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Potential and deflection function = A05-m5bi

We calculate with a simple classical atom-atom model [The interpretation depends on the theoretical methods (link type: DEPENDS ON/detailed in; target:  A05-m3c)] some potential parameters of the K-Br2 system by fitting calculated deflection functions with specific features of the measured differential cross section [The interpretation depends on the theoretical methods (link type: DEPENDS ON/detailed in; target:  A05-m4bi1)], thus determining simultaneously the exact shape of the deflection function.

The ionization deflection function can be calculated from the potential-energy curves [Details in a mesoscopic module (link type: IS DETAILED IN/wider range/project; target:  MESO-m3c-defl)]. Choosing potential parameters by trial and error, the scattering function has been calculated via fitting with the measured differential cross section for the K + Br2 case by a new method [The method this depends on is given elsewhere (link type: IS DETAILED IN/depends on/project; target:  A06-m*]. The numerical calculation of the classical deflection angle is performed with a well-known error. For the calculation we have used the crossing potential curves with the assumption that the charge exchange occurs exactly at the crossing point.

Because the colliding particles are rather heavy and the kinetic energy is not very low, it will be reasonable to compare the measurements primarily with classical calculations. Small quantum-mechanical interference structures on the differential cross-section curves will be washed out by the large energy spread of the alkali beam, the averaging effect of the internal state distribution and the anisotropy of the halogen molecule, the extent of the crossing region and the finite angular resolution of the detector.

For the ionic-potential curve we have chosen a Rittner potential of the form :
Uion(R) = $\displaystyle -\frac{e^2}{R}-\frac{e^2(\alpha_{\Na^+}+\alpha_{\I^-})}{2R^4}$  
    $\displaystyle -\frac{2e^2\alpha_{\Na^+}\alpha_{\I^-}}{R^7}-\frac{C_\ion}{R^6}$  
    $\displaystyle +A_\ion\, \er^{-R/\rho_\ion}+\Delta E.$ (E1)
The covalent potential is given by:
\begin{displaymath}U_\cov(R)=-\frac{C_\cov}{R^6}+A_\cov\,\er^{-R/\rho_\cov} (E2)

Because we use for the ionic and covalent potential formulas many parameters we have looked for effects in the differential cross section that are mainly due to only one of these parameters. The parameters that can be determined rather directly in this way are the endothermicity of the collision ${\Delta }E$, the polarizability of the bromine ion $\alpha _{\zs\Br_2^-}$, the crossing distance Rc, the resonance energy H12, the repulsive steepness coefficient $\rho $ and the ionic-well depth $\varepsilon $.

Firstly ${\Delta }E$ and the electron affinity of Br2 are determined via fitting of the relative shifts of the maxima and the small-angle deflection slopes of the deflection function at different energies with the measured shifts. They are found to be ${\Delta} E=3.1 eV and A (Br2) = 1.2 eV. Unfold Delta E determination
Then $\alpha _{\zs\Br_2^-}$ is determined via fitting the scattering angle for collisions with b=RC with the measured minimum: $\alpha _{\zs\Br_2^-}\approx 150\ \AA^3$.Unfold alpha  determination
Substituting the obtained values of $\alpha _{\zs\K^+} +\alpha _{\zs\Br_2^-}$ and ${\Delta }E$ in the assumed potential, Rc can be determined to be Rc= 5.8 Å.Unfold  Rc determination
The well depth of the potential curve is determined using the classical rainbow angle: $\varepsilon $ = 1.8eV.Unfold epsilon determination
The repulsive steepness coefficient is determined via a doubtful classical fit: $\rho= 0.3 Å. Unfold rho determination
The resonance energy is determined by fitting the curve height ratio and found to be H12=4.5 x 1012 eV. Unfold H12 determination.
  
[to the FULL figure] Figure A05-m4bii-F2: K + BR2 ionic and covalent potential.


  
[to the FULL figure] Figure A05-m4bii-F3: K + BR2, deflection curves for chemi-ionization scattering (CM system). Full curves represent the classically calculated scattering angle for "ionic" and "covalent" scattering at colliding energies of 10.35 and 6.9 eV. Dashed curves show the "pure inelastic" scattering- angle contribution to the full-line curves.

Summarizing, we have determined for the K-Br2 system most of the potential parameters; the missing parameters have been chosen to construct the potential curves in Fig. A05-m5bi-F2 and the deflection functions in Fig. A05-m5bi-F3 The values used are:

\begin{eqnarray*}\Delta E&=&3.1 \mathrm{eV},\\
\alpha_{\zs\K^+}&=&0.94\ \AA^3\...
...(\mbox{\link{[(link type: \lq input from'; target: Rf(A05)4)]}} ),
\end{eqnarray*}

\begin{displaymath}\begin{array}{l}
\varepsilon =1.8+3.1\ \mathrm{eV},\quad C=1...
... \mathrm{eV},\\
\alpha _{\zs\Br_2^-}=150\ \AA^3,
\end{array}\end{displaymath}

where the value for A has been fixed after the choice of C and a, by requiring the ionic well minimum at -1.8 eV. For simplicity the Van der Waals term and repulsive term of both potential curves are supposed to be the same.

Based on the potential curves of the system, the classical deflection function is calculated [depends on theoretical methods given in a mesoscopic module (link type: DEPENDS ON/wider range/project; target:  MESO-m3c-defl)], [depends on numerical methods given in another paper (link type: DEPENDS ON/project; target:  A06)]. For collisions with two channels, a covalent one and an ionic one [depends on theoretical methods given in a mesoscopic module (link type: DEPENDS ON/wider range/project; target:  MESO-m3c-mod)], the deflection function consists of a covalent and an ionic branch which are joined at the crossing radius Rc, as is shown in figure refA05-m5bi-F3, forming a closed deflection curve.

From the deflection function then, the differential cross section can be calculated [The deflection function is used for that purpose in the Quantitative interpretation (link type: `IS USED IN'; target: A05-m5bii)].