Loosely bound three-body nuclear systems in the J -matrix approach 1 - - PowerPoint PPT Presentation

Microscopic Nuclear Structure Theory INT-04-3 program

Loosely bound three-body nuclear systems in the J-matrix approach

Yuri Lurie and Andrey M. Shirokov The College of Judea and Samaria, Israel Moscow State University

1 2 2 1

• Yu. Lurie, A. Shirokov, Annals of Physics 312, pp. 284-318 (2004)

J-matrix formalism with hyperspherical oscillator basis Application to 11Li = 9Li + n + n Two-neutron halo in 6He = α + n + n Phase-equivalent transformation with continuous parameters and three-body cluster system

J-matrix formalism with hyperspherical oscillator basis

Hyperharmonical coordinates (A particles):

hypermomentum K

∑

− =

=

1 1 2 A i i

s hyperradiu ξ ρ

i

A ξ s coordinate t independen 3 3 −

{ }

Ω − angles 4 3A

( ) ( )

∑

Ω Φ = Ψ

γ γ γ γ

ρ

, K K K K

d Y

• R. I. Jibuti et. al, 1983; 1984

Democratic decay:

A-body harmonic oscillator:

( )

2 1 2 2

2 2 ρ ω ω h r r = − =

∑

= A i i i

R r m U

( ) ( )

Ω =

γ κ

ρ γ κ

K K

K Y R

( ) ( ) ( )

( )

( ) ( )

Ω Ψ = Ψ + ≡       − + = − + =        − + + Γ − =

∑ ∑

∞ = + γ κ γ κ κ κ κ κ

ρ γ κ κ ω ρ ρ ρ κ κ ρ

K K K K K K

K K N A N E A K L Y L L R R

2 2 1 2

2 2 3 3 : y eigenenerg 2 6 3 2 exp 2 3 ! 2 1 h

L

( )( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) (

)

q i C q C q E q d i q q q S q S q S q q C q d q q q S q S q S q q C q L q q q S q E K K

K b K K K K K K K K K K K K K + ∞ ± ∞ + ′ ′

= − = ′ ± ′ − ′ ′ − = ′ ′ − ′ ′ − =        − + + Γ =       =      − = ′ + + − = ′ + + + = ′ + + + − = ′ ′ ′

∫ ∫

κ κ κ κ κ κ κ κ γ γ

ω π π κ κ ω κ κ κ κ κ κ κ κ κ κ κ δ δ ω γ κ γ κ : 2 energy 2 V.P. 2 2 exp 2 3 ! 2 : 2 energy solutions Free 1 , 2 1 , 2 3 2 1 , 2 3 1 2 T : energy Kinetic

2 2 2 2 2 2 2 1 2 2 , ,

h h h

L

L L L L

      − ≡ > > > + > ′ + ′ = ′ ′ ′

′

2 ~ ~ , ~ , ~ ~ 2

• r

~ 2 for V : potential the

• f

Truncation

A

K N N K N K K K

K K K

κ κ κ κ κ κ κ γ κ γ κ V T :

~ 2 A

= ′ ′ ′ ′ ′ ′ − + ⇒

∑

≤ ′ + ′ ′ ′ ′

λ γ κ γ κ γ κ

κ γ κ λ

K K E K approach l variationa

N K K

Yu.F. Smirnov and A. M. Shirokov, Preprint ITF-88-47R (Kiev, 1988); A.M. Shirokov, Yu.F. Smirnov, and S.A. Zaytsev,

• Teoret. Mat. Fiz. 117, 227 (1998) [Theor. Math. Phys. 117, 1291 (1998)]

J - matrix method:

( ) ( )

( ) ( ) ( ) ( )

( )

( ) ( )

( ) ( ) ( )

