008)  g 9/2 -1  h 11/2 1

16. Observation in 135 Nd S. Mukhopadhyay et al. PRL (2007) S. Zhu et al. PRL (2003) 2010-09-24 17th Nuclear Physics Workshop in Kazimierz  h 11/2 2  h 11/2 -1

17. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz PRM description of 135 Nd B(M1) & B(E2) E(I) Both energies and transition ratios are well reproduced! β= 0.235 and γ= 22.4 ◦ B.Qi, SQZ, J. Meng, S.Y. Wang, S. Frauendorf. Phys. Lett. B(2009)  h 11/2 2  h 11/2 -1,

18. Observation in 103 Rh 2010-09-24 17th Nuclear Physics Workshop in Kazimierz  h 9/2 -1  h 11/2 2

19. Observation in 105 Rh 2010-09-24 17th Nuclear Physics Workshop in Kazimierz  h 9/2 -1  h 11/2 2

20. PRM description of 103,105 Rh 2010-09-24 17th Nuclear Physics Workshop in Kazimierz B.Qi, SQZ, S.Y. Wang, J. Meng,T. Koike. in preparation.  h 9/2 -1  h 11/2 2

21. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Chiral Geometry in 135 Nd Components of angular momenta

22. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Chiral Geometry in 135 Nd Length and Orientation of angular momenta Static chiral geometry are well developed around I~39/2 !

23. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Distribution of AM Projection Chiral vibration

24. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Distribution of AM Projection Static Chirality

25. Chirality evolution Chiral vibration (I=29/2)  Static chirality (I=39/2)  Chiral vibration (I=45/2) 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

26. An Analysis of Chiral Doublet States with Orientation Operator A Naive Question ： For chiral doublet bands, which state is | L  ? 2010-09-24 17th Nuclear Physics Workshop in Kazimierz - Not correct Naive Question becomes: Which state is | +  ? Which is |   ? Intrinsic frame Lab. frame

27. An Analysis of Chiral Doublet States with Orientation Operator 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Possible Answer is : To judge it from the sign of Orientation parameter?

28. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz An Analysis of Chiral Doublet States with Orientation Operator Before the calculation, one must constrain the phase of wave functions in lab. frame, because the sign of  L|  |L  will be changed accordingly if one change the sign of |+  or |  ? Constraint of the phase of |+  or |  by: 1. For same spin I with different variable  : 2. For different spin I: “reduced E2 transition matrix at axial symmetry case” “Parallel transport principle”

29. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz An Analysis of Chiral Doublet States with Orientation Operator Results:  1p1h PRM,  h   h   =0.23, J=20MeV -1  2 – Picture of three perpendicular angular momenta can be approximately realized. (same as: K. Starosta et.al., NPA 682(2001)357c ) – In the yrast (or yrare) band of chiral doublet bands, the states are the same |+  or |  state, linear combined by | L  and | R . – Such order of |+  or |  state is different from the states with A quantum numbers, discussed by Koike, et al., PRL 93, 172502 (2004).

30. Summary Quantitative description have been carried out by PRM for doublet bands, in odd-odd and odd-A nuclei, in A~100 and 130 mass region, and with different quasiparticle configurations. Static chirality and chiral vibration are shown in the framework of PRM, which have been discussed before in the framework of TAC with RPA. An analysis of chiral doublet states with orientation operator is preformed. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

31. Thank you for your attention! 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

32. PRM description of 126,128 Cs 2010-09-24 17th Nuclear Physics Workshop in Kazimierz

33. Static Chirality and Strong B(M1) Staggering Static: Strong B(M1) Staggering Vibration: Weak/No B(M1) Staggering Static: Strong B(M1) Staggering Vibration: Weak/No B(M1) Staggering

34. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Chiral Vibration and Weak B(M1) Staggering Static: Strong B(M1) Staggering Vibration: Weak B(M1) Staggering Static: Strong B(M1) Staggering Vibration: Weak B(M1) Staggering

36. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Selection Rules of …

37. 2010-09-24 17th Nuclear Physics Workshop in Kazimierz Fingerprints ideal chiral bands 1. nearly degenerate doublet bands 2. S(I) independent of spin 3. staggering of B(M1)/B(E2) ratios 5. identical spin alignments 4. identical B(M1), B(E2) values 6. interband B(E2)=0 at high spin Koike et al., PRL. 93, 172502 (2004) Vaman et al., PRL.92 032501 (2004) Petrache et al., PRL.96, 112502 (2006)