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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.

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${\Delta }E$ determination


The rule $\tau= C(b) is a well-known approximation for elastic scattering, indicating that the scattering angle multiplied by the kinetic energy is in first-order approximation only a function of the impact parameter b. The use of this rule for inelastic scattering might be a method to determine the inelasticity ${\Delta }E$. However, even for elastic scattering the reduced scattering angle $\tau $ only in a first approximation [This has been shown by Smith et al. (link type: IS ARGUED IN/external; target: R(A05)8) is independent of the kinetic energy. Fig. A05-m5aii-F1 shows the calculated scattering-angle curves for two different values of Ei. So the very obvious shift of the `` covalent'' as well as of the ``ionic'' parts of the deflection curves relative to each other along the ``reduced'' angle $\tau $ scale is not only due to the inelasticity of the collision, but partly to the incorrectness of the elastic $\tau = C(b) rule. To fix a value of ${\Delta }E$ from this shift we had to separate carefully these two effects. Therefore we separate the total collision into an elastic and a ``purely inelastic'' part. Up to the second passing of Rc the collision process is elastic because at Rc we are again at zero level of the potential energy due to the flatness of the covalent potential curve for $R\gtrsim R_\mathrm{c}$. The part of the collision from the second passing of Rc up to infinity we call purely inelastic scattering. The dashed curves of A05-m5aii-F1 show for the two relevant different initial energies the pure inelastic contribution to the total scattering angle, that is the hypothetical deflection curve for particles following a straight line until the second crossing where ionization takes place. As already stated, this scattering angle as a function of b is a contribution due especially to the inelasticity, irrespective of whether the diabatic transition takes place in the incoming or in the outgoing branch of the collision. The elastic contributions can be found by subtracting the total and pure inelastic scattering angle. Comparing the relative total scattering-angle shift and the pure inelastic scattering-angle shift it is obvious that generally the larger part of the former one is due not to the inelastic effect but to the incorrectness of the $\tau= C(b) rule, seen most clearly at small values of b where indeed higher-order terms become important. Only for ``covalent'' scattering with b> 4.5 Å (about straight line trajectories up to the second passing of Rc) the shift is mainly due to the inelasticity, so the measured differential cross sections of covalent scattering at scattering angles belonging to this b range ( $\tau\approx
100$ eV degrees range) are suitable to determine ${\Delta }E$. Fortunately, the interesting contribution from this range to the scattering-angle region is dominant over the three other contributions originating from smaller impact parameters. The calculated relative shifts of the maxima and small-angle slopes of the ``covalent'' differential cross section as a function of Ei, fit the measured shifts if we take ${\Delta} E=3.1 eV. Because ${\Delta}E = I(K) - A(Br2) and the ionization potential of potassium is known to be 4.3 eV [Input of that value (link type: INPUT FROM/external; target:  R(A05)9, the electron affinity of Br2 had to be A (Br2) = 1.2 eV, the same value as the one suggested by Person[Input of Persons value (link type: INPUT FROM/external; target:  R(A05)2)] for the vertical electron affinity.
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Then $\alpha _{\zs\Br_2^-}$ is determined via fitting the scattering angle for collisions with b=RC with the measured minimum: $\alpha _{\zs\Br_2^-}\approx 150\ \AA^3$.

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$\alpha _{\zs\Br_2^-}$ determination

Assuming
 
$\displaystyle U_\mathrm{ion}(R\gtrsim R_\crm)\simeq -\frac{e^2}{R}-\frac{e^2(\alpha
_{\zs\K^+}+\alpha _{\zs\Br_2^-})}{2(R^4+a^4)}+{\Delta} E,$     (E3)
where ${\Delta} E=3.1 eV, the value $\alpha _{\zs{K^+}} +\alpha _{\zs\Br_2^-}= 150\ \AA^3$ fits the calculated scattering angle for collisions with b = Rc with the measured minimum at $t\approx 140$. The place of the minimum is only weakly dependent on the value of ${\Delta }E$ and a, but strongly dependent on the value of $\alpha _{\zs\K^+} +\alpha _{\zs\Br_2^-}$. A theoretical value of the polarizability of K+ is $\alpha _{\zs\K^+} = 0.94\ \AA$ [Input of that value (link type: INPUT FROM/external; target:  R(A05)3)] [Input of that value (link type: INPUT FROM/external; target:  R(A05)4)], from which follows then $\alpha _{\zs\Br_2^-}\approx 150\ \AA^3$.

