76
6 Three-Body Approach to Structural Properties …
D(
p 12 ,
p 3 ; E)ψ(
p 12 ,
p 3 ; E) = −
i j,k
p i j
V i j
p
i j
ψ
p
i j ,
p k ; E
d
p i j
(i j, k) = (12, 3), (23, 1), (31, 2)
(6.6)
the analytical structure of three-body wave function in the cm system is written as:
ψ(
p 12 ,
p 3 ; E) = N D −1 (
p 12 ,
p 3 ; E)
⎡
⎢
⎣
g( p 12 )G 1 ( p 3 ) + v 0
23 ( p 23 )G 2 ( p 1 )
+v
(1)
23 ( p 23 )
ˆ
p 23 · ˆ
p 1
G 3 ( p 1 ) + v
(0)
31 ( p 31 )G 4 ( p 2 )
−v
(1)
31 ( p 31 )
ˆ
p 31 · ˆ
p 2
G 5 ( p 2 )
⎤
⎥
⎦
(6.7)
The spectator functions G 2 ( p 1 ) and G 4 ( p 2 ) describe the dynamics of one halo
neutron when the other halo neutrons and the core are interacting through s-wave.
Hence, their structure is exactly the same. Similarly, the functions G 3 ( p 1 ) and
G 5 ( p 2 ), which have the same structure, describe the dynamics of one neutron when
the second neutron and the core are interacting through p-wave. The function G 1 ( p 3 )
represents the dynamics of
9 Li core in the presence of the two neutrons. We thus have
three independent functions which satisfy three coupled integral equations. In order
to find solutions for these spectator functions, we substitute three-body function
Eq. (6.7) in the Schrodinger Eq. (6.6) and compare the similar terms on both sides.
This leads to a set of coupled integral equations that can be written in the closed form
as
G i ( p) = i
⎡
⎣ h i ( p)G i ( p) +
3
j=1
d q K i j (
p,
q; E)G j ( q)
⎤
⎦ , i = 1, 2, 3 (6.8)
The detailed structure of constants i , integrals h i and the kernels k i j has been
given in Ref. [54]. To symmetrize the kernel in Eq. (6.8), we use the following
transformations:
G 1 ( p) →
2 1
χ 1 ( p)
p
√
d p
, G 2 ( p) →
2
χ 2 ( p)
p
√
d p
, G 3 ( p) →
3
χ 3 ( p)
p
√
d p
(6.9)
The final symmetric eigen system can now be written as
3
j=1
q
K
i j (
p,
q; E)χ j (q) = η(E)χ i ( p), i = 1, 2, 3
(6.10)
where we introduce a parameter η(E), which corresponds to the eigenvalue of the
kernel of the above integral equation and K
i j ( p, q; E) are the integral operators:
6 Three-Body Approach to Structural Properties …
D(
p 12 ,
p 3 ; E)ψ(
p 12 ,
p 3 ; E) = −
i j,k
p i j
V i j
p
i j
ψ
p
i j ,
p k ; E
d
p i j
(i j, k) = (12, 3), (23, 1), (31, 2)
(6.6)
the analytical structure of three-body wave function in the cm system is written as:
ψ(
p 12 ,
p 3 ; E) = N D −1 (
p 12 ,
p 3 ; E)
⎡
⎢
⎣
g( p 12 )G 1 ( p 3 ) + v 0
23 ( p 23 )G 2 ( p 1 )
+v
(1)
23 ( p 23 )
ˆ
p 23 · ˆ
p 1
G 3 ( p 1 ) + v
(0)
31 ( p 31 )G 4 ( p 2 )
−v
(1)
31 ( p 31 )
ˆ
p 31 · ˆ
p 2
G 5 ( p 2 )
⎤
⎥
⎦
(6.7)
The spectator functions G 2 ( p 1 ) and G 4 ( p 2 ) describe the dynamics of one halo
neutron when the other halo neutrons and the core are interacting through s-wave.
Hence, their structure is exactly the same. Similarly, the functions G 3 ( p 1 ) and
G 5 ( p 2 ), which have the same structure, describe the dynamics of one neutron when
the second neutron and the core are interacting through p-wave. The function G 1 ( p 3 )
represents the dynamics of
9 Li core in the presence of the two neutrons. We thus have
three independent functions which satisfy three coupled integral equations. In order
to find solutions for these spectator functions, we substitute three-body function
Eq. (6.7) in the Schrodinger Eq. (6.6) and compare the similar terms on both sides.
This leads to a set of coupled integral equations that can be written in the closed form
as
G i ( p) = i
⎡
⎣ h i ( p)G i ( p) +
3
j=1
d q K i j (
p,
q; E)G j ( q)
⎤
⎦ , i = 1, 2, 3 (6.8)
The detailed structure of constants i , integrals h i and the kernels k i j has been
given in Ref. [54]. To symmetrize the kernel in Eq. (6.8), we use the following
transformations:
G 1 ( p) →
2 1
χ 1 ( p)
p
√
d p
, G 2 ( p) →
2
χ 2 ( p)
p
√
d p
, G 3 ( p) →
3
χ 3 ( p)
p
√
d p
(6.9)
The final symmetric eigen system can now be written as
3
j=1
q
K
i j (
p,
q; E)χ j (q) = η(E)χ i ( p), i = 1, 2, 3
(6.10)
where we introduce a parameter η(E), which corresponds to the eigenvalue of the
kernel of the above integral equation and K
i j ( p, q; E) are the integral operators: