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