




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






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

Potential and deflection function = A05-m5bi

We calculate with a simple classical atom-atom model some potential parameters of the K-Br₂ system by fitting calculated deflection functions with specific features of the measured differential cross section , thus determining simultaneously the exact shape of the deflection function.

The ionization deflection function can be calculated from the potential-energy curves . 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 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 :

U_ion(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 R_c, 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 $${\Delta} E$ =3.1 eV and A (Br₂) = 1.2 eV.
Then $\alpha _{\zs\Br_2^-}$ is determined via fitting the scattering angle for collisions with b=R_C with the measured minimum: $\alpha _{\zs\Br_2^-}\approx 150\ \AA^3$ .
Substituting the obtained values of $\alpha _{\zs\K^+} +\alpha _{\zs\Br_2^-}$ and ${\Delta }E$ in the assumed potential, R_c can be determined to be R_c= 5.8 Å.
The well depth of the potential curve is determined using the classical rainbow angle: $\varepsilon$ = 1.8eV.
The repulsive steepness coefficient is determined via a doubtful classical fit: $$\rho$ = 0.3 Å.
The resonance energy is determined by fitting the curve height ratio and found to be H₁₂=4.5 x 10¹² eV. .

Figure A05-m4bii-F2: K + BR₂ ionic and covalent potential.

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 , . For collisions with two channels, a covalent one and an ionic one , the deflection function consists of a covalent and an ionic branch which are joined at the crossing radius R_c, 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 .

U_ion(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)