11
明 孝之 大阪工業大学 阪大 RCNP
Tensor optimized shell model using bare interaction
for light nuclei
共同研究者 土岐 博 阪大 RCNP 池田 清美 理研
RCNP 研究会「少数粒子系物理の現状と今後の展望」 @RCNP 2008.12.23-25
Outline
2
Tensor Optimized Shell Model (TOSM)
Unitary Correlation Operator Method (UCOM)
TOSM + UCOM with bare interaction
• Application of TOSM to Li isotopes
• Halo formation of 11Li
• TM, K.Kato, H.Toki, K.Ikeda, PRC76(2007)024305
• TM, K.Kato, K.Ikeda, PRC76(2007)054309• TM, Sugimoto, Kato, Toki, Ikeda, PTP117(2007)257• TM. Y.Kikuchi, K.Kato, H.Toki, K.Ikeda, PTP119(2008)561• TM, H. Toki, K. Ikeda, Submited to PTP
3
Motivation for tensor force
tensor centralV V
• Structures of light nuclei with bare interaction
tensor correlation + short-range correlation
• Tensor force (Vtensor) plays a significant role in the nuclear structure.
– In 4He,
– ~ 80% (GFMC)V
VNN
R.B. Wiringa, S.C. Pieper, J. Carlson, V.R. Pandharipande, PRC62(2001)
• We would like to understand the role of Vtensor in the nuclear structure by describing tensor correlation explicitly.
model wave function (shell model and cluster model) He, Li isotopes (LS splitting, halo formation, level inversion)
4
Tensor & Short-range correlations
4
• Tensor correlation in TOSM (long and intermediate)
–
– 2p2h mixing optimizing the particle states (radial & high-L)
12 2 1 2 2 0ˆ( ),[ , ] 2S Y L Sr
• Short-range correlation– Short-range repulsion
in the bare NN force
– Unitary Correlation
Operator Method (UCOM)
S
D
H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61
T. Neff, H. Feldmeier, NPA713(2003)311
5
Property of the tensor force
5
Long and intermediate ranges
• Centrifugal potential ([email protected]) pushes away the L=2 wave function.
6
Tensor-optimized shell model (TOSM)
6
• Tensor correlation in the shell model type approach.
• Configuration mixingwithin 2p2h excitationswith high-L orbit TM et al., PTP113(2005) TM et al., PTP117(2007) T.Terasawa, PTP22(’59))
• Length parameters such as are determined independently and variationally.
– Describe high momentum component from Vtensor CPP-HF by Sugimoto et al,(NPA740) / Akaishi (NPA738)CPP-RMF by Ogawa et al.(PRC73), CPP-AMD by Dote et al.(PTP115)
{ }b 0 0 1/2, ,...s pb b
4He
TM, Sugimoto, Kato, Toki, IkedaPTP117(2007)257
7
Hamiltonian and variational equations in TOSM
0k
H E
C
7
1
,A A
i G iji i j
H t T v
: central+tensor+LS+Coulombijv
k kk
C : shell model type configurationk
0H
0,H E
b
• Effective interaction : Akaishi force (NPA738)
– G-matrix from AV8’ with kQ=2.8 fm-1
– Long and intermediate ranges of Vtensor survive.
– Adjust Vcentral to reproduce B.E. and radius of 4He
TM, Sugimoto, Kato, Toki, Ikeda, PTP117(’07)257
8
4He in TOSM
Orbit bparticle/bhole
0p1/2 0.65
0p3/2 0.58
1s1/2 0.63
0d3/2 0.58
0d5/2 0.53
0f5/2 0.66
0f7/2 0.55
8
Length parameters
good convergence Higher shell effect 16 Lmax
Shrink
0 1 2 3 4 5 6
vnn: G-matrix
Cf. K. Shimizu, M. Ichimura and A. Arima, NPA226(1973)282.( ) in q V
9
Configuration of 4He in TOSM
(0s1/2)4 85.0 %
(0s1/2)2JT(0p1/2)2
JT JT=10 5.0
JT=01 0.3
(0s1/2)210(1s1/2)(0d3/2)10 2.4
(0s1/2)210(0p3/2)(0f5/2)10 2.0
P[D] 9.69
Energy (MeV) 28.0
51.0tensorV
• 0 of pion nature.