γ κ γ κ γ κ λ λ γ κ γ κ γ κ κ κ δ δ κ κ δ δ κ κ γ κ γ κ γ κ γ κ

λ λ κ γ γ γ γ γ γ κ κ γ γ γ γ γ

′ ′ + ′ ′       − ′ ′ = ′ ′ +                ≥ < − ≥ ≥       − ≤ Ψ ′ ′ + ′ ′ + = Ψ

′ ′ ′ ′ + − ′ ′ ′ ′ ′ ′

∑ ∑

K K E E K K K K E q C E q C q C K K K K

K K K K K b K K K K K K K K K K K K K K K K K K

in in in in in in in in

1 ~ T ~ ~ 1 ~ region) external ( ~ , S ~ , S 2 1 region) internal ( ~ , 1 ~ 1 ~

, , , ,

P P

( )

fm 10 . 55 . 3 fm 10 . 53 . 3 fm 17 . 10 . 3 MeV 035 . 295 . MeV 080 . 247 . 2

2 1 2

± ± ± = ± ± = r n E

• F. Ajzenberg-Selov, Nucl. Phys. A 490, 1 (1988)
• W. Benenson, Nucl. Phys. A 588, 11c (1995)
• I. Tanihata et al., Phys. Lett. B 206, 592 (1988)

J.S. Al-Khalili et al., Phys. Rev. C 54, 1843 (1996) J.S. Al-Khalili and J.A. Tostevin,

• Phys. Rev. Lett. 76, 3903 (1996)

Application to 11Li = 9Li + n + n

2 1

= = + = + S S s s L l l

y x

r r r r r r

( )

( )

( )

fm . 3 MeV 1 fm 4 . 2 MeV 7 : potential Li fm 8 . 1 MeV 31 : potential Gaussian

2 2 1 1 2 1 9

2 2 2 1 2

= = = = − − = − = = − = −

      −       − −

R V R V e V e V y U n R V e V x U n n

R y R y R x

• L. Johansen, A.S. Jensen, and P.G. Hansen, Phys. Lett. B 244, 356 (1990)

Li

• f

energy state ground The

11

J-matrix method Variational approach

5 10 15 20 25 30

hω, MeV

• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8

E0 , MeV

18 ~ = N 20 ~ = N 22 ~ = N 24 ~ = N convergence limit

5 10 15 20 25 30

hω, MeV

• 0.35
• 0.30
• 0.25
• 0.20
• 0.15
• 0.10
• 0.05

0.00

E0 , MeV

18 ~ = N 20 ~ = N 22 ~ = N 24 ~ = N convergence limit

N ~ with radius matter rms and energy state ground Li the

• f

e Convergenc

11

6 10 14 18 22 26 30 34 38

• 0.4
• 0.3
• 0.2
• 0.1

0.0 0.1 0.2 0.3 0.4

E0 , MeV

hω=6.5 MeV hω=20 MeV

variational J-matrix

N ~

6 10 14 18 22 26 30 34 38 2.2 2.4 2.6 2.8 3.0 3.2 3.4

<r2>1/2 , fm

hω=6.5 MeV hω=20 MeV

variational J-matrix

N ~

2 2 Li 2 / 1 2

9

2 ρ − − = r A A r

( )

2 1 2 11

radius matter rms and 2 energy separation neutron

• two

Li The r n E

38 ~ = N 40 ~ = N 38 ~ = N 40 ~ = N

3.55±0.105 3.53±0.104 0.295±0.0352 3.10±0.173 0.247±0.0801 Experimental data 3.32 0.3 M.V.Zhukov et al. PL B 265 (1991) 19 3.348 3.336 0.336 0.335 J-matrix 3.189 3.176 0.327 0.326 Variational

< r2 >1/2 [fm] E(2n) [MeV] Approximation

1 F. Ajzenberg-Selov,

• Nucl. Phys. A 490, 1 (1988)

2 W. Benenson,

• Nucl. Phys. A 588, 11c (1995)

3 I. Tanihata et al.,

• Phys. Lett. B 206, 592 (1988)

4 J.S. Al-Khalili et al.,

• Phys. Rev. C 54, 1843 (1996)

5 J.S. Al-Khalili and J.A. Tostevin,

• Phys. Rev. Lett. 76, 3903 (1996)

