Chemistry Reference
In-Depth Information
where
T
rel
,
H
int
, and
V
are the relative kinetic energy, the internal energy, and the
interaction energy in the exit channel. Then, in addition to (9.61), the scattering state
|
(
+
)
k
α
obeys also a homogeneous Lippmann-Schwinger equation,
∗
1
(
+
)
k
α
i
η
V
|
(
+
)
k
α
|
=
,
(9.69)
E
−
H
0
+
from which the scattering amplitude in the exit channel (reaction amplitude) can
be deduced by the same procedure as before, except that now the coordinate rep-
resentation with respect to
r
and ρ
, the relative and internal coordinates in the
exit channel, has to be chosen. In this representation, the scattering state becomes a
diverging spherical wave for
r
→∞
. The prefactor (continuum states normalized on
the momentum scale)
μ
2π
k
β
|
V
|
(
+
)
k
α
f
(
k
β,
k
α
)
=−
,
(9.70)
with the reduced mass μ
in the exit channel and an eigenstate
of
H
0
can
be identified with the reaction amplitude. Hence, the differential cross section for
reactive scattering is given by
|
k
β
μ
μ
k
k
|
μμ
(
k
k
|
k
β
|
(
+
)
k
α
d
σ
α
→
β
=
f
(
k
β,
k
α
)
|
2
d
=
V
|
|
2
d
,
(9.71)
2π
)
2
where energy conservation now enforces
E
=
k
2
/(
2μ
)
+
ω
β
=
k
2
/(
2μ
)
+
ω
α
.
Obviously, (9.71) reduces to (9.66) for
V
=
V
, which implies
H
0
=
H
0
and thus
r
=
.
If the interaction
V
in the entrance channel is simpler than the interaction
V
in
the exit channel, it may be more convenient to use the adjoint scattering state,
r
, μ
=
μ, and
=
(
−
)
k
β
,
which describes an incoming wave in the exit channel. The reaction cross section can
then be written as
|
μμ
(
k
k
|
(
−
)
d
σ
α
→
β
=
|
V
|
k
α
|
2
d
,
(9.72)
k
β
2π
)
2
which contains
V
instead of
V
.
The formalism described so far is only applicable to binary collisions, that is,
collisions containing two particles in the entrance and exit channel, respectively.
The theoretical description of collisions involving three (or more) reaction products
(breakup collisions) can also be based on a set of Lippmann-Schwinger equations,
but the increased dimensionality of the relative motion requires a substantial exten-
sion of the formalism (see [68]) beyond the scope of this introductory presentation.
Additional complications arise for identical particles in a given channel, or when the
interaction is long-ranged, as is, for instance, the case for electron-impact ionization.
Ionization cross sections are therefore hardly obtained from rigorous calculations.
They are usually estimated from less ambitious, semiempirical models to be described
in Section 9.2.2.1.
∗
The equation is homogeneous because the incoming wave belongs to a different Hilbert space.
