418
9 The Unification of Structure and Reaction Frameworks
9.1.3.3 Cross Sections of Radiative Capture Reactions
One now presents the calculation of radiative capture reaction cross sections in
coupled-channel Gamow shell model. The differential cross section for one-nucleon
radiative capture reaction from an initial A-body state |E i M i , where the projectile
is a plane wave of energy E i , to a final A-body state |J f M f of energy E f reads
[49–51]
dσ
dΩ γ
=
q
8πk
e 2
¯
hc
μ u c 2
¯
hc
1
2(2J T + 1)
×
M i ,M f ,
M T ,M L ,
P γ ,m s
L
i
L
2π(2L + 1)(L + 1)
L
q L
k
P γ
(2L + 1)!!
×D
L
M L P (ϕ γ , θ γ , 0) f M f |M L,M L |E i M i
2
=
q
8πk
e 2
¯
hc
μ u c 2
¯
hc
1
2(2J T + 1)
×
M i ,M f ,
M T ,M L ,
P γ ,m s
L
g
L
M L ,P (k, q, ϕ γ , θ γ ) f M f |M L,M L |E i M i
2
,
(9.48)
with
g
L
M L ,P γ
(k, q, ϕ γ , θ γ ) = i
L
2π(2L + 1)(L + 1)
L
q L
k
×
P γ
(2L + 1)!!
D
L
M L P γ
(ϕ γ , θ γ , 0) ,
(9.49)
and
|E i M i =
J i ,c e
|(J i M i ) c e c e 0 sm s |j c e m s c e m s J T M T |J i M i ,
(9.50)
where one sums over all entrance channels c e and
|(J i M i ) c e =
c
|(J i M i ) c e c .
(9.51)