Li in mode dipole cluster

• f

rule sum weighted

• energy

and y probabilit Reduced

11

( )

( )

( )

( )

( ) ( )

2 2 2 4 9 d d 1 E d d d 1 E d 1

2 2 2 E1 E1 E1

− = − = − =

∫ ∫

∞

A A Z m e E E E E E E E E E S

f i f f E i f

h π B B S S

( ) ( ) ( )

( )

y Y y A Ze J J J E

f

J i f i

ˆ 2 1 E 1 E 1 2 1 d 1 E d

1 2

r

µ

µ − = + =

∑

M M B

1 2 3 4 5

E, MeV

0.0 0.5 1.0 1.5

dB(E1)/dE , fm2/MeV

V V V J J J B.V.Danilin et al.39 experimental data40

1 2 3 4 5

E, MeV

20 40 60 80 100

SE1(E) , %

V V V J J J

V ― variational approach; J ― J-matrix method

MeV) 0.5 (energy with d ) E1 ( d

• f

e Convergenc = E N E B

MeV 5 . 6 , 21 ~ , 20 ~

. . . .

= = = ω h

s f s g

N N

100 200 300 400 500

N

0.0 0.2 0.4 0.6 0.8 1.0

dB(E1)/dE , fm2/MeV

( )

MeV 6.5 , 500 fm 80 2 = = ≈ ≈ ω ω ρ ω h h h N m N m

( )

fm 10 . 57 . 2 fm 03 . 48 . 2 fm 04 . 33 . 2 MeV 976 . 2

2 1 2

± ± ± = = r n E

A.H. Wapstra et al.,

• At. Data Nucl. Data Tables 39, 281 (1988)
• I. Tanihata et al., Phys. Lett. B 289, 261 (1992)
• I. Tanihata et al., RIKEN Preprint AF-NP-60 (1987)

L.V. Chulkov et al., Europhys. Lett. 8, 245 (1989)

Two-neutron halo in 6He = α + n + n

Phenomenological effective potentials n – α potentials:

• l-dependent Gaussian potential with repulsive s-wave core

(SBB)

B.V. Danilin et al., Yad. Fiz. 53, 71 (1991) [Sov. J. Nucl. Phys. 53, 45 (1991)]

• Woods-Saxon potential (WS)
• J. Bang and C. Gignoux, Nucl. Phys. A 313, 119 (1979)

J.S. Al-Khalili et al., Phys. Rev. C 54, 1843 (1996)

• Majorana splitting potential (MS)

V.I. Kukulin et al., Nucl. Phys. A 586, 151 (1995)

• l-dependent Gaussian potential (GP)

Yu.A. Lurie and A.M. Shirokov,

• Bull. Rus. Acad. Sci., Phys. Ser. 61, 1665 (1997)

state 0s forbidden the to Projection Pauli

1/2

              

n – n potentials:

• Gaussian s-wave potential (Gs)

G.E. Brown and A.D. Jackson, The Nucleon-Nucleon Interaction

• Gaussian potential (G)

B.V. Danilin et al., Phys. Lett. B 302, 129 (1993) L.S. Ferreira et al., Phys. Lett. B 316, 23 (1993)

• Minnesota potential (MN)

/includes central, spin-orbit and tensor components/

• T. Kaneko, M. LeMere, and L.C. Tang, Prep. TPI-MINN-91/32-T (1991)

state) spin 1 (triplet fm 65 . 1 MeV 9 . 60 state) spin (singlet fm 82 . 1 MeV 93 . 30 = = = = = = S R V S R V state) spin 1 (triplet fm 4984 . 1 MeV 09 . 71 state) spin (singlet fm 8 . 1 MeV 00 . 31 = = = = = = S R V S R V

