Nuclear processes in the continuum

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Nuclear processes in the continuum. THE CONTINUUM SHELL MODEL (CSM). The standard formalism to study processes in the nuclear continuum is the Continuum Shell Model. The basis of this formalism is provided by the unity operator written on the real energy axis, i. e. - PowerPoint PPT Presentation

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Nuclear processes in the continuum

The standard formalism to study processes in the nuclear continuum is the Continuum Shell Model. The basis of this formalism is provided by the unity operator written on the real energy axis, i. e.

Where ωn(r) is the wave function of the nth bound state and u(r,E) is the scattering wave at energy E. One discretizes the integral to obtain the CSM representation.

THE CONTINUUM SHELL MODEL (CSM)

The CSM was used, for instance, to study the halo structure in 11Li (Bertsch et. al., PRC 56, 3054 (1997)). By applying the Cauchy theorem Berggren extended the unity to the complex energy plane,

Notice that neither the sum nor the integral include complex conjugates Complex probabilities.

The resonances are stationary wave functions with outgoing boundary conditions,

ϕ n (r)r→∞

⏐ → ⏐ ⏐ Nneiknr

En =(hkn )

2

where

ϕ n (r)r→∞

⏐ → ⏐ ⏐ Nneiknr

Neglecting the contribution from the continuum one gets the unity as

1= | n >< n |n

∑Where n are bound states and resonances. This gave rise to the first application of the Berggren representation, the Resonant RPA(Curutchet, Vertse, RJL, PRC 39, 1020 (1989)).

Meaning of the complex quantities

The energy is E=Er-iΓ/2. The wave function becomes

ϕ (r, t) = ϕ (r,0)e−iEt / h = ϕ (r,0)e−iEr t / he−Γt /(2h )

And the probability is

P(r, t) = P(r,0)e−Γt / h

Which gives a mean life

T = h /Γ

But this has meaning only if the resonance is narrow, in which case all the imaginary parts are small and, e. g., |φ(r)|2 ≈ Re(φ(r)2)

Core 9Li6.

Single particle states: 10Li7 (Ψ(r)eikr).

E [MeV] k [1/fm]

1s1/2 -0.025 (0,-0.033)0p1/2 (0.240,-0.064) (0.103,-0.013)0d5/2 (2.731,-0.545) (0.201,-0.091)0d3/2 (6.396,-4.898) (0.628,-0.342)

312Li9Evaluation of heavy Li isotopes. Three-neutron nucleus

The best way to see whether a resonance has physical meaning is by looking at the wave function. For instanceThe neutron state 0d3/2 in 10Li7 at (6.396,-4,898) MeV

Instead, the state 1g7/2 in 169Sn119 at (1.667,2.8x10-13) MeV i.e. T=2.4x10-9 sec

Going back to the Berggren representation, to include the scattering waves (i. e.the proper continuum) one has toDiscretize the integral

Defining the set of vectors {|φ>} such that for the bound and resonant states it is φn(r) = <r|φn> and for the scattering States φp(r) = (hp)1/2 h(r,Ep) one gets €

1= | n >< n |n

1= | n >< n |n

It was shown that the set {|n>} forms a representation, called the Berggren representation (RL, Maglione, Sandulescu, Vertse, PLB 367, 1 (1996)). This means that

all processes in the complex energy plane can be described by using the Berggren representation.In particular a shell model in the complex energy plane(CXSM) to describe many-body resonances was formulated (IdBetan, RL, Sandulescu Vertse, PRL 89, 042501 (2002)).The CXSM was applied to study the halo structure of11Li8 by assuming that the 3 protons form the inert core9Li6 and that the halo is induced by the two neutrons outside this core (Esbensen, Bertsch, Hencken, PRC56, 3056 (1997)). The single particle states are theresonance 0p1/2 at (0.240,-0.064) and the antibound state 1s1/2 at (-0.025,0) (Zhukov et al, Phys. Rep. 231, 151 (1993)).

(IdBetan, RL, Sandulescu, Vertse, Wyss, PRC72, 054322 (2005)),

To include the antibound state one has to use the integration contour

The antibound state is very close to the continuum threshold. The wave function looks like follows:

E(ant)=-0.050 MeV, E(p1/2)=(0.24,-0.06) MeV

Full line antibound, dashed line bound at the same energy.

E=-0.025 MeV

For the 0p1/2 resonance at (0.240,-0.064) MeV the wave function is

We evaluated 11Li by using the CXSM as a two-neutron system.There are two probable physically meaningful states 0+. The corresponding wave functions are

where

The 11Li8(gs) (two-particle) radial wave function at r1=r2=r and S=0 (singlet) is

E=-0.295 (bound)

≈50% s-state and 50% p-state

For the second 0+ the wave function is

Very recently the ground state and an excited state in 12Li9 (three neutrons outside the core) were measured. The ground state is anantibound state and the excited state is a wide resonance lying at1.5 MeV and is about 1 MeV wide. The spin of this three-particleresonance is probably 5/2+ (Roger et. al., to be published).To analyse this nucleus we used the Multistep Shell Model in thecomplex energy plane (CXMSM). In this model the basis set ofstates have consists of the tensorial product of previously evaluatedstates. In our case the core is

39 Li6

and the basis states are of the

form

| 313Li10 >=| 3

10Li7⊗ 311Li8 >

Or, schematically,

| three − part >=| one − part ⊗ two − part >

The three-body basis is very large because there are many scattering waves. However only very few of these states are relevant.

