Transcript Efimov Effect in 2 n- Core Halo Nuclei V.S.Bhasin Inter

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 THREEBODY 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.
2.
NEUTRONS ASSUMED TO BE IN THE LOW LYING INTRUDER SORBITAL 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)




 

  hn ( p) F ( p)  2 dq K1 ( p, q; E ) G (q )

 

 

1
 c  hc ( p) G( p)   dq K 2 ( p, q; E ) F (q )   dq K 3 ( p, q; E ) G (q )

1
n

[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:  1   1  [  (   ( p 2 / 2a   ) ] 1 ,
n
n
r

1
r
3

2
   c  2a 1  2a( p / 4c   3 ) .
1
c
2
These are the factors appearing on the left hand side of the
above integral equations;
n   n /  , c   c /  , r   / 1
2
3
1
and
2
3
1
 mE/    3 .
2
1
PROCEDURE
Defining
 n1 F ( p)   ( p) and  c1G( p)   ( p)
the two equations are actually reduced to one integral equation
in  ( p) 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; .   5.0
1
___________________________________________________
n- 12Be
2
Energy(keV)

1

1
0
s
a
3

( )
(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
A plot of the three body
Efimov states ( second and
first excited states ) vs the
two body scattering length
( actually, ln a s )
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
19
22
20
Examples : B, C vs C
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
1
any possibility of making  c  0 or  c   , 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.
20C
REVISITED
In 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
  h ( p)
 ( p  2 d   d 
1
1
c
c
2
2
3
2
2
2
2

2 1
3
/ a ) h( p ,  ;  )
2
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 Crosssection 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 crosssection 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 crosssection
•
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.