414
9 The Unification of Structure and Reaction Frameworks
where |u HO
n is a harmonic oscillator state. One thus has to consider the following
channel wave functions:
|J M
HO
c =
+∞
0
u HO
c (r)
r
|c, r r
2 dr =
n
HO
c |u n [a
†
nn c j c
|J T c
J
M
,
(9.33)
where the Berggren basis expansion of the |c, r channel of Eq. (9.5)) has been
used. The expansion in terms of harmonic oscillator states is hereby justified by
the fact that |J T i (f ) targets are localized. The electromagnetic matrix elements
[ c f f || ˆ
O L (p)||J i c i ] HO are calculated using the standard Gamow shell model
formulas as the |J M HO
c
states are expanded in a Berggren many-body basis of
Slater determinants.
The last many-body matrix element of Eq. (9.28), equal to f || ˆ
O L ||J i
HO
nas , is
calculated using Eqs. (9.30), (9.31), and replacing u c i(f) (r) by u HO
c i(f)
(r) in Eq. (9.32).
The total many-body matrix element f || ˆ
O L ||J i of Eq. (9.28) is then obtained by
summing over all c i , c f channels.
9.1.3 Cross Sections
Once the solutions for many-body wave function of the Gamow shell model Hamiltonian in the coupled-channel representation have been calculated, one can directly
apply the standard formulas of reaction theory providing with reactions cross
sections. As the coupled-channel equations of the Gamow shell model Hamiltonian
are defined with cluster orbital shell model coordinates, whose utilization is not
standard in reaction theory, so one will express the reaction cross section formulas
in these new coordinates for clarity.
9.1.3.1 Cross Sections of Direct Reactions
In order to calculate elastic/inelastic scattering cross sections, one has to state
the asymptotic form of many-body scattering wave functions. Indeed, these cross
sections are direct function of the phase shifts of many-body scattering wave
functions.
Let us consider for this a scattering reaction from a given entrance channel
denoted as “e.” All the other c indices stand for the exit channels. Let us rewrite
Eq. (9.19) for a given entrance channel
|Ψ
e,J A
M A
=
c
+∞
0
u
e,J A
c
(r)
r
|(c, r) r
2 dr ,
(9.34)
9 The Unification of Structure and Reaction Frameworks
where |u HO
n is a harmonic oscillator state. One thus has to consider the following
channel wave functions:
|J M
HO
c =
+∞
0
u HO
c (r)
r
|c, r r
2 dr =
n
HO
c |u n [a
†
nn c j c
|J T c
J
M
,
(9.33)
where the Berggren basis expansion of the |c, r channel of Eq. (9.5)) has been
used. The expansion in terms of harmonic oscillator states is hereby justified by
the fact that |J T i (f ) targets are localized. The electromagnetic matrix elements
[ c f f || ˆ
O L (p)||J i c i ] HO are calculated using the standard Gamow shell model
formulas as the |J M HO
c
states are expanded in a Berggren many-body basis of
Slater determinants.
The last many-body matrix element of Eq. (9.28), equal to f || ˆ
O L ||J i
HO
nas , is
calculated using Eqs. (9.30), (9.31), and replacing u c i(f) (r) by u HO
c i(f)
(r) in Eq. (9.32).
The total many-body matrix element f || ˆ
O L ||J i of Eq. (9.28) is then obtained by
summing over all c i , c f channels.
9.1.3 Cross Sections
Once the solutions for many-body wave function of the Gamow shell model Hamiltonian in the coupled-channel representation have been calculated, one can directly
apply the standard formulas of reaction theory providing with reactions cross
sections. As the coupled-channel equations of the Gamow shell model Hamiltonian
are defined with cluster orbital shell model coordinates, whose utilization is not
standard in reaction theory, so one will express the reaction cross section formulas
in these new coordinates for clarity.
9.1.3.1 Cross Sections of Direct Reactions
In order to calculate elastic/inelastic scattering cross sections, one has to state
the asymptotic form of many-body scattering wave functions. Indeed, these cross
sections are direct function of the phase shifts of many-body scattering wave
functions.
Let us consider for this a scattering reaction from a given entrance channel
denoted as “e.” All the other c indices stand for the exit channels. Let us rewrite
Eq. (9.19) for a given entrance channel
|Ψ
e,J A
M A
=
c
+∞
0
u
e,J A
c
(r)
r
|(c, r) r
2 dr ,
(9.34)