Transcript Document

Electromagnetic Properties of
Nuclear Chiral Partners
The Master Equation
For triaxial odd-odd nuclei
Chirality =
Energy [MeV]
Nilsson model +
E [MeV]
6
4
2
0
-2
-4
-6
0.0
0.1


irrotational flow
moment of inertia
0.2
0.3
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
Nuclear single-particle
shell model states.
HSM =
 
V(r) +VLS (r) L  S
112
N=5
126
70
N=4
82
40
N=3
50
20
N=2
20
8
N=1
8
2
N=0
Spher. Harm.
Oscillator
2
+L2
 
+L  S
h11/2
Unique parity h11/2 state in quadrupole-deformed
triaxial potential.
Triaxial shape for
 = 0.3, g = 30º.
H= HSM+ Hdef
Hdef= k [ cos(g)Y20(q,f)+
1/2sin (g){Y22(q,f)+ Y2-2(q,f)}]
HSM = js =0.00
 s =1.36
 ji =0.00  i =2.01
 jl =5.46  l =0.30
6
Energy [MeV]
4
2
0
-2
-4
-6
0.0
0.1
0.2

0.3
 js =5.46  s =0.30
 ji =0.00  i =2.01
 jl =0.00  l =1.36
Semi classical analysis for single-particle Nilsson
hamiltonian in a triaxial nucleus.
E - EF = k ( jx2 - jy2)
j2=jx2+jy2+jz2
E < EF
E > EF
5
5
0
0
5
-5
-5
0
-5
5
-5
0
0
0
-5
5
5
-5
Collective nuclear rotation
resembles that of irrotational liquid but is
different than that of a rigid body. In particular
moments of inertia differ significantly.
irrotational
liquid
rigid
body
laboratory
intrinsic
Angular momentum for rotating triaxial body with irrotational
flow moment of inertia aligns along intermediate axis.
25
Jl
Ji
20
J [2 /MeV]
J[ħ /MeV]
Js
2
J s = 4 B 2 sin 2 (g  120)
15
Js
10
Jl
5
J l = 4 B sin (g )
2
2
0
0
10
20
30
g
40
50
60
Triaxial odd-odd nuclei result in three
perpendicular angular momenta for particlehole configurations built on high-j orbitals .
Results of the Gammasphere GS2K009 experiment.
band 2
134Pr
ph11/2 nh11/2
band 1
Systematics of partner bands in odd-odd A~130 nuclei.
5
Energy [MeV]
4
3
138Eu
2
1
0
5
4
136Pm
3
2
1
0
5
4
3
132Pr
134Pr
130La
132La
134La
128Cs
130Cs
132Cs
2
Energy [MeV]
1
0
5
4
3
2
1
0
5
4
3
2
1
0
8
10
12
14
16
18
20
8
Spin [ħ]
10
12
14
16
Spin [ħ]
18
20
8
10
12
14
16
18
20
Chirality is a general phenomenon in triaxial nuclei:
• two mass regions identified up to date,
• partner bands in odd-odd and odd-A nuclei.
General electromagnetic properties of chiral partners.
| IR
| IL
jn
long
jp
long
short
 IL | E 2 | IR  0
 IL | M 1 | IR  0
jn
B( EM ; I i   I f )  B( EM ; I i   I f )
B( EM ; I i   I f )  B( EM ; I i   I f )
R
Int
Int
1
| I  =
(| IL  | IR )
2
i
| I  =
(| IL  | IR )
2
R
I+2
I+1
I
| I 
| I 
General particle plus triaxial rotor model
H = Vsp + Hrot


1
Vsp (,g,q,f) = k(r)  cos gY 20 (q , f ) 
sin g (Y 22(q , f )  Y 2  2 (q , f )) 
2


2 2
=  Rk
k =1 2 J k
3
Hrot
Moment of inertia:




R = I  j p  jn
4
2p
2
J k = J 0 sin (g  k ) k =1,2,3
3
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
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)
for g=60º (oblate shapes)
Jl=Ji=J0 Js=0
Ji
20
2
J [2/MeV]
/MeV]
J[ħ
Js=Ji=J0 Jl=0
25
15
Js
10
Jl
5
triaxiality
for g=30º (triaxial shapes)
Jl=Js=J0 Ji=4J0.
0
0
10
20
30
g
40
50
60
Symmetric rotor with a triaxial shape at g=90 o
•l2<l3<l1, but J1=J2=1/4J3 , Q20=0, Q22 =Q2-2 ~ at g=90
o
2 2
2
2
=
Rk 
( R3  4( R12  R22 ))
8J 0
k =1 2 J k
3
H rot
•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
1
g=90
2
3
3
2
1
Calculated Level Scheme
B1
A1
A2
B2
Energy vs Spin: two pairs of degenerate bands
Calculated B(M1) and B(E2)
Particle-rotor Hamiltonian for triaxial odd-odd nuclei
(
)
p
n
H g = 90 = H rot  Vsp  Vsp
Core
H rot

2
2
=
( R3  4( R12  R22 ))
8J 0
Single proton-particle in j (=h11/2 ) shell
Vspp  ( jp21  jp22 )
Single neutron-particle in j (=h11/2 ) shell
Vspn  ( jn22  jn21 )
Quantum Number A: invariance properties of H=H rot+V p+V
n
•D2 symmetry → R3 = 0,±2,±4,±6,…..
•Invariant under the operation A consisting of
→ Rotation
p 
 p  or R  3p  = exp  i 3p R 
R3   = exp  i R3 
3
3


2
2
 


2
 2 
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
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
( M 1)
=
 ( g l  g R ) l

3

N 
eff
  ( g s  g R ) s 
4p


gl+gR =0.5 (-0.5)
gseff-gR=2.848 (-2.792) for p (n)
Electromagnetic properties of chiral partners with
A symmetry
C | IL | IR R(q ) | IL | IL
C | IR | IL R(q ) | IR | IR
A = R (q )  C
where
p 3p
q= ,
2 2
A | I  = 1 | I   A | I  = 1 | I 
+1
-1
-1
+1
-1
+1
+1
-1
+1
-1
I+4
I+3
I+2
I+1
I
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.
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
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) 044319 ].
Electromagnetic properties – unexpected B(M1)/B(E2) behavior
for
134Pr
and heavier N=75 isotones.
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
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.
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 128Cs and 130La DSAM results
E. Groedner, J. Srebrny et. al.
Institute of Experimental Physics
Warsaw University, Poland