Isovector scenario for nuclei near the N=Z line

Isovector scenario for nuclei near the N=Z line

Anatoli Afanasjev

S. Frauendorf

Kai Neergard

J. Sheikh

01

10

TS

TS

Mean-field theory of isovector pairing

Mean-field calculations in the A=74 region

Isocranking and RPA

Mean-field theory of isovector pairing

S. F., J Sheikh, NPA 645 (1999) 509

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

Symmetry restoration –Isorotations (strong symmetry breaking)

Bayman, Bes, Broglia PRL 23 (1969) 1299 ( 2 particle transfer)

2

)1(')T E(T,:energy nalisorotatio

|)0,,( :state nalisorotatio

| :state intrinsic

z

0

TTTH

D

z

TTz

1,1 zTT 0,1 zTT0T

The relative strengths of pp, nn, and pn pairing are determined by theisospin symmetry

Moment of inertia for isorotation

||,||)('

:gisocrankin

zTEh

A

MeVxTTTE

75

2

1,

2

)()(

:energysymmetry alexperiment from

exp

Ground states

10 20 30 40 50 60 A

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

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

Model study: half-filled deformed j-shell

4,3 shell,

ˆ)('

2/7

2120

ZNf

JTAGYH xz rr

Full shell model diagonalization

HFB solution: no isoscalar pair field, only isovector

Quasiparticle routhians

4ˆˆ ZN

Mean-field kept at its value at .0

Mean-field calculations in the A=74 region

D. Jenkins, et al. PRC 65 (2002) 064307

C.D. O’Leary et al. PRC 67 (2003) 021301(R)

N. S. Kelsall et al. PRC65 (2002) 044331

N. S. Kelsall et al. Proc. Berkeley 2002, AIP Conf. Proc. 656 (2002) 269

Br70

Kr72

Kr73

Rb74

Realistic calculations

Cranked Relativistic Mean-Field

Pairing: Gogny force, HFB, Lipkin-Nogami

Isocranking: experimental symmetry energy

Quasiparticles around N=Z=36

Tconf ][

2/1][A2/1][F2/1][E

2/1][B

2/1][A

2/1][E

2/1][F

Calculation TRS, R. Wyss

2/3]312[eE

02

)(

bAaB

0][eE

1]0[

Rb74

0][aA

Tconf ][

Rb74

full: experimentdashed: CRMF

1]0[

0][eE

02

)(

bAaB

0][aA

Rb74

full: experimentdashed: CRMF

1]0[

0][eE

02

)(

bAaB 0][aA

no pairing

x

IJ

2/1)1(

Isocranking and RPA

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