• deuteron correlation with (J,T)=(1,0)
4 Gaussians instead of HO
central
71.2 MeV
V 48.6 MeV
T
c.m. excitation = 0.6 MeV
Cf. R.Schiavilla et al. (GFMC) PRL98(’07)132501
10
Tensor & Short-range correlations
10
• Tensor correlation in TOSM (long and intermediate)
–
– 2p2h mixing optimizing the particle states (radial & high-L)
12 2 1 2 2 0ˆ( ),[ , ] 2S Y L Sr
• Short-range correlation– Short-range repulsion
in the bare NN force
– Unitary Correlation
Operator Method (UCOM)
S
D
H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61
T. Neff, H. Feldmeier NPA713(2003)311
TOSM+UCOM
11
Unitary Correlation Operator Method
11H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61
corr. uncorr.C
12
( ) ( )exp( ), r ij ij rij iji j
p s r s r pC i g g
short-range correlator
† H E C HC H E
(
( )(
)( ))
R rR r
s
s r
† 1 (Unitary trans.)C C
rp p p
Bare HamiltonianShift operator depending on the relative distance r
( ) ( )R r r s r
TOSM
2-body cluster expansionof Hamiltonian
12
Short-range correlator : C (or Cr)
: ( ) for Hamiltonian C r R r †
†
( )C r C R r
C lC l
AV8’ : Central+LS+Tensor
3GeV repulsion Originalr2
C
†C sC s
Vc
1E
3E
1O3O
14
4He with AV8’ in TOSM+UCOM
Kamada et al.PRC64 (Jacobi)
AV8’ : Central+LS+Tensor
exact
• Gaussian expansion for particle states (6 Gaussians)• Two-body cluster expansion of Hamiltonian
15
Extension of UCOM : S-wave UCOM
, ( )SsC R r
15
for only relative S-wave wave function
– minimal effect of UCOM†
†
†
central central
LS LS
rel relrel reltensor tensor
for s-wave
for s-wave
N
ˆ( ) ( )
( ) ( )
( ) ( )
o change
S D S S D
s s
s s
s s
C TC T T
C V r C V r
C V r C V r
V r V rC
SD coupling
tensorV 5 MeV gain
16
Different effects of correlation function
16
S
D
rel 0 at 0S r
rel 0 at 0D r due to Centrifugal Barrier
• D-wave
• S-wave
No Centrifugal Barrier
Short-range repulsion
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Saturation of 4He in UCOM
17
T. Neff, H. Feldmeier
NPA713(2003)311Short tensor
Long tensor
TOSM+UCOM
Benchmark cal. Kamada et al. PRC64
UCOM: short-range + tensor
corr. uncorr.rC C
En
erg
y
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4He in TOSM + S-wave UCOM
T
VT
VLS
VC
E
(exact)Kamada et al.PRC64 (Jacobi)
Remaining effect :
3-body cluster term
in UCOM
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Summary• Tensor and short-range correlations
– Tensor-optimized shell model (TOSM)
• He & Li isotopes (LS splitting, Halo formation)
– Unitary Correlation Operator Method (UCOM)
• Extended UCOM : S-wave UCOM
• In TOSM+UCOM, we can study the nuclear structure starting from the bare interaction.
– Spectroscopy of light nuclei (p-shell, sd-shell)
19
2020
Pion exchange interaction vs. Vtensor
Delta interaction
Yukawa interaction
Involve largemomentum
2 2 2
1 2 1 2 122 2 2 2 2 2
2 2 2 2
1 2 122 2 2 2 2 2
ˆ ˆ3( )( ) ( )
( )
q q qq q S
m q m q m q
m q m qS
m q m q m q
12 1 2 1 2ˆ ˆ3( )( ) ( )S q q
Tensor operator
- Vtensor produces the high momentum component.
