Shell Model in Complex Energy Plane

Institute of Atomic Physics, Bucharest

• Resonances and virtual states: Berggren representation

• Shell model with resonances and virtual states

• Application: the structure of 11Li

N. Sandulescu

Espace de Structure Nucleaire Theorique, Saclay

**Similar work: N. Michel, W.Nazarewicz, M. Ploszajczak, K. Bennaceur,…

*Collaborators: R.Id. Betan (Rosario) , R.J.Liotta (Stockholm), T. Vertse (Debrecen)

Resonances and virtual states

10Li

virtual stateresonances

9Li

Single-particle resonant states

79Ni

78Ni

Resonant states

Decaying state (Gamow,1928)

2En

nn iE

divergence !

« Capturing » state: tt *nn kk

tEi n

e ri ne

General defintion ( Siegert, 1939)

0),()](2)1(

[22

22

2

krurVh

M

r

llk

dr

dl

« Resonances »: out-going solutions

Time-reversed solutions : ),(~),( * rkurku nln

),( rku nl

0),()( lll OuWky

;)()(),( lllll IkyOkxrku

Resonant states

Poles of S-matrix

Re k

Im k• k-plane :

• energy plane: ;2

iE );(

222

2

M

M

2

ik

« resonance »

« anti-resonance »

« crazy »

« anti-bound » Re E

Gamow states : normalisation

• Bi-orthogonal set : ),( rku nl ),(),(~ * rkurku nn

• Regularisation: Zeldovich (’60) ; Gyarmati & Vertse (1971)

drrkurkueuu r ),(),(~lim|~ *

0

2

• Matrix elements: uAu ||~ complex quantity !

•Note : Gamow functions rigged Hilbert space

Berggren representation

• Real-energy axis:

• Complex-energy plane:

L

Re k

(T. Berggren, Nucl. Phys. A108,265,1968)

Two-particle resonances

)2,1()2(ˆ)1(ˆˆ VhhH

iiih ˆ)()(),( 2121 rrXrr jiij

kljilkijji XVX ||~~)(

)()( klfXijGf kl

ji

ijf

G )(1 2

;

Re

Im

(R.Betan, R.J.Liotta, N.S., T. Vertse, Phys.Rev. Lett. 89, 042501, 2002)

Single-particle states

Two-particle states

Two-particle resonant states

80Ni

78Ni

Two-particle resonant states

( R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Rev. Lett. 89, 042501, 2002 )

Two-particle resonant states

( R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Rev. Lett. 89, 042501,2002 )

Resonances and anti-bound states

10Li

anti-boundresonances

Anti-bound states

• definition: ;nn ik 22

2 nn ME

Re k

Im k

• wave function:

),()(),( 2/122

riuk

krk n

nscat

);( riu n

(A.B.Migdal et al, Sov.J.Nucl.Phys. 14, 488, 1872 )

Energy contours in Berggren representation

Re

Im

L

Anti-bound state

Resonant states

Re

2/1s

2/1p2/3d

L

Im

Resonances and anti-bound states

10Li

anti-bound

Note: does a unique mean field exist ? NO !

resonances

Effective mean fields for 10Li

J.C.Pacheco, N. Vinh Mau, Pys.Rev.C65(2002)044004

H. Esbensen, G.F. Bertsch, K. Hencken, Phys.Rev.C56(1997)3054

N. Vinh Mau, Nucl. Phys. A592(1995)33

Particle-vibration couplings:

F. Barranco et al, Eur. Phys. J. A11(2001)385

Ground state of 11Li: pole structure

Re

(R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Lett. B584, 48, 2004 )

Two-particle resonant states in 11Li

(R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Lett. B584, 48, 2004 )

Conclusions

Main advantages of shell model in complex energy plane:

• based on relevant continuum configurations

• direct access to multi-particle resonant states

Open problems :

• multi-particle resonant states: decays channels ?

• efficient truncation schemes for large systems ?

- Density Matrix Renormalisation Group

( N. Michel, W. Nazarewicz, M. Ploszajczak, J. Rotureau, nucl-th/0401036)

( G.Hagen, M.Hjorth-Jensen, J. Vagen, nucl-th/0410114 )

- Lee-Suzuki similarity transformation - Multi-reference perturbation method

)()(),( 2121 rrXrr jiij

Rr

ikre iKRe

Decay channels

1r

2r

11rike

22rike

…………….…………….

Localisation of scattering states

( R.Betan, R.J.Liotta, N.S., T. Vertse, Phys. Lett. B584, 48, 2004 )

Anti-bound states: trajectories