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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.

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Rc determination

Assuming the ionic potential is given by Eq.(E3) and
$U_\mathrm{cov}\approx 0$
both for $R\gtrsim R_\mathrm{c}$, then, substituting the determined values of $\alpha _{\zs\K^+} +\alpha _{\zs\Br_2^-}$ and ${\Delta }E$, the crossing distance Rc can be calculated. The dependence on the screening parameter a is rather weak and for an arbitrary value at of a=5 Å five find Rc= 5.8 Å.

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The well depth of the potential curve is determined using the classical rainbow angle: $\varepsilon $ = 1.8eV

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$\varepsilon $ determination

In the case in which the character of the potential curve is known (here we have chosen the Rittner formula for the ionic potential), the classical rainbow angle is a good indication of the well depth of the potential curve. By calculating the value of $(\dr^2{{\mit\Theta} }/\dr b^2)_\mathrm{rainbow}$ from the curve of A05-m5aii-F1, the rainbow parameter q defined by
\begin{displaymath}q=\frac{\hbar^2}{4\mu E_\mathrm{i}}\bigg(\frac{\dr ^2{{\mit\Theta} }}{\dr b^2}
\bigg)_\mathrm{rainbow},\end{displaymath}
can be calculated to be about 0.2 x 10-5 for Ei= 10.35 eV. The Airy approximation of the rainbow structure needs the parameter q and predicts a spacing between the supernumerary rainbows of about 8 eV .degree and a spacing of about 15 eV.degree between the first maximum and the first supernumerary. In the measurements, the energy spread of 20 percent of the alkali beam prevents the resolution of the supernumeraries; only a hint can be found, particularly from the Li+ Br2 differential cross-section measurements in Fig.A05-m4bi2-F1 [This reasoning depends on the figure in the Treated results (link type: DEPENDS ON/is detailed in; target:  A05-m4bi2)]. For K + Br2 (Fig.A05-m4bi1-F1 [This reasoning depends on the figure in the Treated results (link type: DEPENDS ON/is detailed in; target:  A05-m4bi)1]) the differential cross section between $\tau= 150 and $\tau= 300 is expected to be the envelope of the rainbow structure with a principal maximum of low intensity while the intensities of the supernumeraries increase. However, the Airy approximation predicts a continuous decrease of the intensity but the simple Airy method is only a good approximation if the deflection curve is nearly parabolic near the classical rainbow point; our calculated deflection function deviates from parabolic too much to use this approximation. It has been shown by Berry [Arguments for the increase of the intensity of the supernumaries  (link type: IS ARGUED IN/external; target:  R(A05)10)] for a special case (Lennard-Jones potential) that the intensities of the supernumeraries increase calculating the rainbow structure with the uniform approximation or with exact calculations of Hundhausen and Pauly [Arguments for the increase of the intensity of the supernumaries  (link type: IS ARGUED IN/external; target:  R(A05)14)]. However, in all calculations the distances between the maxima are nearly the same.

Taking into account the supernumerary spacings and convolution effects, the classical rainbow can be expected at $\tau\approx275$ eV.degree on the K+Br2 cross-section curve. A Rittner potential with a minimum at - 1.8 eV gives the same calculated classical rainbow angle.

reliability calculated value:
Very recently we have resolved the rainbow structure of the K + Br2 differential cross section (Fig.A05-m5bi-F1) by using an potassium beam of about 15 eV with a very small energy spread. The results have not been used in the calculations described here but the resolved rainbow structure indicate indeed a classical rainbow at $\tau\approx
275$ eV.degree and a supernumerary spacing of about 10 eV .degree. This spacing at a kinetic collision energy of 15 eV is in agreement with the predicted spacing of 8 eV .degree at 10.35 eV by virtue of the proportionality to E1/3.

[to the FULL figure] Figure A05-m5bii-F1:K + Br2, rainbow structure of the chemi-ionization differential cross section at colliding energy of about 15 eV, measured with an alkali beam with a large (open circle curve) and a small energy spread. The unresolved curve has been shifted by 0.5 along the ordinate.

