78
6 Three-Body Approach to Structural Properties …
obtained by solving the integral equation. The analytical expressions which give the
best fit are:
G 1 ( p) =
A1
1 + ( p/ p 0 )
A2
, A1 = 1.58α
−3
, p 0 = 0.9131α, A2 = 2.121
(6.13)
G 2 ( p) = G 4 ( p) =
B1
1 + ( p/ p 0 )
B2
, B1 = 0.343α
−3
, p 0 = 0.721α, B2 = 2.032
(6.14)
G 3 ( p) = G 5 ( p) = C1(1 − exp(− p/C2))
C4 exp(−( p/C3)),
C1 = 0.6246α, C2 = 1.3725α, C3 = 1.9530α, C4 = 1.1309 (6.15)
With these algebraic representations for the spectator functions as input in threebody wave functions, we compute the quadrature of the integral in Eq. (6.12) to
evaluate the normalization constant N, keeping the momentum vectors
p 12 , and
p 3
as the basis vectors to which all the other vectors such as
p 31 ,
p 2 etc., are properly
transformed. The value of N comes out to be 0.0732α
4 . The ground state wave
function is thus completely determined up to the normalization constant, which will
be used to study the probability distribution, momentum distributions and β-decay
to
11 Be
∗ state and d +
9 Li channel.
6.3.2 Probability Distribution in 11 Li
Defining
P
p 12 ,
p 3,
= ψ(
p 12 ,
p 3 ; E)ψ
∗
(
p 12 ,
p 3 ; E)d
p 12 d
p 3
(6.16)
as the probability distribution of finding
9 Li with momentum p 3 in the volume
element p
2
3 d p 3 d and the correlation momentum p 12 of the two halo neutrons in the
element p
2
12 d p 12 d, we can get some insight by plotting the probability density
P( p 12 , p 3 ) == p
2
12 p
2
3
d 12 d 3 |ψ(
p 12 ,
p 3 )|
2
(6.17)
versus the momenta p 12 , and p 13 in a three-dimensional plot shown in Fig. 6.6.
In Figs. 6.7 and 6.8, we depict the two-dimensional plot of the probability density
against the momenta p 12 , and p 3 , respectively, (keeping other momenta fixed). The
curves in Figs. 6.7 and 6.8 are the ones that pass through the peak in Fig. 6.6.
These figures show that the most probable values of momenta p 12 , and p 13 are
18.28 MeV/c and 30.16 MeV/c, respectively. The full width half maximum (FWHM)
12 in Fig. 6.7 is 27.42 MeV/c, and by uncertainty principle, this gives the most
probable distance between the two halo neutrons as
6 Three-Body Approach to Structural Properties …
obtained by solving the integral equation. The analytical expressions which give the
best fit are:
G 1 ( p) =
A1
1 + ( p/ p 0 )
A2
, A1 = 1.58α
−3
, p 0 = 0.9131α, A2 = 2.121
(6.13)
G 2 ( p) = G 4 ( p) =
B1
1 + ( p/ p 0 )
B2
, B1 = 0.343α
−3
, p 0 = 0.721α, B2 = 2.032
(6.14)
G 3 ( p) = G 5 ( p) = C1(1 − exp(− p/C2))
C4 exp(−( p/C3)),
C1 = 0.6246α, C2 = 1.3725α, C3 = 1.9530α, C4 = 1.1309 (6.15)
With these algebraic representations for the spectator functions as input in threebody wave functions, we compute the quadrature of the integral in Eq. (6.12) to
evaluate the normalization constant N, keeping the momentum vectors
p 12 , and
p 3
as the basis vectors to which all the other vectors such as
p 31 ,
p 2 etc., are properly
transformed. The value of N comes out to be 0.0732α
4 . The ground state wave
function is thus completely determined up to the normalization constant, which will
be used to study the probability distribution, momentum distributions and β-decay
to
11 Be
∗ state and d +
9 Li channel.
6.3.2 Probability Distribution in 11 Li
Defining
P
p 12 ,
p 3,
= ψ(
p 12 ,
p 3 ; E)ψ
∗
(
p 12 ,
p 3 ; E)d
p 12 d
p 3
(6.16)
as the probability distribution of finding
9 Li with momentum p 3 in the volume
element p
2
3 d p 3 d and the correlation momentum p 12 of the two halo neutrons in the
element p
2
12 d p 12 d, we can get some insight by plotting the probability density
P( p 12 , p 3 ) == p
2
12 p
2
3
d 12 d 3 |ψ(
p 12 ,
p 3 )|
2
(6.17)
versus the momenta p 12 , and p 13 in a three-dimensional plot shown in Fig. 6.6.
In Figs. 6.7 and 6.8, we depict the two-dimensional plot of the probability density
against the momenta p 12 , and p 3 , respectively, (keeping other momenta fixed). The
curves in Figs. 6.7 and 6.8 are the ones that pass through the peak in Fig. 6.6.
These figures show that the most probable values of momenta p 12 , and p 13 are
18.28 MeV/c and 30.16 MeV/c, respectively. The full width half maximum (FWHM)
12 in Fig. 6.7 is 27.42 MeV/c, and by uncertainty principle, this gives the most
probable distance between the two halo neutrons as