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Isovector pairing in nuclei near the N=Z line
Anatoli Afanasjev
S. Frauendorf
Kai Neergard
J. Sheikh
Mean-field theory of isovector pairingFrauendorf SG, Sheikh JACranked shell model and isospin symmetry near N=Z NUCLEAR PHYSICS A 645 (4): 509-535 JAN 25 1999
Simple model: deformed potential+monopole isovector pairing
x
z
ipipiipipniniinini
i ipiippiinniip
iniin
xz
J
ZNT
ZNA
cccccccch
ccPccccPccP
JTAGhH
:projection momentumangular
)ˆˆ(2
1 :projection isospin
ˆˆˆ :number particle
)(
:potential deformed
)(2
1
:pairingisovector
ˆ'
101
PP
Mean-field approximation
:fieldpair
ˆ)('
||'0'
| :state Boguljubov
P
PP
G
JTAhh
EhH
xzmf
mf
a
a
aa
a
xz
xz
V
UE
V
U
JTAh
JTAh
ˆ)(
)(ˆ
PP
PP
Spontaneous breaking of isospin symmetry
Mean field does not have these symmetries.
0ˆ,'0,'0,' 2 AHTHTH z
0ˆ,'0,'0,' 2 AhThTh mfmfzmf
Degenerate mf-solutions: gauge angle
constEHe Ai |'|,||ˆ
02
ˆ
0
ˆ
np
ppnn
np
ppnn
y
z
constEH
HTz
,|'|,
.equivalent are of directions All
isospace. in rotations all to
respect withinvariant is ' then0 i.e. 0 If
constEH
HTz
|'|
.equivalent are planey -x thein of directions All
plane.y -x thein rotations to
respect only withinvariant is ' then0 i.e. 0 If
plane.y -x thein are solutions mf The
0! e. i. ,ˆ chose always can We npy
Intrinsic excitation spectrum
0,ˆ,ˆˆ,0 , npppnnZN y
0',,0',ˆˆ mfZi
mfNi hehe Parities of proton and neutron
numbers are good.
Symmetries
0,,0,however ,0',ˆˆ Zi
yNi
ymfy eTeThT
Symmetry restoration –Isorotations (strong symmetry breaking)
Bayman, Bes, Broglia PRL 23 (1969) 1299 ( 2 particle transfer)
2
)1(' :energy nalisorotatio
|)0,,( :state nalisorotatio
| :state intrinsic
0
TTTH)E(T,T
D
zz
TTz
Organize into bands with even or odd N
1,1 zTT 0,1 zTT0T
The relative strengths of pp, nn, and pn pairing are determined by theisospin symmetry
Comparison with shell model calculation
SM calculation: J. Engel et al. PLB 389, 211 (96)
101011111111
011 ][
SSNSSNSSN
ccS
npppnn
JTTj
jjT zz
pnppnnz
pnppnnz
pnppnnz
NNNTT
NNNTT
NNNTT
2113
101
00
Strong symmetry breaking
pnppnn
pnppnn
pnppnn
NNN
NNN
NNN
08.21.3
1
Shell model
Isorotation
||,||)('
:gisocrankin
zTEh
2/1
,2
)1()(
:breakingsymmetry strong -tion requantizi
T
dT
dETTTEisorot
2/1|| :generalin TTz
bands with even or odd N
0 2 4 6 8 10-9000
-8800
-8600
-8400
-8200
-8000
A=72
E
[keV
]
T
Ee72 Eo72
N even
N odd
bands with even or odd N
Remove Coulomb energy!
TAaAaTTA
TAa
A
ZaE cccz
zccc
3/13/23/1
2
3/1
2
2,,)2/(
MeVac 741.0MeVac 662.0
T
-1 0 1 2 3 4 5 6 7 8 9 10
-14000-12000-10000
-8000-6000-4000-2000
02000400060008000
10000
N odd
N even
A=72
c=12461keV exp 11125keV
=
(E(T
+1)
-E(T
-1))
/2 [
keV
]
T
Very regularBand!