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Characteristics of Li-isotopes
21
Breaking of magicity N=8• 10-11Li, 11-12Be• 11Li … (1s)2 ~ 50%. (Expt by Simon et al.,PRL83)
• Mechanism is unclear
Halo structure
11Li
2222Pairing-blocking : K.Kato,T.Yamada,K.Ikeda,PTP101(‘99)119, Masui,S.Aoyama,TM,K.Kato,K.Ikeda,NPA673('00)207. TM,S.Aoyama,K.Kato,K.Ikeda,PTP108('02)133, H.Sagawa,B.A.Brown,H.Esbensen,PLB309('93)1.
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11Li in coupled 9Li+n+n model• System is solved based on RGM
9 11 9
1
( Li) ( Li) ( Li) ( ) 0N
j i ii
H E nn
A
23
11 9( Li) ( Li) nnH H H 11 9
1
( Li) ( Li) ( )N
i ii
nn
A
9 shell model type configu( Li) rat: on ii
• Orthogonality Condition Model (OCM) is applied.
91 2 1 2 12
1
( Li) ( ) ( ) ( )N
ij c c ij j ii
H T T V V V nn E nn
9 99( Li) : Hamilt ( Li onian for ) Liij i jH H
9 Orthogonality to the Pauli0 -, forbidden { Li} : st t s a ei
1 2 2 neutrons with Gaussian expa( ) nsion me: od th nn A{
TOSM
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11Li G.S. properties (S2n=0.31 MeV)
Tensor+Pairing
Simon et al.
P(s2)
Rm
E(s2)-E(p2) 2.1 1.4 0.5 -0.1 [MeV]
Pairing correlation couples (0p)2 and (1s)2 for last 2n
2525
2n correlation density in 11Li
nDi-neutron type config.
9Li
Cigar type config.
n
s2=4% s2=47%
K. Hagino and H. Sagawa, PRC72(2005)044321
9Li
H.Esbensen and G.F.Bertsch, NPA542(1992)310
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Short-range correlator : C (or Cr)
: ( ) for hamiltonian,
( ) for relative wave function
C r R r
r R r
† 1 1
( ) ( )r rC p C p
R r R r
†C lC l
† ( )C r C R r †12 12C S C S †C sC s
corr. uncorr.( ) ( )r C r
†( ) i cm iji i j
H T V t T v r C HC H T V
2, ij
i j
T T T T u
( ( ))iji j
V v R r
2-body approximation in the cluster expansion of operator
( ( ))R R r r
Hamiltonian in UCOM
†C tC t
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LS splitting in 5He with tensor corr.
5 4( He) ( He) relH H H 5 4
1
( He) ( He) ( )
N
i ii
nA
27
44H
1
( He) ( ) ( ) ( )N
ij rel e n ij j ii
H T V n E n
4Orthogonarity to the Pauli-forbidden states 0 : { ; He}i
• Orthogonarity Condition Model (OCM) is applied.
•T. Terasawa,PTP22(’59)
•S. Nagata, T. Sasakawa, T. Sawada R. Tamagaki, PTP22(’59)
• K. Ando, H. Bando PTP66(’81)
• TM, K.Kato, K.Ikeda PTP113(’05)
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6He in coupled 4He+n+n model• System is solved based on RGM
6 4( He) ( He) nnH H H 6 4
1
( He) ( He) ( )N
i ii
nn
A
4 6 4
1
( He) ( He) ( He) ( ) 0N
j i ii
H E nn
A
4 shell model type configu( He) rat: on ii
• Orthogonality Condition Model (OCM) is applied.
41 2 1 2 12
1
( He) ( ) ( ) ( )N
ij c c ij j ii
H T T V V V nn E nn
4 44( He) : Hamilt ( He onian for ) Heij i jH H
4 Orthogonality to the Pauli0 -, forbidden { He} : st t s a ei
1 2 2 neutrons with Gaussian expa( ) nsion me: od th nn A{
TOSM
30
Tensor correlation in 6He
30
Ground state Excited state
TM, K. Kato, K. Ikeda, J. Phys. G31 (2005) S1681