104
6 Three-Body Approach to Structural Properties …
Fig. 6.19 Argand plot of
Im(f k ) versus Re(f k ) within
a unit circle in the n – 19 C
scattering when the binding
energy of n – 18 C system is
taken as 250 keV
viz. k = k r − ik i and the width = (k r k i )/μ, we compute by using the values of
resonance energy and the width, the corresponding values of k r and k i , which are
found to be 3.75 × 10
−2 fm
−1 and 1.53 × 10
−3 fm
−1 , respectively. This ensures that
not only does the resonance lie in the fourth quadrant in the k-plane but also k r k i .
(2) The second criterion is to study the behavior of Re( f k ) and Im( f k ) on the Argand
diagram as a function of energy around the resonance position. In Fig. 6.19, we depict
a plot of Im( f k ) versus Re( f k ) within a unit circle as we increase the incident energy
of the neutron from 1.0 to 2.75 keV in steps of 0.25 keV, where the binding energy of
the n−
18 C system is taken to be 250 keV. We find that as the energy is increased, the
point described by (Re f k , Im f k ) indicates an anticlockwise motion in the Argand
plot, thereby satisfying an important condition to be obeyed for the occurrence of a
resonance.
It may be interesting to ask whether the scattering cross section would still predict
the resonance-type structure if the pair energy of the bound system is increased to,
say, 200 keV or even less than that in which case the bound Efimov state in the
three-body system is found to occur. To answer this, we depict by a dashed curve the
plot of scattering cross section for n−
19 C scattering in Fig. 6.18a taking the (n−
18 C)
binding energy to be 200 keV. We find that the scattering cross section does not
produce the resonance-like peak structure and remains more or less constant with
the incident energy of the neutron.
6.8 Efimov States and Their Fano Resonances
in a Neutron-Rich Nucleus
In continuation of our investigation studying the appearance of resonances in elastic
scattering cross section as the Efimov states just become unbound, we have attempted
[102] to tie this result to yet another interesting but not fully familiar result that a
6 Three-Body Approach to Structural Properties …
Fig. 6.19 Argand plot of
Im(f k ) versus Re(f k ) within
a unit circle in the n – 19 C
scattering when the binding
energy of n – 18 C system is
taken as 250 keV
viz. k = k r − ik i and the width = (k r k i )/μ, we compute by using the values of
resonance energy and the width, the corresponding values of k r and k i , which are
found to be 3.75 × 10
−2 fm
−1 and 1.53 × 10
−3 fm
−1 , respectively. This ensures that
not only does the resonance lie in the fourth quadrant in the k-plane but also k r k i .
(2) The second criterion is to study the behavior of Re( f k ) and Im( f k ) on the Argand
diagram as a function of energy around the resonance position. In Fig. 6.19, we depict
a plot of Im( f k ) versus Re( f k ) within a unit circle as we increase the incident energy
of the neutron from 1.0 to 2.75 keV in steps of 0.25 keV, where the binding energy of
the n−
18 C system is taken to be 250 keV. We find that as the energy is increased, the
point described by (Re f k , Im f k ) indicates an anticlockwise motion in the Argand
plot, thereby satisfying an important condition to be obeyed for the occurrence of a
resonance.
It may be interesting to ask whether the scattering cross section would still predict
the resonance-type structure if the pair energy of the bound system is increased to,
say, 200 keV or even less than that in which case the bound Efimov state in the
three-body system is found to occur. To answer this, we depict by a dashed curve the
plot of scattering cross section for n−
19 C scattering in Fig. 6.18a taking the (n−
18 C)
binding energy to be 200 keV. We find that the scattering cross section does not
produce the resonance-like peak structure and remains more or less constant with
the incident energy of the neutron.
6.8 Efimov States and Their Fano Resonances
in a Neutron-Rich Nucleus
In continuation of our investigation studying the appearance of resonances in elastic
scattering cross section as the Efimov states just become unbound, we have attempted
[102] to tie this result to yet another interesting but not fully familiar result that a