In the measured differential cross section of the chemi-ioniation reaction M + X2 M+ + X- a small maximum is observed in the small-angle region, seen in all cross-section curves,
Quantum mechanically this phenomenon can be explained by constructive interference of some contributions, for instance the contributions of the two net repulsive branches of the b- curve. Because of the nearly equal slopes and about equal collision parameters of these branches at small scattering angles, the wavelength of the interference is rather long; a rough estimate has shown that the wavelength has a value of a few degrees at the energies used, which is of the same order of magnitude as the widths of the measured maxima.
Quantum mechanically the simple addition of contributions to the
differential cross section according to Eq. (E1), i.e.
must be replaced by a
formula adding the scattering amplitudes. Especially in the case of
an interference structure with a long wavelength the differential
cross sections, calculated classically and quantum mechanically could
be very different. The composition of the steep ``repulsive'' branches
of the deflection curve (see Fig. A05-m5aii-F2)1 it is expected to give such a
long-wavelength structure on the differential cross section. Now we
shall give a rough estimation of this wavelength. Following the Ford
and Wheeler semiclassical treatment of the scattering amplitude
and using the notation of Bernstein the amplitudes of the
contributions for repulsive scattering to the differential cross
section is given by
where the indices 1 and 2 indicate the ``covalent'' and ``ionic'' contributions. The combined amplitude of these two contributions can be written by
From Fig. A05-m5aii-F2 it can be seen that the branches for net repulsive scattering have about equal values of b and about equal slopes for a certain value of . Considering the relations
from which we obtain directly
In an elementary way the intensity can be found as follows:
and, according to ,
Now we compare two scattering angles and . Then the difference in the argument of the cosine is given by:
Fig. A05-m5aii-F2 shows that in the interesting region the curves are about straight lines so we simplify the last equation into
For two not very different scattering angles and we can roughly estimate a value of by . Then
Using the relation b= L/k (where Å-1 for K + Br2 at 10.35 eV) it can be seen graphically that at small positive scattering angles the interference of these two contributions has a wavelength of the order of a few degrees at colliding energies of about 10 eV.
Although we have only considered a two-contribution interference instead of four contributions, this interference can explain the observed bump in the differential cross section at small angle.