Role of tensor force in He and Li isotopes with tensor optimized shell model
description
Transcript of Role of tensor force in He and Li isotopes with tensor optimized shell model
Role of tensor force in He and Li isotopes with
tensor optimized shell model
Hiroshi TOKI RCNP, Osaka Univ. Kiyomi IKEDA RIKENAtsushi UMEYA RIKEN
Takayuki MYO Osaka Institute of Technology
The Fifth Asia-Pacific Conference on Few-Body Problems in Physics @ Seoul, Korea, 2011.8.22-26
Purpose & Outline
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• We would like to understand role of Vtensor
in the nuclear structure by describing strong tensor correlation explicitly.
• Tensor Optimized Shell Model (TOSM)to describe tensor correlation
• Unitary Correlation Operator Method (UCOM)to describe short-range correlation
• TOSM+UCOM to He & Li isotopes with Vbare
TM, K. Kato, H. Toki, K. Ikeda, PRC76(2007)024305TM, H. Toki, K. Ikeda, PTP121(2009)511TM, A. Umeya, H. Toki, K. Ikeda, PRC(2011) in press.
S
D
Energy -2.24 MeV
Kinetic 19.88
Central -4.46
Tensor -16.64
LS -1.02
P(L=2) 5.77%
Radius 1.96 fmVcentral
Vtensor
AV8’
Deuteron properties & tensor force
Rm(s)=2.00 fm
Rm(d)=1.22 fmd-wave is “spatially compact”(high momentum)
r
AV8’
TM, Sugimoto, Kato, Toki, Ikeda PTP117(2007)257
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Tensor-optimized shell model (TOSM)
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4He• Configuration mixing
within 2p2h excitationswith high-L orbits. TM et al., PTP113(2005) TM et al., PTP117(2007) T.Terasawa, PTP22(’59))
• Length parameters such as b0s, b0p , … are optimized
independently (or superposed by many Gaussian bases).
– Describe high momentum component from Vtensor
– Spatial shrinkage of relative D-wave component as seen in deuteron HF by Sugimoto et al,(NPA740) / Akaishi (NPA738) RMF by Ogawa et al.(PRC73), AMD by Dote et al.(PTP115)
Hamiltonian and variational equations in TOSM
0,k
H E
C
1
,A A
i G iji i j
H t T v
k k
k
C : shell model type configurationk
0n
H E
b
c.m. excitation is excluded by Lawson’s method
(0p0h+1p1h+2p2h)
TOSM code : p-shell region
1, ,( ) ( )
N
nlj lj n lj nC
r r
2/1/2,
ˆ( ) ( ) ( ),nr b
zl
nlj n l l j tN b r e Y r r
Particle state : Gaussian expansion for each orbit
0,lj
H E
C
Gaussian basis function
Configurations of 5He in TOSM
proton neutron
Gaussian expansion
nlj
particle states
hole states (harmonic oscillator basis)
c.m. excitation isexcluded by Lawson’s method
Application to Hypernuclei by Umeya
C0 C1 C2 C3
Tomorrow
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Unitary Correlation Operator Method
7H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61
corr. uncorr.C
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( ) ( )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
† 1 (Unitary trans.)C C
rp p p
Bare Hamiltonian Shift operator depending on the relative distance
† 12( ) ( ) ( )C rC r s r s r s r
TOSM
2-body cluster expansion
Amount of shift, variationally determined.
(short-range part)
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T
VT
VLS
VC
E
(exact)Kamada et al.PRC64 (Jacobi)
• Gaussian expansion with 9 Gaussians
• variational calculation
TM, H. Toki, K. IkedaPTP121(2009)511
4He in TOSM + S-wave UCOM
good convergence
4-8He with TOSM+UCOM
• Difference from 4He in MeV
• No VNNN
• No continuum
~6 MeV in 8He using GFMC
~7 MeV in 8He using Cluster model (PLB691(2010)150 , TM et al.)
4p4h in TOSM
4-8He with TOSM+UCOM
• Excitation energies in MeV
• No VNNN
• No continuum • Excitation energy spectra are reproduced well
5-9Li with TOSM+UCOM
• Excitation energies in MeV
• No VNNN
• No continuum
Preliminary results
• Excitation energy spectra are reproduced well
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Matter radius of He isotopes
I. Tanihata et al., PLB289(‘92)261 G. D. Alkhazov et al., PRL78(‘97)2313O. A. Kiselev et al., EPJA 25, Suppl. 1(‘05)215. P. Mueller et al., PRL99(2007)252501
TOSM
Expt
TM, R. Ando, K. KatoPLB691(2010)150
Halo Skin
Cluster model
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Configurations of 4He
(0s1/2)4 83.0 %
(0s1/2)−2JT(p1/2)2
JT JT=10 2.6
JT=01 0.1
(0s1/2)−210(1s1/2)(d3/2)10 2.3
(0s1/2)−210(p3/2)(f5/2)10 1.9
Radius [fm] 1.54
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Cf. R.Schiavilla et al. (VMC) PRL98(’07)132501
• 4He contains p1/2 of “pn-pair”.
