Title: Efimov Effect in 2 n- Core Halo Nuclei V.S.Bhasin Inter University Accelerator Centre, New Delhi 110067 and Department of Physics
1Efimov 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)
2CONTENTS
- 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
3EFIMOV 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).
4PROCEDURE
- 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
5PROCEDURE
- 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 )
8IMPACT 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) )
).
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10Contrast in the Occurrence of Efimov States in
Borromean Non-Borrmean Type HaloNuclei
Examples 19B, 22C vs 20C
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14The 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.
1520C 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.
16TABLE I. 20 C ground and excited states three
body energy fordifferent two body input
parameters.
17WHERE 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
19Appearance 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.
21Plot of Elastic Scattering Cross-section vs
incident neutron kinetic energy
22Resonances 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
23Explanation
- 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.
24General 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.
25Asymmetric 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 .
26Fit 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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28Conclusion
- 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.