Recent Applications of the Stochastic Variational Method (SVM)brown/p-shell-2004/pdf/ysuzuki.pdf ·...
Transcript of Recent Applications of the Stochastic Variational Method (SVM)brown/p-shell-2004/pdf/ysuzuki.pdf ·...
Recent Applications of the Stochastic Variational Method (SVM)
Y. Suzuki (Niigata)Outline
1. Motivation of the SVM2. Algorithm of the SVM3. Structure of 16C
--- Hindered E2 transition ---Suzuki, Matsumura, Abu-Ibrahim, submitted
4. α-condensation in 12CMatsumura, Suzuki, Nucl. Phys. A739 (2004) 238
5. Summary
Motivation of the SVMThe solution of A-particle Schroedinger equation HΨ=EΨ is complicated
∑=
Φ=ΨK
iii AC
1)(
( ) 01
=−∑=
j
K
jijij CEBH
)(||)( jiij AHAH ΦΦ=
One of the simplest approach is a variational solution
The problems of direct approach with mesh1. lead to prohibitively large dimension D~Kpn for
Ai =(a1,a2,…,an)2. full optimization of parameters is time-consuming
(full diagonalization + recalculation of all the matrix elements)
Algorithm of the SVMPossibility of the stochastic optimization1. increase the basis dimension one by one2. set up an optimal basis by trial and error procedures3. fine tune the chosen parameters until convergence
),,,( 21 mkkk EEE LL2. Get the eigenvalues
4. k → k+1
),,,( 21 mkkk AAA LL1. Generate randomly
3. Select nkE
nkA corresponding to the lowest
and Include it in a basis set
Y. Suzuki and K. Varga, Stochastic variational approach to quantum-mechanical few-body problems, LNP 54 (Springer, 1998).K. Varga and Y. Suzuki, Phys. Rev. C52, 2885 (1995).
Trial function1. Gaussians as combinations of different Jacobi coordinates
=Φ
2. Combinations of Correlated Gaussians : Gexp)(exp 2 xAxrrG
jijiij −=−−= ∑
<
β'exp yyA−=
3. With appropriate angular, spin, isospin functions
Advantages:1. can be generalized to A-particle system2. The matrix elements can be calculated analytically3. can approximate even rapidly changing functions
Suzuki, Lovas, Yabana, Varga, Structure and reactions of light exotic nuclei(Taylor & Francis, 2003)
Two-neutron halo in 6He=α+n+n
100
1000
10000
0.01 0.02 0.03 0.04 0.05dσ
/dt [
mb/
(GeV
/c)2 ]
-t [(GeV/c)2]
Fig. 7
(a)
Tp=717 MeV
p+6He
Expmicroclustershell
Arai, Suzuki, Varga,Phys. Rev C 51,(1995) 2488
( )( )2
12 ∫ −=
Ω⋅ bibiq edbeiK
dd χ
πσ
Abu-Ibrahim, Fujimura, SuzukiNucl. Phys. A657 (1999) 391
Structure of 16CHindered E2 transition from 2+ to 0+ states in 16C
Imai et al, Phys. Rev. Lett. 92 (2004) 02501Elekes et al, Phys. Lett. 586B (2004) 34
Model16C=14C+n+n 15C=14C+n
Some experimental evidences 1. 14C is as stable as 16O2. one-neutron halo structure of 15C from 15C+12C 14C+x3. 14C(t,p) data suppot 14C+2n model for 16C4. α threshold is high
15C1/2+5/2+
14C+n
16C0+
2+
15C+n
14C+2n
12Be+α
0.741.281.77
4.25
5.47
13.81
11Be+α12.73
17F5/2+1/2+
16O+p
18Ne0+
2+17F+p
16O+2p
14O+α
0.500.601.89
3.924.525.11
16O+p+p13N+α
5.82
17O5/2+1/2+
16O+n
18O0+
2+
16O+2n
17O+n
14C+α
0.87
4.141.98
12.19
8.04
6.23
16O+n+n
13C+α6.36
E2 operator in core+valence-nucleons modelA=Ac+Av, Z=Zc+Zv
Mμ(A)=Mμ(c)+Mμ(v)+qeR2Y2μ(R)+…
Examples:1. Valence part forms a cluster
16O: 2+(6.92) 0+(6.05) q2=2.25 B(E2)=52 e2fm4 Exp. 65±7
2. Zv =016C: q2=0.0088, qeff
2=0.16 for δ=0.2
3. Zv≠ 018Ne: q2=2.92, qeff
2=3.70 for δ=0.2
V
2C
C
2V Z
AAZ
AAq
+
= δV
2C
eff AAAqq
+=
R
core
r
x2
x1
Formulation
1221 vUUTTH rR ++++=Spin singletU fits s.p. energies of 15CCentral forces
( )∑=
=−−=ΨK
iSiLMii vyxxAPC
1012 )(exp1 χ
2211
22222112
2111 2xuxuv
xAxxAxAxAx+=
++=
Pauli principle
0|)( =Ψinljm xu for all occupied s.p. states in 14C
2.0
1.5
1.0
0.5
B(E
2)
[e2fm
4]
0.200.150.100.050.00
δ
16C
15C
Exp.
