Candidate multiple chiral doublet bands in A 100 mass region

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1 Candidate multiple chiral doublet bands in A 100 mass region Bin Qi (R) School of Space Science and Physics, Shandong University, Weihai Seventh International Symposium on Chiral Symmetry in Hadrons and Nuclei Beijing

2 Outline Introduction 1 Introduction 2 3

3 Chirality in Atomic Nuclei theoretical prediction of chirality in nuclear structure S. Frauendorf and J. Meng, Nucl. Phys.A617, 131 (1997). Chiral symmetry breaking in the body-fixed frame L & R Restore the symmetry in lab frame: chiral doublet bands Chiral vibration and static Chirality

4 Chirality in Atomic Nuclei theoretical prediction of chirality in nuclear structure S. Frauendorf and J. Meng, Nucl. Phys.A617, 131 (1997). Chiral symmetry breaking in the body-fixed frame L & R Restore the symmetry in lab frame: chiral doublet bands Chiral vibration and static Chirality more than 30 candidate chiral nuclei have been reported odd-odd nuclei A 100 A 130 A 190 A 80 πg 1 9/2 νh 11/2; 102,104,106 Rh, 104,106 Ag, 98,100 Tc πh 11/2 νh 1 11/2 ; Cs, La, 132,134 Pr, 136 Pm, 138 Eu πh 9/2 νi 1 13/2 ; 194,198 Tl πg 9/2 νg 1 9/2 ; 78,80 Br

5 Chirality in Atomic Nuclei theoretical prediction of chirality in nuclear structure S. Frauendorf and J. Meng, Nucl. Phys.A617, 131 (1997). Chiral symmetry breaking in the body-fixed frame L & R Restore the symmetry in lab frame: chiral doublet bands Chiral vibration and static Chirality more than 30 candidate chiral nuclei have been reported odd-odd nuclei A 100 A 130 A 190 A 80 πg 1 9/2 νh 11/2; 102,104,106 Rh, 104,106 Ag, 98,100 Tc πh 11/2 νh 1 11/2 ; Cs, La, 132,134 Pr, 136 Pm, 138 Eu πh 9/2 νi 1 13/2 ; 194,198 Tl πg 9/2 νg 1 9/2 ; 78,80 Br odd-mass nuclei, 103,105 Rh 107 Ag 135 Nd, 133 Ce; even-even nuclei, 106 Mo, 108,110 Ru, 136 Nd

6 Multiple chiral doublet bands (MχD) In 2006, based on the adiabatic and configuration-fixed constrained triaxial relativistic mean field (RMF) theory calculation, triaxial shape coexistence with high-j proton-hole and neutron-particle configurations was found in odd-odd nuclei 106 Rh, which demonstrates the possibility of having multiple chiral doublet bands (acronym MχD). J. Meng, J. Peng, S. Q. Zhang, and S. G. Zhou, Phys. Rev. C 73, (2006).

7 Experimental evidence for MχD 1 In experiment, a pair of candidate chiral bands in 105 Rh with πg 1 9/2 νh2 11/2 configuration, and another pair with tentatively suggested πg 1 9/2 νh 11/2(d 5/2 /g 7/2 ) configuration were respectively reported in J. A. Alcántara-Núñez et al., Phys. Rev. C 69, (2004). J.Tim ar et al.,phys. Lett. B 598, 178 (2004). 2 Very recently, two distinct sets of chiral double bands have been identified in the odd-mass nucleus 133 Ce and are regarded as strong experimental evidence for the existence of MχD. A.D.Ayangeakaa et al., Phys. Rev. Lett. 110, (2013). 3 The observation of MχD represents important confirmation of triaxial shape coexistence and its geometrical interpretation. 4 Is there any other experimental evidence of MχD except these two nuclei?

8 Possible πg 9/2-1 υh 11/2 2 πg 9/2-1 υh 11/2 (d 5/2 g 7/2 ) Band 1&2: D. Jerrestam, Nucl. Phys. A (1994). Band 3&4: B. Zhang, Chin. Phys. C (2011).

