Electromagnetic Properties of Electromagnetic Properties of Nuclear Chiral PartnersNuclear Chiral Partners

The Master EquationThe Master Equation

For triaxial odd-odd nucleiFor triaxial odd-odd nuclei

Chirality = Chirality =

Nilsson model +Nilsson model +

irrotational flow irrotational flow moment of inertiamoment of inertia

0.0 0.1 0.2 0.3-6

-4

-2

0

2

4

6

En

erg

y [M

eV]

E [

MeV

]

Valence nucleons behave as gyroscopes.Valence nucleons behave as gyroscopes.

•Pairing interactions couple single particle states to Cooper pairs with Pairing interactions couple single particle states to Cooper pairs with no net angular momentum.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 The properties of valence nucleons can be derived from the Nilsson modelmodel

Nuclear single-Nuclear single-particle shell model particle shell model states.states.

HSM =

V(r) +VLS (r) L

S

Spher. Harm. Oscillator +L2 +L S

h11/2

HSM =

0.0 0.1 0.2 0.3-6

-4

-2

0

2

4

6

En

erg

y [M

eV]

Triaxial shape for = 0.3, = 30º.

js =0.00 s =1.36 ji =0.00 i =2.01jl =5.46 l =0.30

js =5.46 s =0.30 ji =0.00 i =2.01jl =0.00 l =1.36

Unique parity hUnique parity h11/2 11/2 state in quadrupole-state in quadrupole-

deformed triaxial potential.deformed triaxial potential.

H= HSM+ Hdef

Hdef= kcos(Y20+

1/2sin (Y22+ Y2-2

Semi classical analysis for single-particle Semi classical analysis for single-particle

Nilsson hamiltonian in a triaxial nucleusNilsson hamiltonian in a triaxial nucleus..

j2=jx2+jy

2+jz2 E - EF = ( jx

2 - jy2)

-5

0

5

-5

0

5-5

0

5

-5

0

5

-5

0

5 -5

0

5-5

0

5

-5

0

5

E < EFE > EF

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

laboratory intrinsic

irrotationalliquid

rigidbody

Collective nuclear rotationCollective nuclear rotation

Angular momentum for rotating triaxial body with Angular momentum for rotating triaxial body with

irrotational flow moment of inertia aligns along intermediate irrotational flow moment of inertia aligns along intermediate

axis.axis.

0 10 20 30 40 50 60

0

5

10

15

20

25

Jl

Js

Ji

J [

2 /MeV

]

)120(sin4 22 BJs

)(sin4 22 BJ l

sJ lJ

J[ħ2 /

MeV

]

Triaxial odd-odd nuclei result in three Triaxial odd-odd nuclei result in three perpendicular angular momenta for perpendicular angular momenta for

particle-hole configurations built on high-particle-hole configurations built on high-j orbitals .j orbitals .

Results of the Gammasphere GS2K009 experiment.Results of the Gammasphere GS2K009 experiment.

band 2 band 1134Pr

h11/2 h11/2

Spin [ħ]

0

1

2

3

4

5

8 10 12 14 16 18 20

8 10 12 14 16 18 20

8 10 12 14 16 18 200

1

2

3

4

5

0

1

2

3

4

5

0

1

2

3

4

5

134Pr

136Pm

138Eu

132La

130Cs

132Pr

130La

128Cs

134La

132Cs

Ene

rgy

[MeV

]

Ene

rgy

[MeV

]

0

1

2

3

4

5

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

Spin [ħ]

Chirality is a general phenomenon in triaxial nuclei: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.

General electromagnetic properties of chiral General electromagnetic properties of chiral partners.partners.

)|(|2

|

)|(|2

1|

IRILi

I

IRILI

0|1|

0|2|

IRMIL

IREIL

);();(

);();(

fifi

fifi

IIEMBIIEMB

IIEMBIIEMB

long

Int

short

jj

R

long j

RInt

IR|IL|

I+1

I+2

I

I| I|

H = Vsp + Hrot

Vsp ()

Hrot

Moment of inertia: k =1,2,3

Model for odd-odd nuclei follows the model developed for odd-A Model for odd-odd nuclei follows the model developed for odd-A nuclei by J. Meyer-ter-Vehn in Nucl. Phys. A249 (1975) 111nuclei by J. Meyer-ter-Vehn in Nucl. Phys. A249 (1975) 111

