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