9 Motivation of this work 1 The mechanism of these reported near-degenerate doublet bands in 107 Ag were not given in the References D. Jerrestam, Nucl. Phys. A (1994). B. Zhang, Chin. Phys. C (2011). 2 It is interesting to study whether two such pairs of doublet bands in 107 Ag are associated with the nuclear chirality and further to verify whether or not the observations in 107 Ag are MχD. 3 For this purpose, we study two pairs of doublet bands in 107 Ag via the triaxial RMF theory and multiparticle plus rotor model (PRM).

10 Configuration and deformation of 107 Ag in RMF The detailed formulism of RMF theory can be found in Refs. J. Meng, Peng, Zhang, and Zhou, Phys. Rev. C 73, (2006). J. Peng, Sagawa, Zhang, Yao, Zhang, and Meng, PRC (2008). J. M. Yao, Qi, Zhang, Peng, Wang, and Meng, Phys. Rev. C 79, (2009). In the present calculations, each Dirac spinor is expanded in terms of a set of three-dimensional harmonic oscillator bases in Cartesian coordinates with 12 major shells and meson fields with 10 major shells. The pairing correlations are neglected here. The effective interaction parameter set PK1 is applied. W. Long, J. Meng, N. Van Giai, and S. G. Zhou, Phys. Rev. C 69, (2004).

11 Configuration and deformation of 107 Ag in RMF A g F A : (0.2 0,1 5.1 o ) B : (0.1 7,0.7 7 o ) C : (0.3 0,2 4.1 o ) D : (0.3 8,7.7 0 ) E : (0.2 2,2 8.1 o ) F : (0.2 3, ) E [M e V ] E C D B A b The energy surfaces in adiabatic (open circles) and configuration-fixed (solid lines) constrained triaxial RMF calculations for 107 Ag. The minima in the energy surfaces are labeled as A, B, C, D, E, and F. Their corresponding triaxial deformation parameters β and γ are also given.

12 Configuration and deformation of 107 Ag in RMF N e u tr o n P ro to n d 3 /2 1 g 7 /2-5 2 d 5 /2-5 1 h 1 1 /2 2 p 1 /2 e S.P. [M e V ] g 7 /2 1 g 9 /2 A E F e S.P. [M e V ] g 9 /2 2 p 3 /2 1 f 5 /2 A E F 2 p 1 / p 3 / f 7 /2 1 f 5 / b b Neutron and proton single-particle levels obtained in constrained triaxial RMF calculations with PK1 as functions of deformation β. Positive (negative) parity states are marked by solid (dashed) lines.

13 Configuration and deformation of 107 Ag in RMF Table: The total energies E tot, triaxial deformation parameters β and γ, and their corresponding valence nucleon configurations of minima for states AõF in the configuration-fixed constrained triaxial RMF calculations, and compared with the experimental excitation energies. Configuration E tot E x (cal.)e x (exp.) State Valence nucleons Unpaired nucleons (MeV) (β, γ) (MeV) (MeV) A π(g 2 9/2 p1 2 ) υ(g 1/2 7/2 d4 5/2 ) πp1 1/ (0.20,15.1) 0 0 B πg 3 2 υ(g 9/2 7/2 d4 ) 5/2 1 πg9/ (0.17,0.8 ) C π(g 2 9/2 g 7/2 1 2 ) υ(g7/2 5/2 h2 11/2 ) πg 7/ (0.30,24.1) D π(g7/2 2 g 3 4 ) υ(g 9/2 7/2 d2 5/2 h4 ) 11/2 1 πg9/ (0.38,7.7 ) E πg 1 9/2 υ(h1 11/2 d3 5/2 g 2 ) 7/2 1 πg9/2 11/2 d1 ) (0.22,28.1) 5/ F πg 1 9/2 υ(h1 11/2 h1 11/2 d2 5/2 g 2 1 ) πg 7/2 9/2 υ(h1 11/2 h1 ) (0.23,27.2) 11/ the excitation energy of band head I π = 19/2 of band 3 the excitation energy of I π = 23/2 + of band 1, where backbending occurs.

