Qualitative interpretation=A05-m5ai






[Legenda]
[Contents of the thesis]
[Comments]






[Legenda]
[Contents of the thesis]
[Comments]

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

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.

that cannot be explained 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]

Figure A05-m5aii-F2: The general shape of the deflection curve for chemi-ionization scattering.

and the equation for the differential cross section

$\displaystyle I(\theta)=\frac{1}{\sin \, \theta}\sum_{i=1,2,\ldots}\ P_{b_{i}}(1-P_{b_{i}})b_{i}\bigg\vert\frac{\dr b_{i}}{\dr {{\mit\Theta}. }}\bigg\vert.$ (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.

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.

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.