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**Electromagnetic Properties of**

Nuclear Chiral Partners

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**For triaxial odd-odd nuclei**

The Master Equation For triaxial odd-odd nuclei Chirality = Nilsson model + irrotational flow moment of inertia b E [MeV]

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**Valence nucleons behave as gyroscopes.**

Pairing interactions couple single particle states to Cooper pairs with no net angular momentum. Valence odd nucleons are unpaired. The properties of valence nucleons can be derived from the Nilsson model

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**HSM = V(r) +VLS (r) L S +L2 +L S**

Nuclear single-particle shell model states. HSM = V(r) +VLS (r) L S 2 8 20 40 70 112 50 82 126 Spher. Harm. Oscillator +L2 +L S N=0 N=1 N=2 N=3 N=4 N=5 h11/2

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**Unique parity h11/2 state in quadrupole-deformed triaxial potential.**

HSM = Unique parity h11/2 state in quadrupole-deformed triaxial potential. H= HSM+ Hdef Triaxial shape for b = 0.3, g = 30º. Hdef= kb [ cos(g)Y20(q,f)+ 1/2sin (g){Y22(q,f)+ Y2-2(q,f)}] js = s =1.36 ji = i =2.01 jl =5.46 l =0.30 js =5.46 s =0.30 ji = i =2.01 jl = l =1.36

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**j2=jx2+jy2+jz2 E - EF = k ( jx2 - jy2) E < EF E > EF**

Semi classical analysis for single-particle Nilsson hamiltonian in a triaxial nucleus. j2=jx2+jy2+jz2 E - EF = k ( jx2 - jy2) E < EF E > EF

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**Collective nuclear rotation**

resembles that of irrotational liquid but is different than that of a rigid body. In particular moments of inertia differ significantly. laboratory intrinsic irrotational liquid rigid body

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Angular momentum for rotating triaxial body with irrotational flow moment of inertia aligns along intermediate axis. J[ħ2/MeV]

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Triaxial odd-odd nuclei result in three perpendicular angular momenta for particle-hole configurations built on high-j orbitals .

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**Results of the Gammasphere GS2K009 experiment.**

band 2 band 1 134Pr ph11/2 nh11/2

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**Systematics of partner bands in odd-odd A~130 nuclei.**

Spin [ħ] 134Pr 136Pm 138Eu 132La 130Cs 132Pr 130La 128Cs 134La 132Cs Energy [MeV] Systematics of partner bands in odd-odd A~130 nuclei.

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**Chirality is a general phenomenon in triaxial nuclei:**

two mass regions identified up to date, partner bands in odd-odd and odd-A nuclei.

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**General electromagnetic properties of chiral partners.**

long Int short jp jn R I+1 I+2 I

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**General particle plus triaxial rotor model**

H = Vsp + Hrot Vsp (b,g,q,f) Hrot Moment of inertia: k =1,2,3 Model for odd-odd nuclei follows the model developed for odd-A nuclei by J. Meyer-ter-Vehn in Nucl. Phys. A249 (1975) 111 The model discussion. The starting point is rather general. The total hamiltonian consists of single particle part and rotor part. The single particle term is quadrupole deformed mean field with its shape parameterized by beta and gamma. The rotor contribution can be expressed with moment of inertia and the core rotation which is the total minus the single particle angular momenta. For the moment of inertia we consider an irrotational-flow type. The formalism developed for odd-A nuclei is followed in the current study for odd-odd case, and details can be referred to the paper by Meyer-Ter-Vehn.

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**A useful limit of the particle rotor model for triaxial nuclei**

For irrotational flow moment of inertia there are two special cases for which two out of three moments are equal: axial symmetry for g=0º (prolate shapes) Js=Ji=J0 Jl=0 for g=60º (oblate shapes) Jl=Ji=J0 Js=0 triaxiality for g=30º (triaxial shapes) Jl=Js=J0 Ji=4J0. J[ħ2/MeV]

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**Symmetric rotor with a triaxial shape at g=90 o**

l2<l3<l1, but J1=J2=1/4J3 , Q20=0, Q22 =Q2-2 ~b at g=90 o Intermediate axis is an effective symmetry axis of the core, a good choice for the quantization axis. Core rotation orients along the intermediate axis to minimize the rotational energy. g=30 3 1 2 g=90

