Efimov Effect in 2 n- Core Halo Nuclei

V.S.Bhasin Inter University Accelerator Centre, New Delhi 110067

and Department of Physics & Astrphysics,

University of Delhi. Delhi 110007. ( INDIA)

CONTENTS:1. INTRODUCTION

2. EFIMOV EFFECT IN 14Be (n-n-12Be System)

3. CONTRAST IN THE OCCURRENCE OF EFIMOV STATESIN BORROMEAN and NON-BORROMEAN TYPE HALO NUCLEI, EXAMPLES: 19B, 22C vs. 20C

4. EFIMOV STATES IN THE CONTINUUM IN 20 C BELOW THE THREE- BODY THRESHOLD

5. SIMILARITY BETWEEN THE EFIMOV RESONANT STATES AND FANO RESONANCES

6. CONCLUSION

EFIMOV EFFECT IN 14Be AS THREE BODY ( n-n- 12Be ) SYSTEM

ASSUMPTIONS:1. NEUTRONS ASSUMED TO BE IN THE LOW LYING INTRUDER S-

ORBITAL STATE WITH 12 Be AS CORE; AND

2. ASSUMING SEPARABLE POTENTIALS FOR THE n-n AND n- 12Be PAIRS IN MOMENTUM SPACE, THE THREE- BODY SCHRODINGER EQUATION IS SOLVED TO GET THE INTEGRALEQUATIONS FOR THE SPECTATOR FUNCTIONS F(p) AND G(p)

)();,(2)()( 11 qGEqpKqdpFphnn

)();,()();,()()( 321 qGEqpKqdqFEqpKqdpGphcc

[The notations and the symbols used here are from S. Dasgupta et.al, PRC 50, R550,(1994).]

PROCEDURE• For studying the sensitive computational details of the

Efimov effect, the above equations are recast involving only the dimension-less quantities by redefining the terms: ,])2/(([ 1

3211 aprrnn

.)4/(2122

3211

cpaacc

These are the factors appearing on the left hand side of the above integral equations;

131

231

2 /,/,/ rccnn

and ./ 321 mE

PROCEDUREDefining the two equations are actually reduced to one integral equationin and can then be computed as an eigen value problemafter performing the angular integration and properly symmetrizing the kernels. By feeding the parameters of the n-nand n-c potentials in the kernels, we seek the solution of the three- body binding energy parameter when the eigen value approaches to 1, accurate to at least three significant figures.Table I [ From I.Mazumdar & V S Bhasin P. R.C 56, R5,1997]Summarizes the results of 14Be ground and excited states threebody energy for the two body input parameters; .

)()()()( 11 ppGandppF cn

)( p

0.51

___________________________________________________n- 12BeEnergy(keV)

( ) (fm) (keV) (keV) (keV)

___________________________________________________50 11.71 -21 13505.8 12.32 -61.6 1408 0.0532.0 12.46 -105 1450 2.56 0.0611.0 12.52 -149 1456 3.8 0.220.1 12.62 -483 1488 6.1 0.620.05 12.63_ -658 1490 6.4 0.680.01 12.65 -1491 1490 6.9 0.72

3

2101 sa

A plot of the three body Efimov states ( second and first excited states ) vs the two body scattering length ( actually, ln )sa

IMPACT OF THIS INVESTIGATION As a follow-up of this study,

experimental group of M.Thoennessen, S.Yokoyama and

P.G.Hansen from M. S. U reported the first experimental evidence of a low lying intruder neutron unbound

to 12Be from the fragmentation of 18O suggesting a virtual state with

scattering length a < -10 fm ( Phys. Rev.C 63, 014308, (2000) )

Contrast in the Occurrence of Efimov States in

Borromean & Non-Borrmean Type HaloNuclei

Examples : 19B, 22C vs 20C

The point to realize here is that the Efimov region resulting from the universal character, independent of the detailed nature of the two body interaction, is essentially governed by the two body propagators given above. Expressed in terms of the two body scattering length, the above expression for the n-core system, for instance, can be rewritten as

So long as anc is negative (representing a virtual state, and the third term being always smaller than the first there is hardly any possibility of making or , except when anc approaches a large value and goes to the zero limit. On the other hand, if the binary subsystem is bound corresponding to positive scattering length, there is a clear possibility of allowing to be large enough.

