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Electromagnetic Properties of Nuclear Chiral Partners

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The Master Equation For triaxial odd-odd nuclei Chirality = Nilsson model + irrotational flow 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.Pairing interactions couple single particle states to Cooper pairs with no net angular momentum. Valence odd nucleons are unpaired.Valence odd nucleons are unpaired. The properties of valence nucleons can be derived from the Nilsson modelThe properties of valence nucleons can be derived from the Nilsson model

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Nuclear single-particle shell model states. H SM = V(r) +V LS (r) L S Spher. Harm. Oscillator +L 2 +L S h 11/2

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H SM = Triaxial shape for = 0.3, = 30º. j s =0.00 s =1.36 j i =0.00 i =2.01 j l = 5.46 l =0.30 j s = 5.46 s =0.30 j i =0.00 i =2.01 j l =0.00 l =1.36 Unique parity h 11/2 state in quadrupole-deformed triaxial potential. H= H SM + H def H def = k cos( Y 20 + 1/ 2sin ( Y 22 + Y 2-2

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Semi classical analysis for single-particle Nilsson hamiltonian in a triaxial nucleus. j 2 =j x 2 +j y 2 +j z 2 E - E F = ( j x 2 - j y 2 ) E < E F E > E F

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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 Collective nuclear rotation

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Angular momentum for rotating triaxial body with irrotational flow moment of inertia aligns along intermediate axis. J[ħ 2 /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 Pr h 11/2 h 11/2

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

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Chirality is a general phenomenon in triaxial nuclei: two mass regions identified up to date, two mass regions identified up to date, partner bands in odd-odd and odd-A nuclei. partner bands in odd-odd and odd-A nuclei.

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General electromagnetic properties of chiral partners. long Int short jj j R long j R Int I+1 I+2 I

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H = V sp + H rot V sp ( ) H rot 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 General particle plus triaxial rotor model

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For irrotational flow moment of inertia there are two special cases for which two out of three moments are equal: axial symmetry for =0º (prolate shapes) for =0º (prolate shapes) J s =J i =J 0 J l =0 J s =J i =J 0 J l =0 for =60º (oblate shapes) for =60º (oblate shapes) J l =J i =J 0 J s =0 J l =J i =J 0 J s =0triaxiality for =30º (triaxial shapes) for =30º (triaxial shapes) J l =J s =J 0 J i =4J 0. J l =J s =J 0 J i =4J 0. J[ħ 2 /MeV] A useful limit of the particle rotor model for triaxial nuclei

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l 2

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Calculated Level Scheme A1 A2 B2B1

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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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Core Single proton-particle in j (=h 11/2 ) shell Single neutron-particle in j (=h 11/2 ) shell Particle-rotor Hamiltonian for triaxial odd-odd nuclei

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

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Quantum number A and selection rules for transition rates [H,A]=0 A 2 =1 Quantum number A=±1 A=+1 R 3 =0,±4,±8,… & C=+1 R 3 =±2,±6,±10 …& C=-1 A=-1 R 3 =0,±4,±8,… & C=-1 R 3 =±2,±6,±10 …& C=+1 B(E2;I i →I f )≠0 for A i ≠ A f Core contribution only ⇔ ΔC=0 Q 20 =0 for γ=90º [B(M1;I i →I f ) with A i ≠A f ] >> [B(M1;I i →I f ) with A i =A f ] |ΔR 3 |≤1 B(M1;I i →I f ) ≈0 for C i =C f due to the isovector character of M1 operator g l +g R =0.5 (-0.5) g s eff -g R =2.848 (-2.792) for p ( n )

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Electromagnetic properties of chiral partners with A symmetry where where I+4 I+3 I+2 I+1 I

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

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Based on the above fingerprints 104 Rh provides the best example of chiral bands observed up to date. doubling of states doubling of states S(I) independent of I S(I) independent of I B(M1), B(E2) staggering 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 134 Pr 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.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: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.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.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 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 128 Cs and 130 La DSAM results E. Groedner, J. Srebrny et. al. Institute of Experimental Physics Warsaw University, Poland

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