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**Calculated Level Scheme**

B2 B1

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**Energy vs Spin: two pairs of degenerate bands**

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**Calculated B(M1) and B(E2)**

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**Particle-rotor Hamiltonian for triaxial odd-odd nuclei**

Core Single proton-particle in j (=h11/2 ) shell Single neutron-particle in j (=h11/2 ) shell

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**Quantum Number A: invariance properties of H=Hrot+V p+V n**

D2 symmetry → R3 = 0,±2,±4,±6,….. Invariant under the operation A consisting of Rotation or R3(p/2) [1→2,2→-1,3→3], R3(3p/2) [1→-2,2→-1,3→3] Exchange symmetry between valence proton and neutron C: p↔n C= +1 symmetric C= -1 anti-symmetric

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**Quantum number A and selection rules for transition rates**

[H,A]=0 A2=1 Quantum number A=±1 A=+1 R3=0,±4,±8,… & C=+1 R3=±2,±6,±10 …& C=-1 A=-1 R3=0,±4,±8,… & C=-1 R3=±2,±6,±10 …& C=+1 B(E2;Ii→If )≠0 for Ai ≠ Af Core contribution only ⇔ ΔC=0 Q20=0 for γ=90º [B(M1;Ii→If ) with Ai≠Af ] >> [B(M1;Ii→If ) with Ai=Af ] |ΔR3 |≤1 B(M1;Ii→If ) ≈0 for Ci=Cf due to the isovector character of M1 operator gl+gR =0.5 (-0.5) gseff-gR=2.848 (-2.792) for p (n)

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**Electromagnetic properties of chiral partners with A symmetry**

where +1 -1 I+4 I+3 I+2 I+1 I

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**Chiral fingerprints in triaxial odd-odd nuclei:**

near degenerate doublet D I=1 bands for a range of spin I ; S(I)=[E(I)-E(I-1)]/2I independent of spin I; chiral symmetry restoration selection rules for M1 and E2 transitions vs. spin resulting in staggering of the absolute and relative transition strengths.

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**Based on the above fingerprints 104Rh provides the best example of chiral bands observed up to date.**

doubling of states S(I) independent of I B(M1), B(E2) staggering C. Vaman et al. PRL 92(2004)032501

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Electromagnetic properties – pronounced staggering in experimental B(M1)/B(E2) and B(M1)in / B(M1)out ratios as a function of spin [T.Koike et al. PRC 67 (2003) ].

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**Electromagnetic properties – unexpected B(M1)/B(E2) behavior for 134Pr and heavier N=75 isotones.**

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**Absolute transition rates measurements in A~130 nuclei**

J. Srebrny et al, Acta Phys. Polonica B46(2005)1063 E. Grodner et al, Int. J. Mod. Phys. E14(2005) 347

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**Conclusions and future**

Electromagnetic properties of nuclear chiral partners in triaxial odd-odd nuclei have been identified from a symmetry of a particle-rotor Hamiltonian. A simple ( but limited ) model has been developed which describes uniquely triaxial features with a new quantum number A: Chiral doublet bands, Selection rules for electromagnetic transitions, Chiral wobbling mode. Model predictions are not consistent with the experimental absolute transition rate measurements reported in the mass 130 region. Absolute lifetime measurements are of crucial importance for chiral partner identification and investigation of doublet bands in odd-odd nuclei.

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**Credits T. Koike I. Hamamoto C.Vaman for 128Cs and 130La DSAM results**

Tohoku University, Sendai, Japan I. Hamamoto LTH, University of Lund, Sweden and NBI, Copenhagen, Denmark C.Vaman National Superconducting Cyclotron Laboratory Michigan State University, USA for 128Cs and 130La DSAM results E. Groedner, J. Srebrny et. al. Institute of Experimental Physics Warsaw University, Poland

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原子核配对壳模型的相关研究 Yanan Luo( 罗延安 ), Lei Li( 李磊 ) School of Physics, Nankai University, Tianjin Yu Zhang( 张宇 ), Feng Pan( 潘峰 ) Department of Physics, Liaoning.

原子核配对壳模型的相关研究 Yanan Luo( 罗延安 ), Lei Li( 李磊 ) School of Physics, Nankai University, Tianjin Yu Zhang( 张宇 ), Feng Pan( 潘峰 ) Department of Physics, Liaoning.

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