Chemistry Reference
In-Depth Information
The differential cross section for this process can be obtained from the resonance
model, using an incoming plane wave in the
(
A
−
,
AB
)
channel and an outgoing spher-
ical wave in the
channel as boundary conditions. Normalizing continuum
states on the energy scale and applying (9.71) leads to
(
e
−
,
A
2
B
)
dR
χ
ν
f
(
=
(
2π
)
4
2
d
σ
A
ν
f
V
dk
(
(
+
)
dE
R
)
R
)
(
R
)
d
k
f
,
(9.96)
K
2
(
+
)
dE
satisfies (9.79)
where χ
ν
f
(
R
)
is the vibrational state of the
A
2
B
molecule and
with
V
d
(
R
)
=
V
A
2
B
−
(
R
)
,
V
0
(
R
)
=
V
A
2
B
(
R
)
, and
V
dk
(
R
)
is the interaction between the
quasi-bound
A
2
B
−
state and the
e
−
−
A
2
B
scattering continuum. The inhomogeneity
is
J
(
R
)
=[
V
d
(
∞
)
−
V
d
(
R
)
]
K
(
R
)
, where
K
(
R
)
is a plane wave for the relative
motion of the
(
A
−
,
AB
)
system. Energy conservation enforces
E
=
K
2
/(
2
M
)
+
V
d
(
∞
)
=
k
2
/(
2
m
)
+
ω
ν
f
, where
M
and
m
are the reduced masses of the
(
A
−
,
AB
)
and
the
system, respectively. Note that since
d
σ
A
ν
f
is a differential cross section
for reactive scattering, the interaction in the exit channel,
V
dk
, appears in (9.96) as it
should.
The most favorable situation for associative detachment is when the potential
energy surface of the neutral molecule in the exit channel supports a bound state
whose dissociation energy
E
A
2
d
is larger than the electron affinity
E
a
of the
A
atom.
This situation is shown in Figure 9.15. Provided detachment is mediated by an
attractive state of the collision compound, it takes place even for vanishing initial
kinetic energy in the
(
e
−
,
A
2
B
)
channel. If this is the case, detachment is a very efficient
loss channel for negative ions, even at low temperatures. If, on the other hand, the
compound state is antibonding, associative detachment occurs only when the initial
kinetic energy of the colliding particles is larger than
V
d
(
(
A
−
,
AB
)
R
x
)
−
V
d
(
∞
)
, where
R
x
is
the point where the repulsive potential energy surface crosses
V
0
(
R
)
.
9.2.2.3 Excitation of Internal Degrees
The chemistry and charge balance of a gas discharge is also affected by inelastic
collisions, that is, collisions that increase the internal energy of molecules, atoms,
and ions. Excited (metastable) particles are reactive and participate in basically all
particle number changing collisions. In low-temperature gas discharges, vibrationally
excited molecules play a particularly important role, because, for typical operating
conditions, they are efficiently produced by resonant electron-molecule scattering.
The kinetic energy of electrons in a low-temperature gas discharge is typically a
few electron volts. At these energies, the electron-molecule collision time is rather
long, therefore favoring resonant enhancement of the collision. The cross section for
vibrational excitation
of molecules is thus given by (9.66) with the scattering state
obtained from the resonance model. Specifically, for continuum states normalized on
the energy scale, the cross section becomes
dR
χ
ν
f
(
=
(
2π
)
4
2
d
σ
V
ν
i
→
ν
f
V
d
k
f
(
(
+
)
dE
R
)
R
)
(
R
)
d
f
,
(9.97)
k
i
(
+
)
dE
where χ
ν
f
(
R
)
is the vibrational state of the molecule after the collision and
(
R
)
is the solution of the effective Lippmann-Schwinger equation (9.79) with
V
d
(
R
)
=