14 Multiple chiral doublet bands (MχD) It represents two possible types of chiral three-quasiparticle configurations in the odd-mass nuclei. 1 One is formed by a high-j hole and an aligned pair of high-j particles;

15 Multiple chiral doublet bands (MχD) It represents two possible types of chiral three-quasiparticle configurations in the odd-mass nuclei. 1 One is formed by a high-j hole and an aligned pair of high-j particles; 2 the second occurs when a low-j particle (which acts as a spectator) is coupled to the neighboring odd-odd chiral configuration.

16 Calculate the rotational excitation via PRM The detailed formulism of multiparticle plus rotor model can be seen in B. Qi, S. Q. Zhang, J. Meng, and S. Frauendorf, Phys. Lett. B 675, 175 (2009). B. Qi, S. Q. Zhang, S. Y. Wang, J. Meng, and T. Koike, Phys. Rev. C 83, (2011). Parameters for Band 1 and 2 1 πg 1 9/2 νh2 11/2 with deformation β = 0.23, γ = 27.2 (form RMF) 2 J 0 = 25.0MeV / 2 (adjusted to the experimental energy spectra) Parameters for Band 3 and 4 1 πg 1 9/2 ν(h1 11/2 d 3 5/2 ) with β = 0.22,γ = 28.1 (form RMF). 2 J 0 = 20MeV / 2 (adjusted to the experimental energy spectra) Parameters to calculate electromagnetic transition 1 the intrinsic quadrupole moment Q 0 = (3/ 5π)R 2 0 Zβ,R 0 = 1.2A 1/3 fm 2 gyromagnetic ratios g p(n) = g l + (g s g l )/(2l + 1) with g s = 0.6g s(free), g R = Z/A.

17 Reproduce the energy spectra of band 1 and 2 via PRM 9 E n e r g y [M e V ] B a n d 1 (P R M ) B a n d 2 (P R M ) B a n d 1 (E x p.) B a n d 2 (E x p.) S p in [ ] The excitation energies calculated by PRM with configuration πg 1 9/2 νh2 11/2 for the doublet bands, in comparison with the corresponding data of the bands 1, 2 in 107 Ag.

18 Energy crossing of band 1 and 2 in PRM B (E 2 )[e 2 b 2 ] A g lo w e s t b a n d (P R M ) 1 s t e x c ite d b a n d (P R M ) lo w e s t to 1 s t (P R M ) 1 s t to lo w e s t (P R M ) B (E 2 )[e 2 b 2 ] A g B a n d 1 (P R M ) B a n d 2 (P R M ) B a n d 1 to B a n d 2 (P R M ) B a n d 2 to B a n d 1 (P R M ) B (M 1 ) [m 2 N ] B (M 1 )[m 2 N ] S p in [ ] S p in [ ] The calculated partner bands in PRM with πg 1 9/2 νh2 11/2 conf. Left panel: lowest band and 1st excited band. Right panels: Arrange band based on in-band E 2 transition

19 Reproduce the energy spectra of band 1 and 2 via PRM 9 E n e r g y [M e V ] B a n d 1 (P R M ) B a n d 2 (P R M ) B a n d 1 (E x p.) B a n d 2 (E x p.) S p in [ ] The excitation energies calculated by PRM with configuration πg 1 9/2 νh2 11/2 for the doublet bands, in comparison with the corresponding data of the bands 1, 2 in 107 Ag.

