6.9 Generalizing the Study of Efimov Effect in Neutron-Rich …
111
The overall behavior is very much similar to the results obtained earlier for the
20 C
nucleus with minor differences in the binding energies of the ground state. In the
present case, the ground state appears to be little more strongly bound compared
to the 2n separation energy in
20 C. This can most likely be attributed to the much
heavier mass of the core. In addition, the first and second Efimov states move into the
continuum at a lower two-body energy compared to that in
20 C. In the case of 2-n halo
nuclei, like
38 Mg and
32 Ne, the 2n separation energies are comparable to that of
20 C
(2570 and 1970 keV, respectively) with the n-core systems of
37 Mg and
31 Ne which
are bound by 250 and 330 keV, respectively [90]. The results of the calculations for
38 Mg and
32 Ne are summarized in columns 5, 6, 7 and 8, 9, 10, respectively. For
38 Mg, the second excited state disappears in the continuum at around 140 keV n-core
energy, while the first excited state disappears at around 220 keV. In the case of
32 Ne,
same trend continues to appear.
To probe the evolution of the Efimov states into the resonant states, we set out
to study these cases in the scattering sector using the formalism developed earlier
[99, 102]. Thus, the integral Eq. (6.61) for the scattering amplitude of n-(core + n)
scattering below the three-body breakup threshold is computed for these cases. The
results of elastic scattering cross sections for scattering by n off the bound n-core
systems are depicted in the plots given in Fig. 6.23: for the nucleus with a heavy core
of mass 100 [shown in (a) and (b)], by (n +
36 Mg as core) shown in (c and d) and by
(n +
30 Ne as core) depicted in (e and f) for two different n-core interaction energies
of 250 keV and 150 keV. A similar asymmetric resonance structure with the centroid
around 1.5–1.6 keV and width of around 0.5–0.6 keV is observed in all these cases.
This behavior is remarkably similar to that obtained in n −
19 C elastic scattering.
We have also checked that the scattering length of the n-(core + n) system to be
positive and large, thereby supporting a bound state. The results presented for the
hypothetical nucleus with a heavy core (mass = 100) with two valence halo neutrons
exhibit a similar behavior to that of realistic 2-n halo nuclei thereby confirming that
these features are general enough to persist over a wide mass range and therefore
can offer, from the experimental standpoint, a wide range of halo nuclear systems to
search for the occurrence of the Efimov states.
6.10 Efimov Effect in Halo Nuclei Using Effective Field
Theory
The groundwork for deriving the equations for two- and three-body scattering
problem within the framework of effective field theory has already been laid down
in the section on effective field theory. In the context of halo nuclear systems, we
also set up there the equations for the s-wave scattering amplitude of neutron off the
n−
18 C
bound state target.
Here, we shall illustrate the subtraction procedure for renormalizing these equations in order to compute the scattering amplitude at finite energies. Let us consider,
111
The overall behavior is very much similar to the results obtained earlier for the
20 C
nucleus with minor differences in the binding energies of the ground state. In the
present case, the ground state appears to be little more strongly bound compared
to the 2n separation energy in
20 C. This can most likely be attributed to the much
heavier mass of the core. In addition, the first and second Efimov states move into the
continuum at a lower two-body energy compared to that in
20 C. In the case of 2-n halo
nuclei, like
38 Mg and
32 Ne, the 2n separation energies are comparable to that of
20 C
(2570 and 1970 keV, respectively) with the n-core systems of
37 Mg and
31 Ne which
are bound by 250 and 330 keV, respectively [90]. The results of the calculations for
38 Mg and
32 Ne are summarized in columns 5, 6, 7 and 8, 9, 10, respectively. For
38 Mg, the second excited state disappears in the continuum at around 140 keV n-core
energy, while the first excited state disappears at around 220 keV. In the case of
32 Ne,
same trend continues to appear.
To probe the evolution of the Efimov states into the resonant states, we set out
to study these cases in the scattering sector using the formalism developed earlier
[99, 102]. Thus, the integral Eq. (6.61) for the scattering amplitude of n-(core + n)
scattering below the three-body breakup threshold is computed for these cases. The
results of elastic scattering cross sections for scattering by n off the bound n-core
systems are depicted in the plots given in Fig. 6.23: for the nucleus with a heavy core
of mass 100 [shown in (a) and (b)], by (n +
36 Mg as core) shown in (c and d) and by
(n +
30 Ne as core) depicted in (e and f) for two different n-core interaction energies
of 250 keV and 150 keV. A similar asymmetric resonance structure with the centroid
around 1.5–1.6 keV and width of around 0.5–0.6 keV is observed in all these cases.
This behavior is remarkably similar to that obtained in n −
19 C elastic scattering.
We have also checked that the scattering length of the n-(core + n) system to be
positive and large, thereby supporting a bound state. The results presented for the
hypothetical nucleus with a heavy core (mass = 100) with two valence halo neutrons
exhibit a similar behavior to that of realistic 2-n halo nuclei thereby confirming that
these features are general enough to persist over a wide mass range and therefore
can offer, from the experimental standpoint, a wide range of halo nuclear systems to
search for the occurrence of the Efimov states.
6.10 Efimov Effect in Halo Nuclei Using Effective Field
Theory
The groundwork for deriving the equations for two- and three-body scattering
problem within the framework of effective field theory has already been laid down
in the section on effective field theory. In the context of halo nuclear systems, we
also set up there the equations for the s-wave scattering amplitude of neutron off the
n−
18 C
bound state target.
Here, we shall illustrate the subtraction procedure for renormalizing these equations in order to compute the scattering amplitude at finite energies. Let us consider,