Efimov Effect in 2 n- Core Halo Nuclei V.S.Bhasin Inter University Accelerator Centre, New Delhi 110067 and Department of Physics - PowerPoint PPT Presentation

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Efimov Effect in 2 n- Core Halo Nuclei V.S.Bhasin Inter University Accelerator Centre, New Delhi 110067 and Department of Physics

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Title: Efimov Effect in 2 n- Core Halo Nuclei V.S.Bhasin Inter University Accelerator Centre, New Delhi 110067 and Department of Physics


1
Efimov Effect in 2 n- Core Halo Nuclei
V.S.Bhasin Inter University Accelerator
Centre, New Delhi 110067and Department of
Physics Astrphysics, University of Delhi.
Delhi 110007. ( INDIA)

2
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

3
EFIMOV EFFECT IN 14Be AS THREE BODY ( n-n- 12Be
) SYSTEM
  • ASSUMPTIONS
  • NEUTRONS ASSUMED TO BE IN THE LOW LYING INTRUDER
    S-ORBITAL STATE WITH 12 Be AS CORE AND
  • 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)

The notations and the symbols used here are from
S. Dasgupta et.al, PRC 50, R550,(1994).
4
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

These are the factors appearing on the left hand
side of the above integral equations
and
5
PROCEDURE
  • Defining
  • the two equations are actually reduced to one
    integral equation
  • in and can then be computed as an eigen
    value problem
  • after performing the angular integration and
    properly
  • symmetrizing the kernels. By feeding the
    parameters of the n-n
  • and 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 three
  • body energy for the two body input parameters .

6
  • __________________________________________________
    _
  • n- 12Be
  • Energy(keV)
  • ( ) (fm) (keV)
    (keV) (keV)
  • __________________________________________________
    _
  • 50 11.71 -21
    1350
  • 5.8 12.32 -61.6
    1408 0.053
  • 2.0 12.46 -105
    1450 2.56 0.061
  • 1.0 12.52 -149
    1456 3.8 0.22
  • 0.1 12.62 -483
    1488 6.1 0.62
  • 0.05 12.63_ -658
    1490 6.4 0.68
  • 0.01 12.65 -1491
    1490 6.9 0.72

7
A plot of the three body Efimov states (
second and first excited states ) vs the two
body scattering length ( actually, ln )
8
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 lt -10 fm ( Phys. Rev.C 63, 014308,
    (2000) )

).
9
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10
Contrast in the Occurrence of Efimov States in
Borromean Non-Borrmean Type HaloNuclei
Examples 19B, 22C vs 20C
11
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12
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14
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.
15
20C REVISITED
  • In view of the results of the latest experimental
    study setting the value of n-18 C binding energy
    to be 530130 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 160100 keV, a detailed
    investigation was carried out to study the effect
    on the behaviour of Efimov states in 20 C.

16
TABLE I. 20 C ground and excited states three
body energy fordifferent two body input
parameters.
17
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.

18
  • In the present model, the singularity in the
    present model appears in the two body propagator

19
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

20
  • Without going into the computational details, the
    results can be summarized as
  • The zero energy scattering length parameter for
    n-19 C for different values of the two body
    (n-18 C) binding energy ,i.e,.e2 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.

21
Plot of Elastic Scattering Cross-section vs
incident neutron kinetic energy
22
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

23
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
    n19 C that we observe the resonance.

24
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.

25
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
  • s s 0 (qe)2 / (1e2)
  • for asymmetric resonance profile in terms of
    three parameters, the energy position E r , width
    G, and a so-called profile index q .

26
Fit to the Fano formula
  • Here e(E-Er)/(G/2) and s0 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

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28
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.
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