Isovector scenario for nuclei near the N=Z line
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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
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