20 chiral geometry Introduction A n g u la r M o m e n tu m [ ] B a n d 1 R i B a n d i l s l s J p l (g 9 /2 ) -1 i s i s l (h 1 1 /2 ) 2 s S p in [ ] The rms components of the angular momenta calculated as functions of spin by PRM for the positive parity doublet bands with configurationπg 1 9/2 νh2 11/2 in 107 Ag. J n i s i l l R k = ˆR k 2 J pk = ĵpk 2 J nk = (ĵ (n1)k + ĵ (n2)k ) 2 intermediate (i-), short (s-,) and long (l-) axis

21 Probability distributions for projection of I B a n d 1 B a n d 2 l-a x is i-a x is s -a x is I= 4 5 /2 I= 4 1 /2 I= 3 9 /2 I= 3 7 /2 I= 3 3 /2 P K I= 2 9 / K l K i K s The probability distributions for projection of total angular momentum on the long (l-), intermediate (i-) and short (s-) axis in PRM for band 1 & 2 in 107 Ag.

22 How to understand the evolution of chiral mode? chiral vibration = static chirality l I P o te n tia l Ψ i s -9 0 o Ψ 9 0 o

23 How to understand the evolution of chiral mode? V (0 )= 0 V (0 )= 5 V (0 )= E 4 0 V (0 )= 2 0 V (0 )= 4 0 V (0 )= E 1 -E O 9 0 O -9 0 O Y 9 0 O -9 0 O 9 0 O V (0 ) Extract the potential barrier between chiral doublets in particle rotor model, Bin Qi et al. in progress.

24 Probability distributions for projection of I B a n d 1 B a n d 2 l-a x is i-a x is s -a x is I= 4 5 /2 I= 4 1 /2 I= 3 9 /2 I= 3 7 /2 I= 3 3 /2 P K I= 2 9 / K l K i K s The probability distributions for projection of total angular momentum on the long (l-), intermediate (i-) and short (s-) axis in PRM for band 1 & 2 in 107 Ag.

25 Reproduce the energy spectra of band 3 and 4 via PRM left: πg 1 9/2 ν(h1 11/2 d 1 5/2 ) A g B a n d 3 (P R M ) B a n d 4 (P R M ) B a n d 4 (E x p.) B a n d 4 (E x p.) right:πg 1 9/2 ν(h1 11/2 d 3 5/2 ) 6 5 B a n d 3 (P R M ) B a n d 4 (P R M ) B a n d 3 (E x p.) B a n d 4 (E x p.) E [M e V ] 4 3 E n e r g y [M e V ] S p in [ ] S p in [ ] The excitation energies calculated by PRM for the doublet bands, in comparison with the data of the bands 3, 4 in 107 Ag.

26 Chiral geometry Introduction B a n d 3 R i J p (g 9 /2 ) -1 J n (h 1 1 /2 ) 1 J n (d 5 /2 ) 3 8 A n g u la r M o m e n tu m [ ] B a n d 4 2 l i s s l i s i s l l S p in [ ] Same as Band1,2, but for the negative parity doublet bands with configuration πg 1 9/2 νh1 11/2 d 5/2 3 in 107 Ag. neutron in d 5/2 sub-shell act as spectators i i l s l s s s i i l l

27 Probability distributions for projection of I B a n d 3 B a n d 4 l-a x is i-a x is s -a x is I= 3 1 /2 I= 2 9 /2 I= 2 7 /2 I= 2 5 /2 I= 2 3 /2 P K I= 2 1 / K l K i K s The probability distributions for projection of total angular momentum on the long (l-), intermediate (i-) and short (s-) axis in PRM for band 3 & 4 in 107 Ag.