General particle plus triaxial rotor modelGeneral particle plus triaxial rotor model

23

1

2

2 kk k

RJ

)),(),((sin

2

1),(cosk(r) 2222 20 YYY

)3

2(sin

34 2

0 kJJk

R I j j

For irrotational flow moment of inertia there are two special For irrotational flow moment of inertia there are two special cases for which two out of three moments are equal:cases for which two out of three moments are equal:

axial symmetryaxial symmetry for for =0=0ºº (prolate shapes)(prolate shapes)

JJss=J=Jii=J=J0 0 JJll=0=0 for for =60=60ºº (oblate shapes)(oblate shapes)

JJll=J=Jii=J=J0 0 JJss=0=0

triaxialitytriaxiality

forfor =30=30ºº (triaxial shapes)(triaxial shapes)

JJll=J=Jss=J=J0 0 JJii=4J=4J0.0.

0 10 20 30 40 50 60

0

5

10

15

20

25

Jl

Js

Ji

J [

2 /MeV

]

J[ħ2 /

MeV

]

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

•ll22<l<l33<l<l11, but J, but J11=J=J22=1/4J=1/4J33 , Q , Q2020=0, Q=0, Q2222 =Q =Q2-22-2 ~ ~ at at =90=90 oo

•Intermediate axis is an effective symmetry axis of the core, Intermediate axis is an effective symmetry axis of the core,

a good choice for the quantization axis.a good choice for the quantization axis.•Core rotation orients along the intermediate axis to minimizeCore rotation orients along the intermediate axis to minimize

the rotational energy.the rotational energy.

Symmetric rotor with a triaxial shape at Symmetric rotor with a triaxial shape at =90 =90 oo

))(4(82

22

21

23

0

22

3

1

2

RRRJ

RJ

H kk k

rot

Calculated Level SchemeCalculated Level Scheme

A1 A2

B2B1

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

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

spsprot VVHH 90

))(4(8

22

21

23

0

2

RRRJ

H rot CoreCore

Single proton-particle in j (=hSingle proton-particle in j (=h11/2 11/2 ) shell) shell

Single neutron-particle in j (=hSingle neutron-particle in j (=h11/2 11/2 ) shell) shell

)( 22

21

jjVsp

)( 21

22

jjVsp

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

•D2 symmetry → RD2 symmetry → R33 = 0,±2,±4,±6,….. = 0,±2,±4,±6,…..

•Invariant under the operation A consisting of Invariant under the operation A consisting of

→ Rotation orRotation or

RR33((/2) [1→2,2→-1,3→3], R/2) [1→2,2→-1,3→3], R33(3(3/2) [1→-2,2→-1,3→3]/2) [1→-2,2→-1,3→3]

→Exchange symmetry between valence proton and Exchange symmetry between valence proton and neutron neutron

C: C: ↔↔

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

3 3exp2 2

R i R

3 3

3 3exp

2 2R i R

C= +1 symmetricC= +1 symmetric

C= -1 anti-symmetricC= -1 anti-symmetric

Quantum number A and selection rules for transition Quantum number A and selection rules for transition ratesrates

[H,A]=0[H,A]=0

AA22=1=1

Quantum number A=±1Quantum number A=±1

A=+1 A=+1

RR33=0,±4,±8,… & C=+1=0,±4,±8,… & C=+1

RR33=±2,±6,±10 …& C=-1=±2,±6,±10 …& C=-1

A=-1A=-1

RR33=0,±4,±8,… & C=-1=0,±4,±8,… & C=-1

RR33=±2,±6,±10 …& C=+1=±2,±6,±10 …& C=+1

B(E2;IB(E2;Iii→I→If f ))≠≠0 for A0 for Aii ≠≠ A Aff

Core contribution only Core contribution only ⇔ ⇔ ΔΔC=0C=0

QQ2020=0 for =0 for γγ=90º=90º

[B(M1;I[B(M1;Iii→I→If f ) with A) with Aii≠A≠Af f ] >>] >>

[B(M1;I[B(M1;Iii→I→If f ) with A) with Aii=A=Af f ] ]

||ΔΔRR33 | |≤1≤1

B(M1;IB(M1;Iii→I→If f ) ≈0 for C) ≈0 for Cii=C=Cff

due to the isovector character due to the isovector character

of M1 operatorof M1 operator

ggll+g+gR R ==0.50.5 (-0.5) (-0.5)

ggsseffeff-g-gRR==2.8482.848 (-2.792)(-2.792) for for (())