Within the CXMSM basis one can decide the relevant one- and two-particle states by looking at the corresponding wave functions.For the single-particles we have seen that the 1s1/2 antibound stateand that the 0p1/2 resonance are fundamental to describe 11Li. ButAlso the 0d5/2 resonance may be physically relevant.

For the two-particle states we saw that 11Li(gs) is bound. The only other state to be relevant was found to be the state 21

+.

With this small basis set of states we evaluated 12Li

Zhen Xiang Xu, RL, Chong Qi, T. Roger, P. Roussel-Chomaz, H. Savajols and R. Wyss.To be published

SummaryThe treatment of processes occurring in the continuum part of thenuclear spectrum may require time dependent formalisms. Alternatively, one may extend usual (e. g. shell model) formalismsto the energy plane. For this one has to pay the price of dealingwith complex energies and probabilities. To decide whether a evaluated resonant state is physically meaningful one can examine the corresponding wave function. A resonance appears becausethere is a barrier that traps the system inside the nucleus. Thereforethe wave function inside the nucleus has to be real and localized.This formalism can be applied to study many body resonances just applying the same methods developed for bound states butexpended to the complex energy plane.

protons

neutrons

F. Janouch and RJL, PRC 25, 2123 (1982)

PoPbPo 210210212

PbPo 208212

F. Janouch and RJL, PRC 27, 896 (1983)

________________________

______________________________

___________________________________

Po209Pb209

G. Dodig-Crnkovic, F. A. Janouch and RJL, PLB139, 143 (1984)

G. Dodig-Crnkovic, F. A. Janouch, RJL and L. J. Sibanda, NPA 444, 419 (1985)

R. G. Lovas, RJL, A. Insolia, K. Varga and D. S. Delion, Phys. Rep. 294, 265 (1998)

K. Varga, R. G. Lovas and RJL, PRL 69, 37 (1992)

________________________________________

________________________________________________________________________

M. W. Herzog, RJL and L. J. Sibanda, PRC, 259 (1985)

P. Curutchet, T. Vertse and RJL, PRC39, 1020 (1989)

T. Vertse, E. Maglione and RJL, NPA584, 13 (1995)

V(r) = 0, r > R

),'(),(1

)'()()'( qruqrdqurrrr Ln

J. Bang, F. A. Gareev, M. H. Gizzatkulov and S. A. Goncharov, Nucl. Phys. A309, 381 (1978)

RJL, E. Maglione, N. Sandulescu and T. Vertse, PLB367, 1 (1996)

0||||

0||)(0||

qpqp

jijijin

ccnpqVij

ccnccn

R. Id Betan, RJL, N. Sandulescu and T. Vertse, PRL89, 042501 (2002)

A. T. Kruppa, P.-H. Heenen, H. Flocard and RJL, PRL, 79, 2217 (1997)

Resonant HF Binding energies

Binding energy (E) and root mean square radius (r2) in different self-consistent mean-field models for the N=20 isotones.

42Ti 44Cr 46Fe

-E r2 -E r2 -E r2

HFB 350.84 3.50 354.39 3.58 355.51 3.65

HF-BCS-B 350.52 3.50 353.95 3.57 355.15 3.64

HF-BCS-R1 350.65 3.51 354.13 3.58 355.28 3.64

HF-BCS-R2 350.90 3.51 354.44 3.58 355.51 3.65

A. T. Kruppa, P. H. Heenen and RJL, PRC 63, 044324 (2001)

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Im||~

[Re22)(

1)(||)(

nuAnu

n

nEEnuAnu

nnEE

nEAE

ϕϕ

EEn

gE

nnEE

nEHE )(22)(

1)(|

0|)(

ϕϕ

pppp

ppiii

i ccEEgccH )(0

qqppqqp

pqp

scatscat ccccEEgEEgGVpairing

)()(

),()(),( rEEgrE nnnn ϕ

1),(22 rEdrr nn

D

ϕ

N. Sandulescu, N. Van Giai and RJL, PRC 61(R), 061301 (2000)

H. J. Unger, NPA 164, 564 (1967)

antibound

bound

R. Id Betan, RJL, N. Sandulescu, T. Vertse and R. Wyss, PRC 72, 054322 (2005)

MeVs

Ni

731.0)( 2/1

84

M. Grasso, N. Sandulescu, N. Van Giai and RJL, PRC 64, 064321 (2001)