New Symmetry-Adapted ab initio Approach to Nuclear Reactions
75
In the SA-NCSM, the microscopic many-body basis is based on the spherical
harmonic oscillator single-particle basis, and labeled by irreducible representations
according to the group chain:
SU(3) (λμ) ⊃
κ
SO(3) L ⊃ SO(2) M L .
(3)
Consequently, for any given total spin J and its projection M, the wave function
of a nucleus will be described within a basis {|α i (λ i μ i )κ i (L i S i )J M} with each
component weighted by a coefficient C i . Here α i represents additional quantum
numbers needed to enumerate the complete shell-model space.
In the SA-RGM, the channels are defined by coupling each component of the
SA-NCSM wave functions between the projectile and the target. Consequently, the
channels with good SU(3) spin and isospin quantum numbers are given in the case
of one nucleon projectile as
|Φ
ρ(λμ)κ(LS)J MT M T
γ n
=
|α 1 (λ 1 μ 1 )S 1 T 1 ⊗ |(n0)
1
2
1
2
ρ(λμ)κ(LS)J MT M T
,
(4)
where the index γ ≡ {α 1 (λ 1 μ 1 )S 1 T 1 ; (n0)
1
2
1
2 } labels the channel basis, with (n0)
representing the SU(3) labels of the projectile with spin
1
2 and isospin
1
2 . Note that
there is no dependence on the orbital momentum of the target and the projectile.
In this basis, the exchange matrix, which ensures the antisymmetrization in the
kernels, has the following form (in conventional notations [14]):
Φ
ρ (λ μ )κ (L S )J MT M T
γ n
| ˆ
P A,A−1 |Φ
ρ(λμ)κ(LS)J MT M T
γ n
=
1
A − 1
δ ρρ δ (λμ)(λ μ ) δ κκ δ LL δ SS
τρ o (λ o μ o )
S o ¯
ρ
τ S o S
1 T
1
(−1)
n+n −(λ o +μ o )
× (−1)
T 1 +
1
2 +T (−1)
S 1 +
1
2 +S
S 1 S o
S
1
1
2 S
1
2
T 1 τ T
1
1
2 T
1
2
×
dim(λ o μ o )
dim(n0)
U
(λ 1 μ 1 )(λ o μ o )(λ
μ
)(n
0); (λ
1 μ
1 ) ¯
ρρ
(n0)ρ o ρ
× ×α
1 (λ
1 μ
1 )S
1 T
1 | ||
a
†
(n0)
1
2
1
2
⊗ ˜
a ( ˜
0n )
1
2
1
2
ρ o (λ o μ o )S o τ
|||α 1 (λ 1 μ 1 )S 1 T 1 ¯
ρ .
(5)
Clearly, the presence of the delta Kronecker functions in Eq. (5) makes the exchange
matrix diagonal within this SU(3) basis, allowing for several numerical simplifications [15]. Furthermore, matrix calculations avoid complications of dealing with the
75
In the SA-NCSM, the microscopic many-body basis is based on the spherical
harmonic oscillator single-particle basis, and labeled by irreducible representations
according to the group chain:
SU(3) (λμ) ⊃
κ
SO(3) L ⊃ SO(2) M L .
(3)
Consequently, for any given total spin J and its projection M, the wave function
of a nucleus will be described within a basis {|α i (λ i μ i )κ i (L i S i )J M} with each
component weighted by a coefficient C i . Here α i represents additional quantum
numbers needed to enumerate the complete shell-model space.
In the SA-RGM, the channels are defined by coupling each component of the
SA-NCSM wave functions between the projectile and the target. Consequently, the
channels with good SU(3) spin and isospin quantum numbers are given in the case
of one nucleon projectile as
|Φ
ρ(λμ)κ(LS)J MT M T
γ n
=
|α 1 (λ 1 μ 1 )S 1 T 1 ⊗ |(n0)
1
2
1
2
ρ(λμ)κ(LS)J MT M T
,
(4)
where the index γ ≡ {α 1 (λ 1 μ 1 )S 1 T 1 ; (n0)
1
2
1
2 } labels the channel basis, with (n0)
representing the SU(3) labels of the projectile with spin
1
2 and isospin
1
2 . Note that
there is no dependence on the orbital momentum of the target and the projectile.
In this basis, the exchange matrix, which ensures the antisymmetrization in the
kernels, has the following form (in conventional notations [14]):
Φ
ρ (λ μ )κ (L S )J MT M T
γ n
| ˆ
P A,A−1 |Φ
ρ(λμ)κ(LS)J MT M T
γ n
=
1
A − 1
δ ρρ δ (λμ)(λ μ ) δ κκ δ LL δ SS
τρ o (λ o μ o )
S o ¯
ρ
τ S o S
1 T
1
(−1)
n+n −(λ o +μ o )
× (−1)
T 1 +
1
2 +T (−1)
S 1 +
1
2 +S
S 1 S o
S
1
1
2 S
1
2
T 1 τ T
1
1
2 T
1
2
×
dim(λ o μ o )
dim(n0)
U
(λ 1 μ 1 )(λ o μ o )(λ
μ
)(n
0); (λ
1 μ
1 ) ¯
ρρ
(n0)ρ o ρ
× ×α
1 (λ
1 μ
1 )S
1 T
1 | ||
a
†
(n0)
1
2
1
2
⊗ ˜
a ( ˜
0n )
1
2
1
2
ρ o (λ o μ o )S o τ
|||α 1 (λ 1 μ 1 )S 1 T 1 ¯
ρ .
(5)
Clearly, the presence of the delta Kronecker functions in Eq. (5) makes the exchange
matrix diagonal within this SU(3) basis, allowing for several numerical simplifications [15]. Furthermore, matrix calculations avoid complications of dealing with the