31

4

l R

N effs R

g g lM

g g s

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

ILIRC

IRILC

||

||

IRIRR

ILILR

||

||

A R C 2

3,

2

IIAIIA |1||1|

wherewhere

+1

-1

-1

+1

+1

+1

+1

-1

-1

-1

I+4

I+3

I+2

I+1

I

• near degenerate doublet near degenerate doublet II==11 bands bands

for a range of spin for a range of spin I I ;;

• S(I)=[E(I)-E(I-1)]/2I S(I)=[E(I)-E(I-1)]/2I independent of spin independent of spin II;;

• chiral symmetry restoration selection chiral symmetry restoration selection

rules for M1 and E2 transitions vs. spin rules for M1 and E2 transitions vs. spin

resulting in staggering of the absolute resulting in staggering of the absolute

and relative transition strengths.and relative transition strengths.

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

Based on the above fingerprints Based on the above fingerprints 104104Rh provides the best Rh provides the best

example of chiral bands observed up to date.example of chiral bands observed up to date.

doubling of statesdoubling of states

S(I) S(I) independent ofindependent of II

B(M1),B(M1), B(E2)B(E2) staggeringstaggering

C. Vaman et al. PRL C. Vaman et al. PRL 92(2004)03250192(2004)032501

Electromagnetic properties – pronounced staggering in Electromagnetic properties – pronounced staggering in

experimental B(M1)/B(E2) and B(M1)experimental B(M1)/B(E2) and B(M1)in in / B(M1)/ B(M1)out out ratios as a ratios as a

function of spin [function of spin [T.Koike et al. PRC 67 (2003) 044319 ].T.Koike et al. PRC 67 (2003) 044319 ].

Electromagnetic properties – unexpected B(M1)/B(E2) Electromagnetic properties – unexpected B(M1)/B(E2)

behavior for behavior for 134134Pr and heavier Pr and heavier N=75 isotones. N=75 isotones.

Absolute transition rates measurements in A~130 Absolute transition rates measurements in A~130 nucleinuclei

J. Srebrny et al, Acta Phys. Polonica B46(2005)1063J. Srebrny et al, Acta Phys. Polonica B46(2005)1063

E. Grodner et al, Int. J. Mod. Phys. E14(2005) 347E. Grodner et al, Int. J. Mod. Phys. E14(2005) 347

Conclusions and futureConclusions and future

•Electromagnetic properties of nuclear chiral partners in Electromagnetic properties of nuclear chiral partners in triaxial odd-odd nuclei have been identified from a symmetry triaxial odd-odd nuclei have been identified from a symmetry of a particle-rotor Hamiltonian.of a particle-rotor Hamiltonian.

•A simple ( but limited ) model has been developed which A simple ( but limited ) model has been developed which describes uniquely triaxial features with a new quantum describes uniquely triaxial features with a new quantum number A:number A:

→Chiral doublet bands,Chiral doublet bands,→Selection rules for electromagnetic transitions,Selection rules for electromagnetic transitions,→Chiral wobbling mode.Chiral wobbling mode.

•Model predictions are not consistent with the experimental Model predictions are not consistent with the experimental absolute transition rate measurements reported in the mass absolute transition rate measurements reported in the mass 130 region.130 region.

•Absolute lifetime measurements are of crucial importance Absolute lifetime measurements are of crucial importance for chiral partner identification and investigation of doublet for chiral partner identification and investigation of doublet bands in odd-odd nuclei.bands in odd-odd nuclei.

CreditsCredits

T. KoikeT. KoikeTohoku University, Sendai, JapanTohoku University, Sendai, Japan

I. HamamotoI. HamamotoLTH, University of Lund, SwedenLTH, University of Lund, Sweden

and NBI, Copenhagen, Denmarkand NBI, Copenhagen, Denmark

C.VamanC.VamanNational Superconducting Cyclotron LaboratoryNational Superconducting Cyclotron Laboratory

Michigan State University, USAMichigan State University, USA

for for 128128Cs and Cs and 130130La DSAM resultsLa DSAM results

E. Groedner, J. Srebrny et. al.E. Groedner, J. Srebrny et. al.Institute of Experimental PhysicsInstitute of Experimental Physics

Warsaw University, PolandWarsaw University, Poland