Chemistry Reference
In-Depth Information
annihilation to the ionization energy of the
AB
molecule and the electron affinity of
the
A
atom (see Figure 9.14).
The
Landau-Zener model
illustrates quite nicely how semiempirical models
encode complicated processes in a few physically intuitive parameters. In contrast
to the processes discussed so far, where the relevant nuclear dynamics take place
on a single potential energy surface, ion-ion annihilation forces the nuclei to switch
between two potential energy surfaces of the collision compound. The minimal the-
oretical model is therefore based on two coupled Lippmann-Schwinger equations
for the relative motion of the nuclei. The Landau-Zener model is the semiclassi-
cal approximation to this set of equations. Following Olson [66], the total ion-ion
annihilation cross section is then given by
4π
R
x
1
F
+
E
E
σ
II
A
(
E
)
=
(
λ
)
,
(9.92)
with
∞
)
1
)
dx x
3
exp
F
(
λ
)
=
(
−
λ
x
−
exp
(
−
λ
x
(9.93)
1
and
2π
M
2
|
V
(
R
x
)
|
2
λ
=
V
f
|
√
E
,
(9.94)
|
V
i
−
+
E
where
E
and
M
arethekineticenergyandthereducedmassoftherelativemotionofthe
(
A
−
,
AB
+
)
system,
E
is the energy gain due to annihilation,
V
(
R
x
)
is the interaction
between the
(
A
−
,
AB
+
)
and
(
A
,
AB
)
configurations at
R
=
R
x
, and
V
i
,
f
=
dV
i
,
f
(
R
x
)/
dR
with
V
i
(
R
)
and
V
f
(
R
)
as the potential energy surfaces of the
(
A
−
,
AB
+
)
and
(
A
,
AB
)
system, respectively.
Equation 9.92 can be developed further by recalling that
V
i
(
R
)
∼
R
−
1
(Coulomb
interactionbetween
A
−
and
AB
+
)and
V
f
(
R
)
∼
r
−
n
with
n
>
1(polarizationinteraction
|
V
i
−
V
f
|≈
≈−
U
i
(
R
x
)
=
between
A
and
AB
).Hence,forlargeenough
R
x
,
R
−
x
and
E
√
2
M
π
R
5
/
2
x
R
−
x
. Usually,
E
E
. Combining all this leads to λ
≈
|
V
(
R
x
)
|
2
[66].
Since
F
(
λ
)
approaches its maximal value
F
max
≈
0.1 at λ
max
≈
0.424, a rough estimate
for the annihilation cross section is
1.3
R
x
1
1
R
x
E
σ
II
A
(
)
≈
+
E
(9.95)
√
2
M
π
R
5
/
2
x
with
R
x
determined from λ
max
=
|
(
R
x
)
|
V
2
or from empirical cross section
R
−
1
x
(
)
→
(
R
x
)
data for high energies where
E
is not
trivial. Ideally, it can be parametrized in terms of an effective ionization energy
E
A
i
of
the
AB
molecule and the
electro
n
a
ffinity
E
a
of the
A
atom. Olson and coworkers [67]
obtain
V
and σ
E
1.3
R
x
. Determining
V
1.04
4
E
A
i
E
a
√
qR
exp
0.857
R
(
R
)
=
[−
]
for large internuclear distances
(
2
E
a
+
2
E
AB
with
R
=
0.5
)
R
and
q
the Franck-Condon factor that represents
i
