Chemistry Reference
In-Depth Information
the molecule. Thus, a microscopic description of electron-impact ionization, and
likewise of many of the other processes listed in Table 9.1, cannot be based on simple
potentialscatteringtheory(elasticscattering).Ageneralizedscatteringtheoryisrather
required, capable of accounting for changes in the internal energy (inelasticity), for
rearrangement, and for breakup of the scattering fragments.
The appropriate theoretical framework is quantum-mechanical multichannel
scattering theory [68-70]. To introduce its essential ingredients, two colliding frag-
ments are considered. In the center-of-mass frame, the total Hamiltonian of the
system is
=
T
rel
+
H
int
+
=
H
0
+
H
V
V
,
(9.60)
where
T
rel
is the kinetic energy of the relative motion,
H
int
controls the internal degrees
of freedom of both fragments, and
V
is the interaction energy between the two.
The Lippmann-Schwinger equation for the scattering state with the boundary
conditions shown in Figure 9.9 reads in Dirac's bra-ket notation
1
(
+
)
k
α
(
+
)
k
α
=|
k
α
+
|
|
i
η
V
,
(9.61)
E
−
H
0
+
where the first term denotes the incoming plane wave in the entrance channel. The
channel state,
)
+
ω
α
, where
k
is the relative momentum, ω
α
is the internal energy, and μ is the reduced
mass of the fragments in the entrance channel.
Quite generally, the scattering amplitude, which in turn determines the differen-
tial collision cross section, is defined as the amplitude of the outgoing spherical wave
emerging from the right-hand side of (9.61) for large interparticle distances. Hence,
in order to find the scattering amplitude, (9.61) has to be expressed in coordinate
representation, which is here specified by
r
, the interparticle distance, and ρ,the
internal coordinates of both particles; then the limit
r
|
k
α
=|
φ
α
|
k
, satisfies
(
E
−
H
0
)
|
k
α
. Thus,
E
=
E
k
α
=
k
2
/(
2μ
has to be taken.
Normalizing continuum states on the momentum scale then leads to [69,70]
→∞
Incoming plane wave
Outgoing spherical wave
FIGURE 9.9
Schematic representation of the incoming plane wave and the diverging
spherical wave as described by the Lippmann-Schwinger equation (9.61).
