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Qualitative interpretation=A05-m5aii

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,
figure of the cross sections given in A05-m4bi
[to the FULL figure] Figure A05-m5aii-F1: K + BR2, chemi-ionization differential cross section (CM system'; target: , measured at colliding energies of 6.9 and 10.35 eV. [This figure is copied from the experimental Treated results (link type: INPUT FROM; target: A05-m4bi1)]
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that cannot be explained [Compare with the classical interpretation (link type: IS COMPARED WITH/sq-back; target:  A05-m5ai)] by a classical model stating the general shape of the deflection curve as sketched in figure A05-m5aii-F2
[Unfold the general shape of the deflection function as given in MESO-m3c-defl]
[to the FULL figure] Figure A05-m5aii-F2: The general shape of the deflection curve for chemi-ionization scattering. [The figure is input from a mesoscopic theoretical methods module (link type: INPUT FROM; target: MESOm3c-defl)]
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and the equation for the differential cross section [The equation is imported from the mesoscopic theoretical methods (link type: INPUT FROM/is elaborated in/wider range/project; target:  MESO-m3c-defl)]  
$\displaystyle I(\theta)=\frac{1}{\sin \, \theta}\sum_{i=1,2,\ldots}\
P_{b_{i}}(1-P_{b_{i}})b_{i}\bigg\vert\frac{\dr b_{i}}{\dr {{\mit\Theta}.
}}\bigg\vert.$     (E1)

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-$\tau $ 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.

[unfold details estimation 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 [(link type: `external/input from'; target: Rf(A05)12] semiclassical treatment of the scattering amplitude and using the notation of Bernstein [(link type: `external/wider range/input from/elaborate'; target: Rf(A05)13)] the amplitudes of the contributions for repulsive scattering to the differential cross section is given by
\begin{eqnarray*}f_{1,2}(\theta)&=&\frac{1}{k(2\sin\, \theta
)^{1/2}}\bigg(\fr...
...ta _{\zs{L_{1,2}}}-L_{1,2}\theta
-\frac{1}{2}\,\pi\bigg)\bigg],\end{eqnarray*}

where the indices 1 and 2 indicate the ``covalent'' and ``ionic'' contributions. The combined amplitude of these two contributions can be written by
\begin{displaymath}f(\theta )=f_1(\theta )+f_2(\theta ).\end{displaymath}

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 $\dr b/\dr {{\mit\Theta} }$ for a certain value of ${{\mit\Theta} }$. Considering the relations
\begin{displaymath}b=L/k\quad \mbox{and}\quad \eta^{\prime\prime}=\dr {{\mit\Theta} }/2\dr L,\end{displaymath}
we conclude
$L_1\approx L_2\equiv L_0$ and $\eta_{L_1}^{\prime\prime}\approx \eta_{L_2}^{\prime\prime}\equiv
\eta_0^{\prime\prime}$
from which we obtain directly
\begin{eqnarray*}f(\theta)&\simeq &\frac{1}{k}\bigg(\frac{L_0}{-2\eta_0^{\prime\...
...
\bigg(2\eta_{L_2}-L_2\theta-\frac{1}{2}\pi\bigg)\bigg]\bigg\}.
\end{eqnarray*}
In an elementary way the intensity can be found as follows:
\begin{eqnarray*}I(\theta )&=&\frac{2L_0}{-k^2\eta_0^{\prime\prime}\sin\, \theta...
...g[\eta_{L_{1}}-\eta _{L_2}-\frac{1}{2}\,\theta
(L_1-L_2)\bigg],\end{eqnarray*}
and, according to $\eta_{L}^\prime=\frac{1}{2}\, {{\mit\Theta} }_L$,
\begin{eqnarray*}I(\theta )&=&\frac{2L_0}{-k^2\eta_0^{\prime\prime}\sin\, \theta...
...{\mit\Theta}
}(l_2)\dr l-\frac{1}{2}\,\theta (L_1-L_2)\Bigg].
\end{eqnarray*}
Now we compare two scattering angles $\theta _p$ and $\theta _q$. Then the difference ${\Delta }$ in the argument of the cosine is given by:
\begin{eqnarray*}{\mit\Delta }&=&\frac{1}{2}\int_{L_{1q}}^{L_{1p}}{{\mit\Theta} ...
...,\theta _p(L_{1p}-L_{2p})+\frac{1}{2}\theta _q (L_{1q}-L_{2q}).\end{eqnarray*}
Fig. A05-m5aii-F2 shows that in the interesting region the curves are about straight lines so we simplify the last equation into
\begin{eqnarray*}{\mit\Delta }&\approx&\frac{1}{4}(L_{1p}-L_{1q})(\theta _p-\the...
...L_{2p}-L_{1p}+L_{2q}-L_{1q})\\
&&+\theta
_q(L_{1q}-L_{2q}).
\end{eqnarray*}
For two not very different scattering angles $\theta _p$ and $\theta _q$ we can roughly estimate a value of ${\Delta }$ by $\theta _p\approx \theta
_q\equiv \theta _0$. Then
\begin{displaymath}{\mit\Delta }=\frac{1}{2}\,\theta _0(L_{1q}-L_{2q}+L_{2p}-L_{1p}).\end{displaymath}
Using the relation b= L/k (where $k\approx 400$ 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.
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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.