416
9 The Unification of Structure and Reaction Frameworks
behavior, that is [3, 4, 6]
|ϕ
˜
e
m ˜
e M ˜
e
T
−→ exp (i [kz + η ln(k(r − z))]) ·
|χ 1/2 ⊗ |Ψ
J ˜
e
T
T
+
˜
cm ˜
c M ˜
c
T
f ˜
em ˜
e M ˜
e
T → ˜
cm ˜
c M ˜
c
T
(θ, φ)
exp (i [kr − η ln(2kr)])
r
·
|χ 1/2 ⊗ |Ψ
J ˜
c
T
T
,
(9.39)
in which |χ 1/2 stands for the intrinsic spin wave function of the projectile in the
channel ˜
c. Indeed, the state (9.39) exhibits at large distance an incoming plane wave
(modified by the infinite-range Coulomb potential) in the entrance channel ˜
e and
outgoing spherical waves in each outgoing channel ˜
c.
Using Eqs. (9.34), (9.35), (9.37), and (9.38), one can express the scattering
amplitudes in terms of the T-matrix elements [3, 4, 6]:
f ˜
em ˜
e M ˜
e
T → ˜
cm ˜
c M ˜
c
T
(θ, φ) = δ ˜
c ˜
e δ m ˜
c m ˜
e δ M ˜
c
T M ˜
e
T
f C (θ)
+
e j e c j c J A
c
e j e J ˜
e
T c j c J ˜
c
T J A
m ˜
e M ˜
e
T m ˜
c M ˜
c
T
k ˜
e
T
J A
˜
ee e j e , ˜
cc c j c
Y
c
m ˜
e +M ˜
e
T −m ˜
c −M ˜
c
T
(θ, φ),
(9.40)
where f C (θ ) is the Coulomb scattering amplitude given by the relation [3]:
f C (θ ) = −
η
2k sin
2 (θ/2)
e
−i[η ln sin 2 (θ/2)−2σ 0 ]
(9.41)
and:
c
e j e J ˜
e
T c j c J ˜
c
T J A
m ˜
e M ˜
e
T m ˜
c M ˜
c
T
= =j e m ˜
e | e 0
1
2
m ˜
e c m ˜
e + M ˜
e
T − M ˜
c
T | c m ˜
e + M ˜
e
T − m ˜
c − M ˜
c
T
1
2
m ˜
c
× ×J A m ˜
e +M ˜
e
T |j e m ˜
e J ˜
e
T M ˜
e
T A m ˜
e +M ˜
e
T |j c m ˜
e +M ˜
e
T −M ˜
c
T J ˜
c
T M ˜
c
T
× i (( e − c )
4π(2 e + 1)e i(σ e +σ c ) .
(9.42)
From the expression (9.39) of the physical scattering state, the differential cross
section for the scattering process to the channel ( ˜
c, m ˜
c , M
˜
c
T ) at a given angle θ reads
dσ ˜
em ˜
e M ˜
e
T →˜ cm ˜
c M ˜
c
T
d Ω
(θ ) =
k ˜
c
k ˜
e
f ˜
em ˜
e M ˜
e
T →˜ cm ˜
c M ˜
c
T
(θ, φ)
2
.
(9.43)