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 1 2 Yuri Lurie and Andrey M. Shirokov 1 The College of Judea and Samaria, Israel 2 Moscow State University

Yu. Lurie, A. Shirokov , Annals of Physics 312 , pp. 284-318 (2004) � J -matrix formalism with hyperspherical oscillator basis � Application to 11 Li = 9 Li + n + n � Two-neutron halo in 6 He = 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): − A 1 2 ∑ ρ = ξ hyperradiu s i − 3 A 3 independen t = i 1 ξ coordinate s i { } − Ω 3 A 4 angles � hypermomentum K ( ) ( ) Ψ = ∑ Φ ρ Ω R. I. Jibuti et. al, 1983; 1984 d Y γ γ γ K K K γ K , Democratic decay :

A -body harmonic oscillator: 2 A ω r ω ( ) r h ( ) ( ) 2 2 κ γ = ρ Ω = ∑ − = ρ K R Y U m r R κ γ K K i i 2 2 = i 1 ( ) −  2 κ 2 ! ρ + κ ( ) ( ) 1 2 K 2 L   ρ = − ρ ρ 1 exp L R κ κ K ( )   2 Γ κ + + 3 2 L   − 3 A 6 = + K L 2 −  3 A 3  = + ω ≡ κ + eigenenerg y : E  N  N 2 K h κ K  2  ∞ ( ) ( ) ∑ ∑ Ψ = κ γ Ψ ρ Ω K R Y κ γ K K γ κ = K 0

Kinetic energy : ( )( ) ′  − κ + κ + + κ = κ + 1 3 2 , 1 L ω  h 3 ′ ′ ′ ′ κ γ κ γ = δ δ κ + + κ = κ K T K 2 ,  L ′ ′ γ γ K , K , 2 2  ( ) ′ − κ κ + + κ = κ − 1 2 , 1 L  ω   h 2 =   Free solutions energy E q :  2  ( ) − 2  κ 2 ! + q ( ) 1 2 K 2 L   = S q q exp L q κ κ K ( )   2 Γ κ + + 3 2 L   ( ) ( ) ∞ ′ ′ S q S q 2 q κ K K 0 ( ) ′ = − C q V.P. ∫ d q κ K ( ) π 2 2 S q ′ − q q 0 K 0 ( ) ( ) ∞ ′ ′ ( ) ( ) S q S q 2 q κ ± 0 K K ′ = − C q ∫ d q ( ) κ K π 2 2 S q ′ − ± q q i 0 0 K 0 ( ) ( ( ) ( ) ω h + ) b 2 = − = energy E q : C q C i q κ κ K K 2

Truncation of the potential : ~ ~ ′ ′ ′ ′ ′ κ γ κ γ = κ + > κ + > K V K 0 for 2 K N or 2 K N A ~  −  N K ~ ~ ~  κ > κ κ > κ κ ≡  , , ′ K K K 2   ⇒ variationa l approach : ~ ′ ′ κ + ≤ 2 K N ′ ′ ′ ′ ′ ′ ∑ κ γ + − κ γ κ γ λ = K T V E K K 0 λ A ′ ′ ′ κ γ K

J - matrix method: 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)]  γ ( ) ( ) K ~ ~ ~ ′ ′ ′ ′ ∑ κ γ κ + γ κ + γ Ψ in in κ ≤ κ K 1 K 1 K , P ′ ′ K K K  ′ ′ γ K  ( internal region)    ( ) ( ) ( ) ( ) 1 γ  − +  K ( ) ~  in in δ δ − ≥ κ ≥ κ   C q C q S , E 0 γ K ′ ′ γ γ K , K , K κ κ γ κ γ Ψ in in = K K K K    2  ( ) ( )  γ K b ( ) ~ − δ δ < κ ≥ κ S C q , E 0  γ γ K , K , K γ κ K K in in  ( external region)    ~  ′ ′  κ γ λ λ κ γ K K ′ ( ) ( ) ~ ~ ~ K ′ ′ ′ ′ ′ ′ κ γ κ + γ = ∑ κ γ κ + γ K 1 K   K T 1 K P ′ ′ ′ K K K − E E  λ  λ

� Application to 11 Li = 9 Li + n + n ( ) = ± E 2 n 0 . 247 0 . 080 MeV F. Ajzenberg-Selov, Nucl. Phys. A 490 , 1 (1988) ± 0 . 295 0 . 035 MeV W. Benenson, Nucl. Phys. A 588 , 11c (1995) 1 2 2 = ± r 3 . 10 0 . 17 fm I. Tanihata et al., Phys. Lett. B 206 , 592 (1988) ± 3 . 53 0 . 10 fm J.S. Al-Khalili et al., Phys. Rev. C 54 , 1843 (1996) ± 3 . 55 0 . 10 fm J.S. Al-Khalili and J.A. Tostevin, Phys. Rev. Lett. 76 , 3903 (1996)

r r r + = l l L x y r r r + = s s S 1 2 = S 0 ( ) 2 x − ( ) R − = − = = Gaussian n n potential : U x V e V 31 MeV R 1 . 8 fm 0 0 2 2     y y − −     ( ) R R 9     − = − 1 − 2 n Li potential : U y V e V e 1 2 = = = = V 7 MeV R 2 . 4 fm V 1 MeV R 3 . 0 fm 1 1 2 2 L. Johansen, A.S. Jensen, and P.G. Hansen, Phys. Lett. B 244 , 356 (1990)

