74
6 Three-Body Approach to Structural Properties …
Table 6.2 Values of the root mean square radii of neutron-neutron and neutron– 9 Li separations
calculated using Eqs. (6.1) and (6.2) for different binding energies of 11 Li
B.E of 11 Li (MeV) r nn (fm) r nn (fm) (from other
model calculations [4])
r nc (fm) r nc (fm) (from other
model calculations [4])
0.20
10.63
10.93
0.25
9.9
9.86
0.315
8.93
6.24–7.80
8.87
5.47–6.40
analysis, the inclusion of the repulsive s-state n–n interaction in the spin-singlet state
may possibly play a marginal role but it is unlikely to bring about a significant change.
Having calculated r nn and r nc using three-body model, we can also compute the
r matter for
11 Li using the relation given by [71], i.e.,
r
2
matter
=
A c
A
r
2
core
+
1
A
ρ
2
,
(6.3)
where
r
2
core
is the mean square value of the radius of
9 Li core and the radial variable
ρ
2
= x
2
+ y
2 ,
x = ( r 1 − −
r 2 )/
√
2, and
y =
√
(2 A c /A)
( r 1 + +
r 2 )/2 − −
r 3
. Working
in this representation and taking the values of the
9 Li core to be 2.3 fm, we obtain the
value of matter radius to be 3.6 fm, which is to be compared with the experimental
value 3.14 ± 0.06 fm.
Thus, in this work, we have attempted to extract the structural information about
the
11 Li nucleus within the framework of three-body model employing separable
potentials. We find that even with simplified s-state interaction between different
pairs, most of the central gross features of the system can be accounted in reasonable
agreement with experimental data.
6.3 Resonant States of 11 Li, Probability Distributions
and β-decay of Halo Analog States
Having achieved a rather broad qualitative agreement with the experimental data
of some of the important properties of
11 Li using the above-simplified model, we
later carried out detailed studies, in a series of papers [72, 73], extending the model
described above to investigate (i) energy positions and widths of resonant states
of
11 Li above breakup threshold, (ii) probability distributions, momentum distributions and n–n correlations in the ground state, and (iii) β-decay of
11 Li to halo
analog of
11 Be
∗ and to the deuteron +
9 Li channel. Experiments suggest [74, 75]
that β - decay of
11 Li into high-lying state of the daughter nucleus
11 Be
∗ (18.3 MeV)
is the super-allowed Gamow–Teller decay with reduced Gamow–Teller transition
probability B GT > 1. Such a large value of B GT implies a large overlap of
11 Be
∗ state
with the ground state of
11 Li. Following Zhukov et al. [76], we considered here
11 Be
∗
6 Three-Body Approach to Structural Properties …
Table 6.2 Values of the root mean square radii of neutron-neutron and neutron– 9 Li separations
calculated using Eqs. (6.1) and (6.2) for different binding energies of 11 Li
B.E of 11 Li (MeV) r nn (fm) r nn (fm) (from other
model calculations [4])
r nc (fm) r nc (fm) (from other
model calculations [4])
0.20
10.63
10.93
0.25
9.9
9.86
0.315
8.93
6.24–7.80
8.87
5.47–6.40
analysis, the inclusion of the repulsive s-state n–n interaction in the spin-singlet state
may possibly play a marginal role but it is unlikely to bring about a significant change.
Having calculated r nn and r nc using three-body model, we can also compute the
r matter for
11 Li using the relation given by [71], i.e.,
r
2
matter
=
A c
A
r
2
core
+
1
A
ρ
2
,
(6.3)
where
r
2
core
is the mean square value of the radius of
9 Li core and the radial variable
ρ
2
= x
2
+ y
2 ,
x = ( r 1 − −
r 2 )/
√
2, and
y =
√
(2 A c /A)
( r 1 + +
r 2 )/2 − −
r 3
. Working
in this representation and taking the values of the
9 Li core to be 2.3 fm, we obtain the
value of matter radius to be 3.6 fm, which is to be compared with the experimental
value 3.14 ± 0.06 fm.
Thus, in this work, we have attempted to extract the structural information about
the
11 Li nucleus within the framework of three-body model employing separable
potentials. We find that even with simplified s-state interaction between different
pairs, most of the central gross features of the system can be accounted in reasonable
agreement with experimental data.
6.3 Resonant States of 11 Li, Probability Distributions
and β-decay of Halo Analog States
Having achieved a rather broad qualitative agreement with the experimental data
of some of the important properties of
11 Li using the above-simplified model, we
later carried out detailed studies, in a series of papers [72, 73], extending the model
described above to investigate (i) energy positions and widths of resonant states
of
11 Li above breakup threshold, (ii) probability distributions, momentum distributions and n–n correlations in the ground state, and (iii) β-decay of
11 Li to halo
analog of
11 Be
∗ and to the deuteron +
9 Li channel. Experiments suggest [74, 75]
that β - decay of
11 Li into high-lying state of the daughter nucleus
11 Be
∗ (18.3 MeV)
is the super-allowed Gamow–Teller decay with reduced Gamow–Teller transition
probability B GT > 1. Such a large value of B GT implies a large overlap of
11 Be
∗ state
with the ground state of
11 Li. Following Zhukov et al. [76], we considered here
11 Be
∗