1. Electromagnetic Properties of
Nuclear Chiral Partners

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

3. 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

4. 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 

5. 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

6. 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

7. 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

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

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

10. Results of the Gammasphere GS2K009 experiment.
band 2
band 1
134Pr
ph11/2 nh11/2

11. 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.

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

13. General electromagnetic properties of chiral partners.
long
Int
short jp jn R
I+1
I+2 I

14. 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.

15. 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]

16. 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

17. Calculated Level SchemeB2 B1

18. Energy vs Spin: two pairs of degenerate bands

19. Calculated B(M1) and B(E2)

20. 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

21. 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

22. 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)

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

24. 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.

25. 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

26. 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) ].

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

28. 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

29. 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.

30. 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