1. Osaka Institute of Technology
Role of tensor force in light unstable nuclei with tensor-optimized shell model
Takayuki MYO 　明　孝之
Osaka Institute of Technology
大阪工業大学
In collaboration with Atsushi UMEYA (Nippon Inst. Tech.)
Hiroshi TOKI (RCNP)
Kiyomi IKEDA (RIKEN)
RIBF ULIC-Symposium "Perspective in Isospin Physics" - Role of non-central interactions in structure and dynamics of unstable nuclei RIKEN, 1

2. Outline
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, A. Umeya, H. Toki, K. Ikeda PRC84 (2011)
TM, A. Umeya, H. Toki, K. Ikeda PRC, in press

3. Deuteron properties & tensor force
Energy
-2.24 MeV
Kinetic
19.88
Central
-4.46
Tensor
-16.64 LS
-1.02
P(L=2)
5.77%
Radius
1.96 fm S D
AV8’
Vcentral
Rm(s)=2.00 fm
Rm(d)=1.22 fm r
d-wave is
“spatially compact”
(high momentum)
Vtensor

4. Tensor-optimized shell model (TOSM)
TM, Sugimoto, Kato, Toki, Ikeda PTP117(2007)257
Configuration mixing within 2p2h excitations with high-L orbits. 　 TM et al., PTP113(2005) TM et al., PTP117(2007)
Length parameters such as b0s, b0p , … are optimized independently, or superposed by many Gaussian bases.
Spatial shrinkage of D-wave as seen in deuteron HF by Sugimoto et al,(NPA740) / Akaishi (NPA738) RMF by Ogawa et al.(PRC73), AMD by Dote et al.(PTP115)
Satisfy few-body results with Minnesota central force (4,6He)
4He 4 4

5. Hamiltonian and variational equations in TOSM
(0p0h+1p1h+2p2h)
Particle state : Gaussian expansion for each orbit
Gaussian basis function
Hiyama, Kino, Kamimura PPNP51(2003)223
c.m. excitation is excluded by Lawson’s method
TOSM code : p-shell region

6. Configurations in TOSM
Gaussian expansion
particle states C0 C1 C2 C3
nlj
proton
neutron
hole states
(harmonic oscillator basis)
Application to Hypernuclei by A. Umeya to investigate N-N coupling

7. Unitary Correlation Operator Method
(short-range part)
TOSM
short-range correlator
Bare Hamiltonian
Shift operator depending on the relative distance
Amount of shift, variationally determined.
2-body cluster expansion 7 7
H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61

8. Short-range correlator : C (or Cr)
[fm]
Shift function
Transformed VNN (AV8’) 1E
s(r)
Originalr2
3GeV repulsion 3E Vc 1O C 3O
We further introduce
partial-wave dependence in “s(r)” of UCOM
S-wave UCOM 8

9. 4He in TOSM + short-range UCOM
(exact)
Kamada et al.
PRC64 (Jacobi)
TM, H. Toki, K. Ikeda
PTP121(2009)511
variational calculation
VLS E
Gaussian expansion
with 9 Gaussians VC VT
good convergence 9

10. Tensor Optimized Few-body Model (TOFM)
Same as TOSM concept
No use of UCOM
Correlated Gaussian basis + Global vector in SVM K
102.35
95.37
4He with AV8’
S-wave (L=0)
D-wave (L=2) Y2
-4.05
-4.71 LS
-24.05
-25.92 E VC
-54.58
-55.22 VT
-60.79
-68.32 Y2 Y2
TOFM
SVM
Horii, Toki, Myo, Ikeda PTP127(2012)1019 10

11. 5-8He with TOSM+UCOM Excitation energies in MeV Bound state app.
TM, A. Umeya, H. Toki, K. Ikeda PRC84 (2011)
Excitation energies in MeV
Bound state app.
No continuum
No VNNN
Excitation energy spectra are reproduced well

12. 5-9Li with TOSM+UCOM Excitation energies in MeV
TM, A. Umeya, H. Toki, K. Ikeda PRC, in press
Excitation energies in MeV
Excitation energy spectra are reproduced well

13. 5-9Li with TOSM Minnesota force (Central+LS)
Excitation energies in MeV
Too large excitation energy