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The repulsive steepness coefficient is determined via a doubtful classical fit: $\rho= 0.3 Å

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$\rho $ determination

Fig.A05-m5bi-F2 of the potential, Fig.A05-m5bi-F3 of the deflection function and Eq.(E4) [This formula is imported from the mesoscopic theoretical methods (link type: INPUT FROM/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)
indicate, that with increasing steepness of the potential, the steepness of the deflection curve for small b also increases and consequently the contribution of these parts to the differential cross section decreases. On the other hand the steepness of the repulsive potential has nearly no influence on the large contribution of ``covalent'' scattering to the differential cross section at $\tau\approx
100$ eV.degree, but it does influence the decrease to the small value at $\tau\gtrsim 50$. Summarizing, the value of the cross section at $\tau\approx100$ relative to the small values at $\tau > 50 and $\tau >&nbsp300$ and the slope at 50&nbs&nbs100 might be suitable to determine a value for $\rho $, in our classical consideration. However, it can be expected that the difference between classical and quantum calculations could be large especially for the quantitative calculation of the differential cross section. Therefore the value of $\rho= 0.3 Å that gives the best classical fit (supposing equal values of $\rho $ for the covalent as well as the ionic potential) is rather doubtful, in spite of the fact that the classical calculated cross sections are very sensitive to the value of $\rho $. At any rate the value $\rho= 0.3 Å for K+ Br2 is a very realistic one compared with other molecules with a known value of $\rho $.

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The resonance energy is determined by fitting the curve height ratio and found to be H12=4.5 x 1012 eV.

  
[to the FULL figure] Figure A05-m4bii-F2: K + BR2 ionic and covalent potential.
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H12 determination

The differential cross sections of Fig. A05-m4bi1-F1 indicate equal total cross sections for chemi-ionization of
K + Br2 at energies of 10.35 and 6.9 eV, in agreement with total cross-section measurements of Baede and Los [These values agree with those in an earlier paper (link type: AGREES WITH/project; target:  A03-m4b]. However, the remarkable shift of the curve-height ratio as a function of
\begin{displaymath}\tau, R(\tau)=(I\sin \,\theta /10.35)/ (I\sin \, \theta /6.9)\end{displaymath}
is mainly due to the dependence of Pb(1-Pb) on Ei and b and partly due to the unequal values of the quantity $(\dr b/\dr \tau)$ for the two energies. The curve-height ratio does not give a value of the resonance energy H12 directly, because this ratio is also a function of the potential curves. Eq. (E5) for the transition probability [This formula is imported from the mesoscopic theoretical methods (link type: INPUT FROM/wider range/project; target:  MESO-m3c-mod)]  
\begin{displaymath}P_\br=\exp\left( \frac{-2\pi H_{12}^2}{\hbar v_\rr\Big\vert\frac{\dr}{\dr
R}(H_{11}-H_{22})\Big\vert _{R_\crm} }\right),
\end{displaymath}   (E5)
requires explicitely the slopes of the crossing potential curves at Rc, while moreover the exact shapes of the potential curves are important to know the collision parameter b for a certain scattering angle $\tau $. We have calculated the ratios R(120)/R(40) = 0.84 and R(400) = 0.7 using the given potential parameters and a resonance energy H12 = 4.5 x 10-2 eV. Of course there is the difficulty of the ambiguous $\tau\to b$   relation but in the examples considered the scattering at $\tau\approx 120$ has one strongly dominating contribution and scattering at $\tau\approx 40$ and $\tau\approx 400$ has, respectively, four and two contributions with about equal values of b. The ratio at $\tau\approx 120$ is the most important one for the derivation of H12 because [on account of the flat long-range covalent potential curve and the simple form of $U_\mathrm{ion}(R\gtrsim R_\mathrm{c})]$ the value of R(120), where b is only a little bit smaller than Rc, is only a function of $(\alpha _{\zs\K^+}+\alpha _{\zs\Br_2^-})$ and ${\Delta }E$. A second point for the importance of R(120) is, that this ratio is most sensitive with respect to the variation of trial values of H12. The calculated ratios given above are in very good agreement with the measurements. This method to fix H12 seems to be rather sensitive: a resonance energy of 6.5x 10-2 eV would predict the quite different ratios 1.44, 0.95 and 0.73, respectively. The value H12=4.5x 10-2 eV is in rather good agreement with the value of Ref. 1c which would be about 6x 10-2 eV if the same polarizability is taken.

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[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)].