6.3 Resonant States of 11 Li, Probability Distributions and β-decay of Halo Analog States
75
state as the isobaric halo analog state consisting of
9 Li core with halo neutron–proton
pair having isospin T
h
= 0 and the angular momentum J
h
= 1.
We first recapitulate some of the essential steps to write down the wave function
of
11 Li including pair potentials in both the s- and p-states of n−
9 Li core. For the
n–n interaction, we choose the n–n singlet
1 S 0 potential:
p
12 |V 12 | p 12
= −
λ
2μ 12
g( p 12 )g
p
12
, g( p) =
1
p 2 + β 2 ,
(6.4)
where strength parameter λ = 18.6α
3 and range parameter β = 5.8α obtained so
as to reproduce the n–n spin-singlet scattering length a = −23.69 fm and effective
range r = 2.32 fm. Similarly, n−
9 Li separable potential is chosen as
p
31 |V 31 | p 31
=
l=0,1
(2l + 1)V
l
31
p 31 , p
31
P l (cos(θ )
V
l
31
p 31 , p
31
= −
λ
l
2μ 31
v
l
31 ( p 31 )v
l
31
p
l
31
, v
l
31 ( p) =
p
l
(p
2 +β
2
l )
l+1
(6.5)
with l = 0, 1 for the n-core
9 Li
interaction operative respectively in s- and pstates to reproduce a virtual S 1/2 state at an excitation energy of 250 keV [77] and
a p-wave resonance observed experimentally at E r = 0.538 ± 0.062 MeV with
resonance width = 0.358 ± 0.023 MeV [78]. The range and strength parameters
for these potentials are accordingly adjusted to reproduce these observables. These
are summarized in Table 6.3.
Using the pair-wise potential given in Eqs. (6.4) and (6.5) in the three-body
Schrodinger equation,
Table 6.3 Parameters and observables of pair-wise two-body interactions used in the three-body
models of 11 Li and 11 Be ∗
Binary system Partial wave (l) Strength parameter Range parameter Observables
n − 9 Li
0
λ 0 = 2.25α 3
β 0 = 3.0α
a s = −13.39 fm
r s = 4.93 fm
n − 9 Li
1
λ 1 = 2000.0α 5
β 1 = 5.6α
E r = 0.538 MeV
= 0.475 MeV
p − 9 Li
0
λ 0
c = 2.25α 3
β 0 = 3.0α
a sc = −23.98 fm
r sc = 1.14 fm
p − 9 Li
1
λ 1
c = 2000.0α 5
β 1 = 5.6α
E r = 2.42 MeV
= 6.56 MeV
n–p
0 (spin-triplet) λ 12 = 23.7α 3
β 12 = 5.5α
a = 5.43 fm
r = 1.9 fm
75
state as the isobaric halo analog state consisting of
9 Li core with halo neutron–proton
pair having isospin T
h
= 0 and the angular momentum J
h
= 1.
We first recapitulate some of the essential steps to write down the wave function
of
11 Li including pair potentials in both the s- and p-states of n−
9 Li core. For the
n–n interaction, we choose the n–n singlet
1 S 0 potential:
p
12 |V 12 | p 12
= −
λ
2μ 12
g( p 12 )g
p
12
, g( p) =
1
p 2 + β 2 ,
(6.4)
where strength parameter λ = 18.6α
3 and range parameter β = 5.8α obtained so
as to reproduce the n–n spin-singlet scattering length a = −23.69 fm and effective
range r = 2.32 fm. Similarly, n−
9 Li separable potential is chosen as
p
31 |V 31 | p 31
=
l=0,1
(2l + 1)V
l
31
p 31 , p
31
P l (cos(θ )
V
l
31
p 31 , p
31
= −
λ
l
2μ 31
v
l
31 ( p 31 )v
l
31
p
l
31
, v
l
31 ( p) =
p
l
(p
2 +β
2
l )
l+1
(6.5)
with l = 0, 1 for the n-core
9 Li
interaction operative respectively in s- and pstates to reproduce a virtual S 1/2 state at an excitation energy of 250 keV [77] and
a p-wave resonance observed experimentally at E r = 0.538 ± 0.062 MeV with
resonance width = 0.358 ± 0.023 MeV [78]. The range and strength parameters
for these potentials are accordingly adjusted to reproduce these observables. These
are summarized in Table 6.3.
Using the pair-wise potential given in Eqs. (6.4) and (6.5) in the three-body
Schrodinger equation,
Table 6.3 Parameters and observables of pair-wise two-body interactions used in the three-body
models of 11 Li and 11 Be ∗
Binary system Partial wave (l) Strength parameter Range parameter Observables
n − 9 Li
0
λ 0 = 2.25α 3
β 0 = 3.0α
a s = −13.39 fm
r s = 4.93 fm
n − 9 Li
1
λ 1 = 2000.0α 5
β 1 = 5.6α
E r = 0.538 MeV
= 0.475 MeV
p − 9 Li
0
λ 0
c = 2.25α 3
β 0 = 3.0α
a sc = −23.98 fm
r sc = 1.14 fm
p − 9 Li
1
λ 1
c = 2000.0α 5
β 1 = 5.6α
E r = 2.42 MeV
= 6.56 MeV
n–p
0 (spin-triplet) λ 12 = 23.7α 3
β 12 = 5.5α
a = 5.43 fm
r = 1.9 fm