1c 0 c

20C REVISITEDIn view of the results of the latest experimental

study setting the value of n-18 C binding energy to be 530±130 keV

[T.Nakamura et al.,Phys.Rev.Lett 83,1112(1999)] whereas the earlier results

[G. Audi and Wapstra, Nucl. Phys.A56,66(1993)]found the value to be 160±100 keV, a detailed investigation was carried out to study the effect

on the behaviour of Efimov states in 20 C.

TABLE I. 20 C ground and excited states three body energy for

different two body input parameters.

WHERE TO GO FROM HERE?

• At this point to proceed further,we took some time until we noticed the work of Amado & Noble Phys.Rev.D5,1992(1972)] ,who originally pointed out that the Efimov states move into the unphysical sheet associated with the two body unitarity cut on increasing the strength of the binary interaction to bind the two body system. This provided us with a clue to undertake the study for n-19 C scattering.

• In the present model, the singularity in the present model appears in the two body propagator

12

322

222

23

2

11

);,()/2(

)(

phaddp

phcc

Appearance of Resonance in n-19 C Scattering• The equation for the off-shell scattering amplitude in n-19 C

( bound state of n-18 C) can be written as

• where a k(p) is the off-shell scattering amplitude, normalized such that

Without going into the computational details, the results can be summarized as:

1. The zero energy scattering length parameter for n-19 C for different values of the two body (n-18 C) binding energy ,i.e,.ε2 =100 keV, 220

keV and 250 keV retains a positive sign, thereby ruling out the possibility of the Efimov states turning over to the virtual states.

2. At incident energies different from zero, the behaviour of elastic scattering cross-section vs incident energy of neutron on 19 C for three different binding energies, i.e., 250keV, 300 keV and 350 keV is depicted as shown.

Plot of Elastic Scattering Cross-section vs incident neutron kinetic

energy

Resonances by Efimov States & Fano Resonances

• A characteristic feature of the resonances predicted here is that they do not have a symmetric Breit-Wigner shape but rather an asymmetric profile widely observed and studied as Fano resonances.This brings us close to the similarity underlying the two phenomena: Fano Resonances and the resonances by the Efimov states

Explanation

• When studied as a function of energy, a scattering cross-section exhibits a resonance when the energy traverses the position of a discrete state that is embedded in the energy continuum. Here we tune the two body energy so as to make weakly bound states of 20 C gradually get weaker in binding and then disappear to become quasi bound states in n+ 19 C continuum. For the first excited state this happens at 220 keV. It is precisely at such energies above that value of 220 keV, when we have such an Efimov state embedded in the elastic scattering continuum of n+19 C that we observe the resonance.

General Resonance Profile

• Two alternative pathways to the final state, one directly into the continuum and the other through the embedded dicrete state, interfere to give rise to the resonance.

• From general quantum mechanical interference, such a profile may show both constructive and destructive features. Therefore as the energy traverses, the general resonance profile may be asymmetric, with destructive and constructive interference on the two sides of the central maximum.

Asymmetric Resonance Profile

• Fano [ U. Fano, Phys. Rev. 124, 1866 (1961) ] presented this picture in terms of the interference of the two competing amplitudes and described the elastic scattering cross-section

• σ = σ 0 [(q+ε)2 / (1+ε2)]

• for asymmetric resonance profile in terms of three parameters, the energy position E r , width Γ, and a so-called “profile index” q .

Fit to the Fano formula

• Here ε=(E-Er)/(Γ/2) and σ0 is the background cross-section far from the resonance and q is the ratio of the two quantities- the amplitude through the discrete state and the direct amplitude to the underlying continuum.

• The following figure shows the fit to the Fano formula:

Conclusion• We have here demonstrated how by tuning a selected range of the

weak two-body ( n-18 C) binding energy or the large scattering length, dictated by the experimental data, the appearance of the Efimov states move from bound to a continuum character resulting in sharp ( Feshbach-type) resonance with an asymmetric profile. Interestingly, the recent experiment with ultra cold atoms also saw only a couple of Efimov states, again one of these appearing as a resonance as the scattering length was tuned through threshold by changing the magnetic field.

• Indeed, the cleanest demonstration of the Efimov phenomena would be to observe a series of such states n in the limit of very large very large scattering length, with energy positions relative to threshold and widths scaling with a characteristic exponential dependence on n. Clearly a more detailed study of such universal scaling with n will have to await a better knowledge of the binary n-18 C interaction.