74
A. Mercenne et al.
basis, can use drastically reduced configuration spaces with practically the same
accuracy of results, and has been successfully applied up to medium-mass nuclei
[10, 11]. Motivated by the need for calculated nuclear cross sections in experimental
research and astrophysical studies, and following the success of the NCSM/RGM
for light nuclei, we combine the SA-NCSM with the RGM. As a first step, we focus
on reactions of two clusters, in which the projectile is a nucleon.
2 Ab initio Symmetry-Adapted Framework for Nuclear
Reactions
In the RGM framework, the nucleons are organized within different groups, or
clusters, “resonating” through the inter-cluster exchange of nucleons. This antisymmetrization between the different clusters guarantees the Pauli exclusion principle,
which, along with the consideration of the cluster internal structure, is one of the
most important features of the approach. In the case of two clusters, the wave
function is written as (in notations of Ref. [6]):
|Ψ
J π T
=
ν
r
drr
2 g J π T
ν
(r)
r
ˆ
A |Φ
J π T
νr ,
(1)
where the index ν represents all quantum numbers that define channels and
partitions: ν = {(A − a)α 1 I 1 T 1 ; aα 2 I 2 T 2 ; s}, and the cluster states are defined as
|Φ J π T
νr =
(|(A − a)α 1 I 1 T 1 ⊗ |aα 2 I 2 T 2 )
(sT )
× Y (ˆ r A−a,a )
(J π T ) δ(r−r A−a,a )
rr A−a,a
. The
amplitudes g J π T
ν
(r) describe the relative motion between the target and the projectile for all channels ν, and the cross section can be extracted from their asymptotic
behavior. The g J π T
ν
(r) functions are the solutions to the Schrödinger equation:
ν
drr
2
H
J π T
ν ν
r, r
− EN
J π T
ν ν
r
, r
g J π T
ν
(r)
r
= 0 .
(2)
Here, the Hamiltonian H J π T
ν ν (r , r) and norm N J π T
ν ν (r , r) kernels are expressed
as Φ J π T
ν r | ˆ
A ˆ
O ˆ
A |Φ J π T
νr with ˆ
O being the identity and the Hamiltonian operator,
respectively, and where ˆ
A is the antisymmetrizer ensuring the Pauli exclusion
principle. The kernels are computed using the wave functions of the clusters. Eq. (2)
can then be solved using an R-matrix approach [12, 13].
An ab initio application of this approach is the NCSM/RGM [6], which uses
NCSM wave functions and realistic interactions. However, the method becomes
numerically challenging for heavier systems due to the size and complexity of the
configuration space. We address the limitation of the NCSM/RGM by combining the
SA-NCSM with the RGM formalism, where the former allows for the calculation
of the intermediate mass wave functions required by the RGM.
A. Mercenne et al.
basis, can use drastically reduced configuration spaces with practically the same
accuracy of results, and has been successfully applied up to medium-mass nuclei
[10, 11]. Motivated by the need for calculated nuclear cross sections in experimental
research and astrophysical studies, and following the success of the NCSM/RGM
for light nuclei, we combine the SA-NCSM with the RGM. As a first step, we focus
on reactions of two clusters, in which the projectile is a nucleon.
2 Ab initio Symmetry-Adapted Framework for Nuclear
Reactions
In the RGM framework, the nucleons are organized within different groups, or
clusters, “resonating” through the inter-cluster exchange of nucleons. This antisymmetrization between the different clusters guarantees the Pauli exclusion principle,
which, along with the consideration of the cluster internal structure, is one of the
most important features of the approach. In the case of two clusters, the wave
function is written as (in notations of Ref. [6]):
|Ψ
J π T
=
ν
r
drr
2 g J π T
ν
(r)
r
ˆ
A |Φ
J π T
νr ,
(1)
where the index ν represents all quantum numbers that define channels and
partitions: ν = {(A − a)α 1 I 1 T 1 ; aα 2 I 2 T 2 ; s}, and the cluster states are defined as
|Φ J π T
νr =
(|(A − a)α 1 I 1 T 1 ⊗ |aα 2 I 2 T 2 )
(sT )
× Y (ˆ r A−a,a )
(J π T ) δ(r−r A−a,a )
rr A−a,a
. The
amplitudes g J π T
ν
(r) describe the relative motion between the target and the projectile for all channels ν, and the cross section can be extracted from their asymptotic
behavior. The g J π T
ν
(r) functions are the solutions to the Schrödinger equation:
ν
drr
2
H
J π T
ν ν
r, r
− EN
J π T
ν ν
r
, r
g J π T
ν
(r)
r
= 0 .
(2)
Here, the Hamiltonian H J π T
ν ν (r , r) and norm N J π T
ν ν (r , r) kernels are expressed
as Φ J π T
ν r | ˆ
A ˆ
O ˆ
A |Φ J π T
νr with ˆ
O being the identity and the Hamiltonian operator,
respectively, and where ˆ
A is the antisymmetrizer ensuring the Pauli exclusion
principle. The kernels are computed using the wave functions of the clusters. Eq. (2)
can then be solved using an R-matrix approach [12, 13].
An ab initio application of this approach is the NCSM/RGM [6], which uses
NCSM wave functions and realistic interactions. However, the method becomes
numerically challenging for heavier systems due to the size and complexity of the
configuration space. We address the limitation of the NCSM/RGM by combining the
SA-NCSM with the RGM formalism, where the former allows for the calculation
of the intermediate mass wave functions required by the RGM.