16C
Momentum distribution
( )( ) ( )
2
Im2Im2
||
)ˆ()(12
1
12
1
||∑ ∫
∫∫=
−=
−−−−
+×
−=
lm
lmlmlj
zPi
sbbn
ineln
rYrgdzel
dseedbdPd
nFTnnT
h
hχχ
πσ
( ) ( ) ( )rrg CCjmmlj ,|
1615
0 ρρ ΦΦ=−
0+ 1/2+
5/2+
16C 15C
0.5
0.4
0.3
0.2
0.1
dσ
/dp
|| (
arb
. units)
-200 -100 0 100 200p|| (MeV/c)
16C +
12C --->
15C + X
E = 83 MeV/nucleon
5/2+
1/2+
Yamaguchi et al, Nucl. Phys. A724 (2003) 3Maddalena et al,Phys. Rev. C 63 (2001) 024613
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
g jl-m
(r)
[fm-3
/2]
20151050
r (fm)
s1/2
d5/2
α-condensation in 12CSymmetric nuclear matter at very low density4-particle correlation
↓α-condensation ?
G.Ropke et al, Phys.Rev.Lett.80, 3177(1998)
C12+10
+2
+20
Tohsaki et al, Phys.Rev.Lett.87,192501(2001)
3αC12+
20 state in↓
3α-condensation ?
Finite nuclei Dominance of mean field
To investigate the amount of α condensation in 12C by defining 3αboson wave function from a microscopic model
3α microscopic cluster model
])(exp[)()()()( 2321F ∑
<
−−=Φji
jiijΑq RRβαφαφαφ
])(exp[)()()()( 2321C cmi
iΑ RR −−=Φ ∑γαφαφαφγ
),,( 231312 βββ=q
Explicitly correlated 3α cluster wave function
R1
R2
R3
x1
x2
3α condensation-type wave function
CF Φ⊃Φ ( )γβββ31
231312 ===
Minnesota
+10
fm][2 ⟩⟨rE [MeV]
+20
E [MeV] 212 fm])[0M(0 ++ →fm][2 ⟩⟨r
CG -11.95 2.20 0.44 3.69 4.00 CG -7.72 2.36 0.71 4.28 5.02Exp. -7.27 2.35 0.38 5.48±0.22
Volkov No.1
)(FF qCq
qΦ=Ψ ∑)( C γΦP: projector ontoMinnesota
Volkov No.1
+10 +
202
FΨP0.98 0.910.96 0.89
])(exp)e)(()()([),( 2)(321
2
∑∑ −−=Φ−−
< kcmk
p
ji
jiAp RRRR γαφαφαφγ
MinnesotaVolkov No.1
+10 +
202FΨP
0.99 0.990.99 0.99
Linear pair correlation),( pγΦP: projector onto
3-α boson wave function
)()()( 321F χαφαφαφA=Ψ
Fermion w.f. Boson w.f.
χψ N=B
⟩⟨= )()()(|)()()( 321321 χαφαφαφαφαφαφχ ANNorm kernel
χψ N=B ⟩⟨= χχχψ NNN |B
Approximate orthogonalityPauli principle
α-cluster density distribution
Volkov No.1
⟩−−⟨= BB |)(|)( ψδψρ rRRr cmi
dR1
R2
R3
r
3α boson structure
⟩−−⟨= B21B |)(|)( ψδψ dRRdDα-α distance distribution Momentum distribution
⟩−−⟨= BB |)(|)( ψδψρ kkkk cmi
Volkov No.1 Volkov No.1
Pair correlation function⟩−−−−⟨= B0B0 |)()(|),( ψδδψ rRRrRRrr cmjcmiP
r
r0
θ x
y
r|||| 0 == rr
x=2fm x=2fmy=2fm y=2fm
+10 +
20Volkov No.1
Amount of α-condensationDensity matrix Y. Suzuki and M. Takahashi, Phys.Rev.C65 (2002) 064318
)()(],[d 3 rrrrr f'f'' λρ =∫For ideal condensation (S-state) : λ=3 (one) , 0 (othors)
n L (2L+1)λ n L (2L+1)λ
+20+
10
0S 1.119 0S 2.1200D 1.003 0D 0.2090G 0.609 1S 0.177
Volkov No.1
SummaryThe SVM on explicitly correlated basis states has been applied to study the structure of 16C and 12C.
+20Amount of α-condensation in state of 12C is about 70%.
Correlated motion among αclusters is represented perfectly with the linear pair correlation function.
14C+2n model is reasonable to account for the hindrance of B(E2) in 16C.The longitudinal momentum distribution of 15C fragments from 16C breakup is well reproduced with s and d contributions.
Future problemsWhy δ for 16C smaller than for 15C Possibility of α condensation in heavier nuclei