# Theoretical methods = A05-m3c

The experimental differential cross sections for chemi-ionization in
M + X_{2} -> M^{+} + X_{2}^{-}
are analysed using a simple classical atom-atom model for ion-pair formation in molecular collisions
.
The transition to the ionic state takes place *via* crossing of the neutral and ionic ground states. The electron transition probability is calculated applying the Landau-Zener approximation; trajectories are calculated using the impact parameter approximation
.
We assume an isotropic intermolecular potential and neglect the internal states of the X_{2} molecule.

### Applicability of the model to the measured collison:

If we apply this model to the system of
K-Br_{2},
we assume that the following simplifications are valid
#### Simplification 1: no internal states Br_{2}

We simplify the potential surfaces by neglecting the vibrational-state distribution and describe the collision as a two-body problem.
Actually the interatomic *M*-*X*-*X* potential should be described by hypersurfaces to include for instance the vibrational states of the
and X_{2}^{-} molecules. However, we cannot resolve the vibrational structure from the differential cross section, so these effects only have an averaging effect on the measurements.

Unfortunately the effect of vibrational excitation is rather large for the collisions of interest.
From the semi-empirical Br_{2} and Br_{2}^{-} interatomic-potential curves of
Person
it can be seen that even at a very restricted low vibrational-state distribution of Br_{2}, the exothermic Br_{2} -> Br_{2}^{-}
transition can cause a high vibrational excitation or even
dissociation of the Br_{2}^{-} molecule, giving a wide range of possible values of the electron affinity
A(Br_{2}).
As

=I(M) - A(X_{2}) ,

where
I(\rm M)
represents the ionization potential of the alkali atom and A (X_{2})
the electron affinity of the halogen molecule, this means
that in a chemi-ionization process
is very dependent on the
vibrational state of the halogen molecule before and after the
collision.

Indeed, the largest possible bromine-electron affinity
A(Br_{2}), as measured by Baede and Los
,
2.8 eV, is very different
from our value of 1.2 eV for the vertical electron affinity
the Quantitative interpretation.

Therefore this simplification is not really valid.

#### Simplification 2: isotropic intermolecular potential

Another simplification is made by neglecting anisotropy effects. The anisotropic potentials will also have an averaging effect on the measurements.
has determined by classical trajectory calculations the chemi-ionization differential cross section of K + Br_{2} taking into account a nonspherical bromine molecule. His results for collinear and perpendicular collisions are quite different.
Therefore this simplification is not really valid either.