• 0 of pion nature.
• deuteron correlation with (J,T)=(1,0)
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Tensor correlation in 6He
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Ground state Excited state
TM, K. Kato, K. Ikeda, J. Phys. G31 (2005) S1681
Tensor correlation is suppressed due to Pauli-Blocking
val.-n
halo state (0+)
6He : Hamiltonian component
• Difference from 4He in MeV
6He 0+1 2+
1 0+2
n2 config (p3/2)2 (p3/2)2 (p1/2)2
Kin. 53.0 52.4 34.3
Central 27.8 22.9 14.1
Tensor 12.0 12.8 0.2
LS 4.0 5.0 2.1
=18.4 MeV (hole)
bhole=1.5 fm
same trend in 5,7,8He
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Summary
• TOSM+UCOM with bare nuclear force.
• 4He contains “pn-pair” of p1/2 than p3/2.
• He isotopes with p3/2 has large contributions
of Vtensor & Kinetic energy than those with p1/2.
• Vtensor enhances LS splitting energy.
– TM, A. Umeya, H. Toki, K. Ikeda, PRC(2011) in press.
Review Di-neutron clustering and deuteron-like tensor correlation in nuclear structure focusing on 11Li K. Ikeda, T. Myo, K. Kato and H. Toki Springer, Lecture Notes in Physics 818 (2010) “Clusters in Nuclei” Vol.1, 165-221.
4-8He in TOSM+UCOM
• Argonne V8’ w/o Coulomb force.
• Convergence for particle states.– Lmax =10
– 6~8 Gaussian for radial component
• Lawson method to eliminate CM excitation.
• Bound state approximation for resonances.
TM, H. Toki, K. Ikeda, PTP121(2009)511TM, A. Umeya, H. Toki, K. Ikeda, PRC, in press.
5He : Hamiltonian component
• Difference from 4He in MeV
5He 3/2 1/2
n config. p3/2 p1/2
Kin. 24.1 17.9
Central 9.0 7.0
Tensor 5.6 1.1
LS 2.6 1.0
=18.4 MeV (hole)
bhole=1.5 fm
LS splitting in 5He & tensor correlation
• T. Terasawa, A. Arima, PTP23 (’60) 87, 115.• S. Nagata, T.Sasakawa, T.Sawada, R.Tamagaki, PTP22(’59)• K. Ando, H. Bando PTP66 (’81) 227• TM, K.Kato, K.Ikeda PTP113 (’05) 763 (+n OCM)
30% of the observed splitting from Pauli-blocking d-wave splitting is weaker than p-wave splitting
Pauli-Blocking
val.-n
Tensor correlation & Splitting in 5He
Vtensor~exact
in 4He
Enhancement of Vtensor
4-8He with TOSM
• No VNNN
• No continuum
• Difference from 4He in MeV
Minnesota force (Central+LS)
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Short-range correlator : C (or Cr)
3GeV repulsion Originalr2
C
Vc
1E
3E
1O3O
s(r)
[fm]
We further introducepartial-wave dependencein “s(r)” of UCOM
S-wave UCOM
Shift function Transformed VNN (AV8’)
• Tensor force (Vtensor) plays a significant
role in the nuclear structure.
– In 4He, Vtensor ~ Vcentral
(AV18, GFMC)
– Main origin : -exchange in pn-pair
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Importance of tensor force
• We would like to understand the role of Vtensor in the nuclear
structure by describing tensor correlation explicitly.
tensor correlation + short range correlation model wave functions : shell model, cluster model (TOSM: Tensor Optimized Shell Model) He, Li isotopes (LS splitting, halo formation, level inversion)
7He : Hamiltonian component
• Difference from 4He in MeV
7He 3/21 3/2
2 3/23
n3 config (p3/2)3 (p3/2)2(p1/2) (p3/2)(p1/2)2
Kin. 82.9 74.1 65.8
Central 39.1 33.6 27.0
Tensor 19.9 16.0 10.0
LS 8.6 2.8 1.0
bhole=1.5 fm
8He : Hamiltonian component
• Difference from 4He in MeV
8He 0+1 0+
2 2+1
n4 config (p3/2)4 (p3/2) 2(p1/2)2 (p3/2) 3 (p3/2)
Kin. 114.2 101.0 109.6
Central 59.2 51.6 56.2
Tensor 25.2 17.1 23.4
LS 11.6 2.55 7.1
bhole=1.5 fm