70
6 Three-Body Approach to Structural Properties …
6.2 11 Li Halo Nucleus as a Three-Body System: Ground
State Properties (Binding Energy, Matter Radius, n–n
and n-core Correlations)
One of the earliest examples of halo nuclei which attracted a great deal of attention on the experimental and theoretical fronts is the
11 Li nucleus. Experimentally,
its abnormally small two-neutron separation energy (≈ 250 keV), large root mean
square radius (3.14 ± 0.06 fm) and the existence of a neutron halo are some of
the striking features experimentally revealed by measurements of interaction cross
sections and by momentum distribution of projectile fragments. On the theoretical
side, the conventional shell model and the Hartree–Fock calculations [64, 65] met
with only a limited success. On the other hand, there was sufficient experimental
evidence to view
11 Li nucleus having a cluster-like structure as composed of
9 Li
core with two neutrons loosely bound to the core. With this picture in view, we
[66] proposed, as early as in 1994, a three-body model employing a two-body separable potential to present some preliminary results of the calculations on the basic
properties of
11 Li nucleus.
The formulation of three-body (n−n−core) system using separable s-state separable potential has already been developed in Sect. 4, where the coupled integral
equations for the spectator functions, G(
p) and F(
p), [see Eqs. (4.26)–(4.28)], were
derived. This formalism can be easily adapted to the case of
11 Li, considered as a
three-body system of n, n and structure-less
9 Li core. For the n–n potential:
V 12 = −
λ n
2μ 12
g( p 12 )g
p
12
, where μ 12 = m/2(m 1 = m 2 = m, mass of the neutron)
and λ n is the strength parameter of n–n interaction and the momentum-dependent
function g(p) is taken as 1/
p
2
+ β
2
. From the low-energy n–n scattering data, the
values of the strength parameter λ n and the range parameter β are estimated as:
(i) λ n = 23.43α
3 ; β = 6.255α, which reproduce the value of n–n singlet scattering
length, a nn = −23.69 fm and the effective ranger r nn = 2.15 fm .
(ii) λ n = 18.6α
3 , β = 5.8α which gives a nn = −23.69 fm and r nn = 2.32 fm.
As far as the n –
9 Li interaction, the available experimental data are rather limited:
The n –
9 Li system shows a weakly attractive unbound state of 0.8 MeV. Assuming
this to be a virtual state, the parameters λ c = 14.0α
3 and β c = 5.5α reproduce the
value of n-core scattering length a nc = −7.7 fm and r nc = 2.67 fm.
Thus, starting from the coupled integral Eqs. (4.26) and after performing the
angular integrations, we compute these equations numerically, which we solve as
an eigenvalue problem. In fact, we feed the value of three-body binding energy E
as input and solve the equations as eigenvalue for the two-body strength parameter
λ c . The numerical values so obtained are compared with those determined by the
two-body analysis, as depicted in Table 6.1 [66].
It may be noted that the
11 Li system even with 200 keV seems to be little overbound. This may not be surprising if we keep in mind that the singlet n–n interaction
used here is oversimplified in the sense that it does not include a repulsive core which
should in all probability reduce binding marginally. From the table, it is also clear that
6 Three-Body Approach to Structural Properties …
6.2 11 Li Halo Nucleus as a Three-Body System: Ground
State Properties (Binding Energy, Matter Radius, n–n
and n-core Correlations)
One of the earliest examples of halo nuclei which attracted a great deal of attention on the experimental and theoretical fronts is the
11 Li nucleus. Experimentally,
its abnormally small two-neutron separation energy (≈ 250 keV), large root mean
square radius (3.14 ± 0.06 fm) and the existence of a neutron halo are some of
the striking features experimentally revealed by measurements of interaction cross
sections and by momentum distribution of projectile fragments. On the theoretical
side, the conventional shell model and the Hartree–Fock calculations [64, 65] met
with only a limited success. On the other hand, there was sufficient experimental
evidence to view
11 Li nucleus having a cluster-like structure as composed of
9 Li
core with two neutrons loosely bound to the core. With this picture in view, we
[66] proposed, as early as in 1994, a three-body model employing a two-body separable potential to present some preliminary results of the calculations on the basic
properties of
11 Li nucleus.
The formulation of three-body (n−n−core) system using separable s-state separable potential has already been developed in Sect. 4, where the coupled integral
equations for the spectator functions, G(
p) and F(
p), [see Eqs. (4.26)–(4.28)], were
derived. This formalism can be easily adapted to the case of
11 Li, considered as a
three-body system of n, n and structure-less
9 Li core. For the n–n potential:
V 12 = −
λ n
2μ 12
g( p 12 )g
p
12
, where μ 12 = m/2(m 1 = m 2 = m, mass of the neutron)
and λ n is the strength parameter of n–n interaction and the momentum-dependent
function g(p) is taken as 1/
p
2
+ β
2
. From the low-energy n–n scattering data, the
values of the strength parameter λ n and the range parameter β are estimated as:
(i) λ n = 23.43α
3 ; β = 6.255α, which reproduce the value of n–n singlet scattering
length, a nn = −23.69 fm and the effective ranger r nn = 2.15 fm .
(ii) λ n = 18.6α
3 , β = 5.8α which gives a nn = −23.69 fm and r nn = 2.32 fm.
As far as the n –
9 Li interaction, the available experimental data are rather limited:
The n –
9 Li system shows a weakly attractive unbound state of 0.8 MeV. Assuming
this to be a virtual state, the parameters λ c = 14.0α
3 and β c = 5.5α reproduce the
value of n-core scattering length a nc = −7.7 fm and r nc = 2.67 fm.
Thus, starting from the coupled integral Eqs. (4.26) and after performing the
angular integrations, we compute these equations numerically, which we solve as
an eigenvalue problem. In fact, we feed the value of three-body binding energy E
as input and solve the equations as eigenvalue for the two-body strength parameter
λ c . The numerical values so obtained are compared with those determined by the
two-body analysis, as depicted in Table 6.1 [66].
It may be noted that the
11 Li system even with 200 keV seems to be little overbound. This may not be surprising if we keep in mind that the singlet n–n interaction
used here is oversimplified in the sense that it does not include a repulsive core which
should in all probability reduce binding marginally. From the table, it is also clear that