Chemistry Reference
In-Depth Information
(
+
)
α
k
of Lippmann-Schwinger equations satisfied by the scattering state
|
contains
an inhomogeneity,
|
α
k
=|
φ
A
α
|
k
, where
|
φ
A
α
is the initial state of the molecule
and
is the plane wave representing the relative motion of the initial electron and
the molecule.
From a rigorous mathematical point of view, the exact two-electron scattering
state in the field of a positive ion is unknown. Hence, exact calculations of electron
impact ionization cross sections are not available, even for atoms, where compli-
cations due to the nuclear dynamics are absent. Perturbative treatments, based, for
instance, on the distorted Born approximation, which replaces ψ
k
1
k
2
by a plane wave
for the primary electron and a Coulomb wave for the secondary electron, are possible,
provided the velocity of the incident electron is much larger than the velocities of
the electrons bound in the molecule. Exchange and correlation effects can then be
ignored and the calculation of cross sections for electron-impact ionization essen-
tially reduces to the calculation of photoionization cross sections. This approach, due
originally to Bethe [53] and reviewed by Rudge [71] and Inokuti [73], fails at low
electron energies. In that region, however, Vriens' binary-encounter model [54] can
be used. It assumes that the primary electron interacts pairwise with target electrons,
leaving the remaining electrons and the nuclear dynamics undisturbed. The ioniza-
tion cross section is then basically the Mott cross section for the collision of two
electrons, appropriately modified by the binding and kinetic energies of the target
electrons.
Thesemiempirical
Kim-Ruddmodel
forelectronimpactionization[55]combines
the Bethe model with the binary-encounter model. For an individual orbital, the
ionization cross section is then given by
|
k
1
2
Q
1
ln
e
1
,
S
1
e
2
1
e
−
ln
e
σ
I
(
e
)
=
−
+
(
2
−
Q
)
−
(9.88)
e
+
u
+
1
e
+
1
with
e
B
, where
E
is the kinetic energy of the incident electron,
U
is the average kinetic energy of the orbital, and
B
is the binding energy of the
orbital. The remaining quantities are
S
=
E
/
B
and
u
=
U
/
N
, where
N
is
the orbital occupation number and
M
i
is an integral over the differential oscillator
strength defined in [55]. The results obtained from (9.88) are surprisingly good.
Another widely used semiempirical model for electron impact ionization is the
Deutsch-Märk model
[56-58]. It uses cross sections for the atomic constituents of
the molecules and sums them up according to the atomic population of the molecular
orbitals that is obtained from a Mullikan population analysis. The total ionization
cross section can then be written as
=
4π
N
/
B
2
and
Q
=
2
BM
i
/
σ
I
(
E
)
=
g
i
,
nl
π
r
i
,
nl
N
i
,
nl
f
(
e
i
,
nl
)
(9.89)
n
,
l
,
i
with the mean square radius
r
i
,
nl
of the
n
,
l
th sub-shell of the constituent
i
,the
occupancy
N
i
,
nl
of that sub-shell, weighting factors
g
i
,
nl
that have to be determined