60 62 64 66 68 70 72 74 76 789.8
10.010.210.410.610.811.011.211.411.611.812.012.212.412.612.813.013.2
c[M
eV]
A
mcex mc mcfit
TAaAaTTA
TAa
A
ZaE cccz
zccc
3/13/23/1
2
3/1
2
2,,)2/(
MeVac 662.0
MeVac 741.0
2
)2/1()2/1( zz TETE
58 60 62 64 66 68 70 72 74 76 780.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
T
A
Odd A
)( cTT ][MeV 48.0fit overall
0.38 72
0.38 68
0.28 60
][MeV
1
1
A
Wigner Energy:Manifestationof spontaneous isospin symmetrybreaking
T=0 and ½ states
...... ,0 even-even 0
0 even-even 0
0 odd-odd 0
2/1 neutron odd 0
2/1 proton odd 0
0 even-even 0|
T|ββ
T|ββ
T|ββ
T|β
T | β
T
jpip
jnin
jnip
in
ip
inip onsquasineutr nsquasiproto
Restrictions due to the symmetry yT
States with good N, Z –parity are in general no eigenstates of .yT
If they are (T=0) the symmetry restricts the possible configurations, if not (T=1/2) the symmetry does not lead to anything new.
0|:0 yTT
00|)(2
1
00|)(2
1
00|)(2
1
00|)(2
1
00|
00|
jnipjpiny
jnipjpiny
jpipjniny
jpipjniny
inipy
y
T
T
T
T
T
T
Excitation energy of first T=1 state
10 20 30 40 50 60 A
MeVMeV./θ
/θ)E()E(
/θΔ)E(E
TETE
2.1621
o-o 11 20
e-e 121 0)0(
)0()1(10
0|inip
Adding nn pairs to the condensate does not change the structure.
Pair rotational bands are an evidence for the presence of a pair field.
Ordinary nn pair field
60 65 70 75 80
-2000
0
2000
4000
6000
8000
10000
E
-cA
[keV
]
A
E0 E16
60 64 68 72 76 80-10000
-9000
-8000
-7000
-6000
-5000
=(E
(A+
2)-E
(A-2
)/4
A
dEee dEoo
Isoscalar pairing at high spin?
Isoscalar pairs carry finite angular momentum
iJ z 2
total angular momentum
•A. L. GoodmanPhys. Rev. C 63, 044325 (2001)
Predicted by
Which evidence?
Symmetries of the isoscalar pair field
Frauendorf S, Sheikh JASymmetry breaking by proton-neutron pairing PHYSICA SCRIPTA T88: 162-169 2000
which symmetries leave ATHH z ' invariant?
1 )( Aig e R Either even or odd A belong to the band.
1 )( Nin e R Even and odd N belong to the band.
1 )( zJiz e R Both signatures belong to the band.
nNI
e ig
nzg
2
gaugeplex ||
1)()(
S
RRS
If the isoscalar pair field is present,
nii
pii ccpP0
iJ z 2
total angular momentum
Pair rotational bands for an isoscalar neutron-proton pair field
ZNA 22
E
Even-even, even I Odd-odd, odd I
60 65 70 75 80
-2000
0
2000
4000
6000
8000
10000
E-c
A[k
eV]
A
E0 E16
2))17()15(()16(
:odd-odd
EEE
Only quenching ofisovector pairingor evidence for isoscalar pairfield?
Conclusions
• Ground state energies explained by strong isovector pair field
• Very regular isorotational bands• Wigner energy: T(T+1) dependence• Excitation spectra explained by isovector
pair field that is quenched at high spin.• Maybe isoscalar correlations enhanced at
highspin.
Symmetry restoration by RPA
Kai Neergard, PLB 537 (2002) 287, ArXiv nucl-th
zTAGhH ˆ
22TPP
Too small symmetry energyWith realistic level spacing
Ensures the right symmetryEnergy by choice of
2)(
2)0(
2
2)(
ˆˆˆ22
0
22
0
T
GTE
T
GhTE
TTG z
zzTyP
Mean-field approximation
RPA correlation energy
RPA roots 2qp energies
jiji eeTE )(
2
1)(2
jiji eeTE )(
2
1
2)(
12
THT , T T+1/2
T
TdT
dE )(0
ji
ji eeTTEETETE1
220 )(
2)0()(
2)1(
)(
TT
TE
),(),(
22),(),()(
11
22
2
jijiE
ZNjijiTE
jiex
ZNotherv ji
T2
G-
2
T
G
)02.1(033.02
TTMeV