) potentials SBB (Gs He

• f

energy state ground The

6

+

J-matrix method Variational approach

5 10 15 20 25 30 35 40 45 50

hω, MeV

• 1.2
• 1.0
• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

E0 , MeV

14 ~ = N 16 ~ = N 18 ~ = N 20 ~ = N

convergence limit

5 10 15 20 25 30 35 40 45 50

hω, MeV

• 1.0
• 0.8
• 0.6
• 0.4
• 0.2

E0 , MeV

14 ~ = N 16 ~ = N 18 ~ = N 20 ~ = N

convergence limit

) potentials SBB (Gs radius matter rms and energy state ground He the

• f

~ with e Convergenc

6

+ N

6 10 14 18 22 26 30

• 1.0
• 0.8
• 0.6
• 0.4
• 0.2

0.0

E0 , MeV

hω=10 MeV hω=25 MeV

variational J-matrix

N ~

6 10 14 18 22 26 30 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7

<r2>1/2 , fm

hω=10 MeV hω=25 MeV

variational J-matrix

N ~

( )

radius matter rms and 2 energy separation neutron

• two

He The 6 n E

MeV , ) 2 ( n E MeV , ω h 26 ~ = N 28 ~ = N

2.33±0.046, 2.48±0.037 2.57±0.108 0.9765 Experimental data 2.502 0.3051, 1.002, 0.7843, 0.6944 Other potential models 2.445 2.482 0.681 0.684 J-matrix 2.389 2.382 0.672 0.656 Variational 13.00 MN + MS 2.547 2.588 0.516 0.515 J-matrix 2.484 2.477 0.509 0.494 Variational 10.75 MN + GP 2.428 2.451 0.883 0.889 J-matrix 2.393 2.386 0.878 0.875 Variational 11.35 G + GP 2.44 1.00 J.S. Al-Khalili et al., (1996) 2.461 2.444 1.008 1.003 J-matrix 2.404 2.394 0.997 0.993 Variational 11.90 Gs + WS “Correct asymptotic value” Danilin et al. (1991) 2.539 2.554 0.923 0.927 J-matrix 2.510 2.502 0.919 0.917 Variational 10.00 Gs + SBB

r.m.s. matter radius, fm Approx. Potentials

26 ~ = N 28 ~ = N

References to the table:

1 V.I. Kukulin, V.N. Pomerantsev, Kh.D. Razikov, V.T. Voronchev, and G.G. Ryzhikh,

• Nucl. Phys. A 586, 151 (1995)

2 E. Garrido, D.V. Fedorov, and A.S. Jensen, Nucl. Phys. A 617, 153 (1997) 3 K. Kato, S. Aoyama, S. Mukai, and I. Ikeda, Nucl. Phys. A 588, 29c (1995);

• S. Aoyama, S. Mukai, K. Kato, and I. Ikeda, Progr. Theor. Phys. 93, 99 (1995)

4 E. Hiyama, M. Kamimura, Nucl. Phys. A 588, 35c (1995) 5 A.H. Wapstra, G. Audi, and R. Hoekstra, At. Data Nucl. Data Tables 39, 281 (1988) 6 I. Tanihata, D. Hirata, T. Kobayashi, S. Shimoura, K. Sugimoto, and H. Toki,

• Phys. Lett. B 289, 261 (1992)

7 I. Tanihata et al., RIKEN Preprint AF-NP-60 (1987) 8 L.V. Chulkov, B.V. Danilin, V.D. Efros, A.A. Korsheninnikov, and M.V. Zhukov,

• Europhys. Lett. 8, 245 (1989)

MeV) 35 . 11 , potentials GP (G He in mode dipole cluster

• f

y probabilit Reduced

6

= + ω h

23 ~ , 22 ~

. . . .

= =

s f s g

N N

2 4 6 8 10 12

E, MeV

0.0 0.1 0.2 0.3

dB(E1)/dE , fm2/MeV

V V V J J J

1 2 3 4 5 6

E, MeV

0.0 0.1 0.2 0.3 0.4 0.5

dB(E1)/dE , fm2/MeV

12 => 13 16 => 17 20 => 21 14 => 15 18 => 19 22 => 23

g.s. f.s.