28 Conclusion Introduction 1 Two pairs of nearly degenerate doublet bands in 107 Ag are studied by RMF theory and PRM. 2 The triaxial deformations favorable for the construction of the chiral doublet bands with the suggested configurations πg 1 9/2 νh2 11/2 and πg 1 9/2 νh 11/2d 5/2 are obtained from the configuration-fixed constrained triaxial RMF calculations. 3 Adopting the PRM, the data are reproduced excellently for the two pairs of doublet bands, even the energy crossing of bands 1 and 2 is obtained self-consistently. The chiral geometry of the aplanar rotation is further conformed by analyzing the angular momentum components. 4 Thus we suggest two pairs of doublet bands in 107 Ag as two distinct sets of chiral doublet bands, which might be more evidence of MχD after the observed candidate MχD in 105 Rh and 133 Ce B. Qi, H. Jia, N. B. Zhang, C. Liu, S. Y. Wang, Phys. Rev. C, 88, (2013).

29 in A 100 mass region reported chiral nuclei in A 100 mass region: Proton number In Cd Ag Pd Rh Ru Tc Mo 98 Tc Ag Ag 105 Ag Ag Ag A~ Ag Rh Rh Rh Rh Rh 100 Tc 106 Mo Neutron number 110 Ru 112 Ru Ag: S.Ray,PRC(2008) Z.G.Wang,PRC(2013) J.Timár,PRC(2007) Joshi,PRL(2007) B.Zhang,Chin. Phys.C(2011) C.Liu,PRC (2013). Rh:J.Gizon,NPA(1999) J.Timár,PLB(2004) C.Vaman,PRL,(2004) J.Timár,PRC(2006) Joshi,PLB(2004) Ru:Y.X.Luo,PLB(2009). Tc:Joshi,Eur.Phys.J(2005). Mo:S.J.Zhu,Eur.Phys.J.A(2005) open question: MχD in odd-a isotopes of Rh and Ag; in odd-odd nuclei?

30 in A 100 mass region A g A g 7 A g 7 p g 9 /2 u h 1 1 /2 (g 7 /2 d 5 /2 ) p g 9 /2 u h /2 p g 9 /2 u h /2 E (M e V ) p g 9 /2 u h 1 1 /2 (g 7 /2 d 5 /2 ) p g 9 /2 u h 1 1 /2 (g 7 /2 d 5 /2 ) A g A g A g 5 p g 9 /2 u h 1 1 /2 4 E (M e V ) p g 9 /2 u (d 5 /2 g 7 /2 ) p g 9 /2 u h 1 1 /2 p g 9 /2 u h 1 1 / S p in [ ] expect the partner band of 105 Ag; possible MχD in 104 Ag, triplet states in 103 Ag, in progress

31 in A 100 mass region E (M e V ) A g A B A :(0.1 9,0.0 3 o ) p g -3 9 /2 u g 1 7 / b C D B :(0.2 2,0.0 3 o ) p g -3 9 /2 u h /2 E (M e V ) A g A B C E :(0.2 3,3 4.9 o ) p g -1 9 /2 u h /2 (g 7 /2 d 5 /2 ) b F :(0.2 4,1 7.9 o ) p g -1 9 /2 u h /2 h /2 The energy surfaces in triaxial RMF calculations for 104 Ag and 105 Ag. E F D

32 Acknowledgement 1 Peking Univ.: J. Meng, S.Q. Zhang, P.W. Zhao, Q.B. Chen, H.Hua, X.Q. Li, C. Xu 2 Shandong Univ.: S. Y. Wang, C. Liu, H. Jia, N. B. Zhang, P. Zhang, L. Liu 3 Jilin Univ.: J. Li 4 CIAE: X. G. Wu, C. Y. He 5 Stellenbosch Univ.: S. M. Wyngaardt, R. Newman, P. Papka, 6 ithemba Labs: R. Bark, E. Lawrie, J.J. Lawrie, S. Mullins, J. F. Sharpey-Schafer, S. Majola, L.P. Masitery, P. Datta

33 Introduction Thank you for your attentionœ œ

34 føã5 Introduction B (M 1 )/B (E 2 ) [m 2 N /e 2 b 2 ] 1 0 B a n d 1 (P R M ) B a n d 2 (P R M ) B a n d 1 (E x p.) B a n d 2 (E x p.) S p in [ ]