14. Matter radius of He & Li isotopes
Expt
TOSM with AV8’
Halo
Skin
A. Dobrovolsky, NPA 766(2006)1
I. Tanihata et al., PLB289(‘92) G. D. Alkhazov et al., PRL78(‘97)2313
O. A. Kiselev et al., EPJA 25, Suppl. 1(‘05) P. Mueller et al., PRL99(2007)252501 14 14

15. Configurations of 4He with AV8’
TM, H. Toki, K. Ikeda
PTP121(2009)511
(0s1/2)4
83.0 %
(0s1/2)−2JT(p1/2)2JT 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
deuteron correlation with (J,T)=(1,0)
Cf. R.Schiavilla et al. (VMC)
PRL98(2007)132501
R. Subedi et al. (JLab)
Science320(2008)1476
12C(e,e’pN)
S.C.Simpson, J.A.Tostevin
PRC83(2011)014605
4He contains p1/2 of “pn-pair”
12C 10B+pn
– Same feature in 5He-8He ground state 15

16. Selectivity of the tensor coupling in 4HeVT
L=2 s1 s2 s3 s4 s1 s2 s3 s4
1+(pn)
1+(pn) VT
Selectivity of
tensor operator s1 s2 s3 s4
L=2
DL=2, DS=2 16

17. Relativistic chiral mean field model
Ogawa, Toki （RCNP) 2011,NPA
16O a
Vp / A
• with 2p2h excitations.
• Difference between
　12C and 16O is “3 MeV / A”
12C
• It comes from
low pion-spin states (J=0,1,2)
which suffers Pauli blocking.
16O
16O
P1/2
12C a
P3/2
4He
Vp(Jp)/ A
12C
S1/2

18. 4-8He with TOSM+UCOM Excitation energies in MeV No VNNN No continuum
Excitation energy spectra are reproduced well

19. Tensor correlation in 6He
val.-n
Ground state
Excited state
halo state (0+)
Tensor correlation is suppressed due to Pauli-Blocking
TM, K. Kato, K. Ikeda, J. Phys. G31 (2005) S1681 19 19

20. 6He : Hamiltonian component in TOSM
Difference from 4He in MeV
bhole=1.5 fm
6He
0+1
0+2
n2 config
(p3/2)2
(p1/2)2
DKin.
53.0
34.3
DCentral
27.8
14.1
DTensor
12.0
0.2
DLS
4.0
2.1
=18.4 MeV
(hole)
LS splitting
energy in 6He
same trend
in 5-8He
Terasawa, Arima PTP23 (’60)
Nagata, Sasakawa, Sawada, Tamagaki, PTP22(’59)
Myo, Kato, Ikeda, PTP113 (’05)

21. Configurations of “pn in 6Ligs” and VNN
AV8’
Minnesota
(0p1/2)(0p3/2)
43%
29%
(0p3/2)2
38%
56% LS jj
S=0
S=1
(0p1/2)(0p3/2)
11%
89%
(0p3/2)2
56%
44%
Enhancement of S=1 component with AV8’ (Vtensor)
Indication of the a+d clustering 21
(1+)

22. Ground states : p-shell configurations
Weight
6Li (1+)
(0p1/2)(0p3/2)
43%
7Li (3/2-)
(0p3/2)3
48%
8Li (2+)
(0p3/2)4
41%
9Li (3/2-)
(0p3/2)5
46% LS jj jj jj
Tensor-type excitations
6Li … LS coupling
7-9Li … jj coupling
0s → 0p1/2,
0p3/2 → sd & higher shells 22

23. Summary TOSM+UCOM using Vbare.
Reproduce the excitation energy spectra.
4He contains “pn-pair of p1/2” than p3/2.
He isotopes with p3/2 has large contributions of Vtensor & Kinetic energy.
6Li : S=1 component, LS coupling.
7-9Li : jj coupling.
Foundation to the analyses of 10Li & 11Li. 23 23

24. 4He in UCOM (Afnan-Tang, Vcentral only)24 24

25. 4-8He with TOSM Minnesota force (Central+LS)
Difference from 4He in MeV
No VNNN
No continuum

26. Configurations of “nn in 6Hegs” and VNN
AV8’
Minnesota
(0p3/2)2
72%
81%
(0p1/2)2
10% 4% jj
S=0
S=1
(0p3/2)2
33%
66%
(0p1/2)2 26