N ~

He in mode dipole cluster

• f

y probabilit Reduced

6

1 2 3 4 5 6

E, MeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6

dB(E1)/dE , fm2/MeV

G + SBB G + GP MN + GP MN + MS S.Funada et al.61 B.V.Danilin et al.62

Phase-equivalent transformation (PET) with continuous parameters and three-body cluster system

Idea: To improve a description of a three-body system by continuous varying of off-shell properties of two-body subsystem(s)

• Local PETs

(binary subsystem has at least one bound state; can't be applied to the n-α subsystem):

• Supersymmetry PETs

(transformation of a potential with Pauli forbidden state into a phase-equivalent potential with a repulsive core; does not have free parameters): e.g.: F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995)

• E. Garrido, D.V. Fedorov, and A.S. Jensen, Nucl. Phys. A 650, 247 (1999)
• PET based on unitary transformation of the Hamiltonian
• F. Coester, S. Cohen, B. Day, and C.W. Vincent, Phys. Rev. C 1, 769 (1970);

M.I. Haftel and F. Tabakin, Phys. Rev. C 3, 921 (1971) H.C. Pradhan, P.U. Sauer, and J.P. Vary, Phys. Rev. C 6, 407 (1972)

{ }

( ) ( ) ( ) ( )

[ ] [ ] [ ][ ]

[ ] [ ] [ ] [ ] [ ]

[ ]

[ ] { } [ ] [ ] [ ] [ ][ ] [ ]

{ }

[ ] [ ]

          − − =       + − = − + = − ⇒ ∆ + Ψ = Ψ         = ⊕ = = = Ψ = ′

+ − = + ∞ = ∞ = ′ ′

∑ ∑ ∑

β γ β γ β γ γ β γ β γ β γ γ γ γ

λ λ λ λ λ λ

cos sin sin cos sin cos sin sin sin cos cos cos cos sin sin cos H H V V : potential equivalent

• Phase

matrix and shifts phase scattering same the ~ ; spectrum same the : H H ~ H H ~ H : basis in problem body

• Two

1 2

U U U U S k C E and I U I U U U U k C C E C k k k

PET N n k k k k k k

L

He

6

in the variational approach

MeV 13 21 ) 1 ( ~ , 20 ) ( ~ potentials MS MN = = = +

− +

ω h N N

PET

) 20 ~ , 7 γ ( energy state ground = = N

• 180
• 120
• 60

60 120 180

β, deg.

• 1

1 2 3

E , MeV

G + GPPET MN + GPPET MN + MSPET

• 180
• 120
• 60

60 120 180

γ, deg.

• 1

1 2 3 4 5

E , MeV

0+ (ground state) 1- 0+

energy binding the

• f

increasing 12 7 %

• binding

4 2 additional %

γ parameter vs matrix)

• (

energy state ground He The 6 J

20 ~ = N

• 15
• 10
• 5

5 10 15 20 25

γ , deg.

• 1.0
• 0.9
• 0.8
• 0.7
• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.0

E , MeV

MN + GPPET MN + MSPET G + GPPET G + SBBPET

∆E

approach) (JJ potentials MS MN He in mode dipole cluster

• n

PET

• f

Results

PET 6

+

MeV 13 , 23 ~ , 22 ~

. . . .

= = = ω h

s f s g

N N

1 2 3 4 5 6

E, MeV

0.0 0.1 0.2 0.3 0.4

dB(E1)/dE , fm2/MeV

• 100 0.384

00 0.662 6.50 0.695 200 0.374

MeV , γ

b

E

Conclusion

The hyperspherical J-matrix formalism makes it possible to study both the discrete and continuum spectra of A-body systems in an unique approach. This approach appears to be a very powerful tool for calculations of the bound state properties as well as the few-body disintegration of the system in the case of democratic decay. The suggested J-matrix motivated non-local phase equivalent transformation may be used to fit binding energies of many-body systems in the case when all information about two-body inter- action is extracted from the scattering data only. The transfor- mation can be easily utilized in the studies of many-body systems with any L² basis.