11 The ground state energy of Li Variational approach J -matrix method 0.8 0.00 ~ = N 18 ~ = ~ = -0.05 N 18 N 20 0.6 ~ = ~ = N 20 N 22 ~ = ~ = N 22 N 24 ~ = N 24 -0.10 0.4 E 0 , MeV E 0 , MeV -0.15 0.2 -0.20 0.0 -0.25 -0.2 -0.30 convergence limit convergence limit -0.4 -0.35 0 5 10 15 20 25 30 0 5 10 15 20 25 30 h ω , MeV h ω , MeV

11 Convergenc e of the Li ground state energy ~ and rms matter radius with N 0.4 variational h ω=6.5 MeV J-matrix 3.4 0.3 0.2 3.2 < r 2 > 1/2 , fm 0.1 E 0 , MeV 3.0 h ω=20 MeV 0.0 2.8 -0.1 2.6 -0.2 h ω=20 MeV 2.4 variational -0.3 J-matrix h ω=6.5 MeV -0.4 2.2 ~ 6 10 14 18 22 ~ 26 30 34 38 6 10 14 18 22 26 30 34 38 N N 1 / 2 − A 2 2 2 2 = − ρ r r 9 Li A

( ) 11 The Li two - neutron separation energy E 2 n 1 2 2 and rms matter radius r Approximation E (2 n ) < r 2 > 1/2 [fm] [MeV] ~ = ~ = ~ = ~ = N 38 N 40 N 38 N 40 Variational 0.326 0.327 3.176 3.189 1 F. Ajzenberg-Selov, Nucl. Phys. A 490 , 1 (1988) J -matrix 0.335 0.336 3.336 3.348 2 W. Benenson, M.V.Zhukov et al. 0.3 3.32 Nucl. Phys. A 588 , 11c (1995) PL B 265 (1991) 19 3 I. Tanihata et al., Phys. Lett. B 206 , 592 (1988) Experimental data 0.247±0.080 1 3.10±0.17 3 4 J.S. Al-Khalili et al., 0.295±0.035 2 3.53±0.10 4 Phys. Rev. C 54 , 1843 (1996) 5 J.S. Al-Khalili and J.A. Tostevin, 3.55±0.10 5 Phys. Rev. Lett. 76 , 3903 (1996)

Reduced probabilit y and energy - weighted sum rule 11 of cluster dipole mode in Li ( ) E ( ) ( ) 1 d E 1 B ( ) d E 1 1 2 B = − ( ) ∫ S E E E d E = ∑ J E 1 J E1 f i f M f i d E S + d E 2 J 1 E1 0 i J f ∞ ( ) ( ) d E 1 B = − ∫ E E d E S E1 f i f ( ) d E 2 Ze r ( ) 0 ˆ µ = − E 1 y Y y M µ 1 2 2 2 A 9 e 2 Z h = ( ) π − 4 2 m A A 2 1.5 V V 100 V J J J B.V.Danilin et al. 39 d B (E1) / d E , fm 2 /MeV 80 experimental data 40 1.0 S E1 ( E ) , % 60 40 0.5 V V 20 V J J J 0.0 0 0 1 2 3 4 5 0 1 2 3 4 5 E , MeV E , MeV V ― variational approach; J ― J -matrix method

d B ( E1 ) = Convergenc e of with N (energy E 0.5 MeV) d E ~ ~ = = ω = N 20 , N 21 , 6 . 5 MeV h g . s . f . s . 1.0 d B (E1) / d E , fm 2 /MeV 2 N h h 0.8 ρ ≈ ≈ 80 fm ω ω m m 0.6 ( ) = ω = N 500 , 6.5 MeV h 0.4 0.2 0.0 0 100 200 300 400 500 N

� Two-neutron halo in 6 He = α + n + n ( ) = A.H. Wapstra et al., E 2 n 0 . 976 MeV At. Data Nucl. Data Tables 39 , 281 (1988) 1 2 2 = ± r 2 . 33 0 . 04 fm I. Tanihata et al., Phys. Lett. B 289 , 261 (1992) ± 2 . 48 0 . 03 fm I. Tanihata et al., RIKEN Preprint AF-NP-60 (1987) ± 2 . 57 0 . 10 fm L.V. Chulkov et al., Europhys. Lett. 8 , 245 (1989)

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)    Pauli Projection • Majorana splitting potential (MS)  to the forbidden  V.I. Kukulin et al., Nucl. Phys. A 586 , 151 (1995)  0s state 1/2   • l -dependent Gaussian potential (GP)   Yu.A. Lurie and A.M. Shirokov,   Bull. Rus. Acad. Sci., Phys. Ser. 61 , 1665 (1997)

n – n potentials: • Gaussian s -wave potential (Gs) G.E. Brown and A.D. Jackson, The Nucleon-Nucleon Interaction = = = V 30 . 93 MeV R 1 . 82 fm (singlet S 0 spin state) 0 = = = V 60 . 9 MeV R 1 . 65 fm (triplet S 1 spin state) 0 • 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) = = = V 31 . 00 MeV R 1 . 8 fm (singlet S 0 spin state) 0 = = = V 71 . 09 MeV R 1 . 4984 fm (triplet S 1 spin state) 0 • 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)