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 :
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 , the polarizability of the bromine ion , the crossing distance Rc, the resonance energy H12, the repulsive steepness coefficient and the ionic-well depth .
Firstly 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 =3.1 eV and A (Br2) = 1.2 eV.
The rule = 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 . However, even for elastic scattering the reduced scattering angle only in a first approximation 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 scale is not only due to the inelasticity of the collision, but partly to the incorrectness of the elastic = C(b) rule. To fix a value of 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 . 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 = 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 ( eV degrees range) are suitable to determine . 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 =3.1 eV. Because = I(K) - A(Br2) and the ionization potential of potassium is known to be 4.3 eV , the electron affinity of Br2 had to be A (Br2) = 1.2 eV, the same value as the one suggested by Person for the vertical electron affinity.
Then is determined via fitting the scattering angle for collisions with b=RC with the measured minimum: .
Assuming the ionic potential is given by Eq.(E3)
The well depth of the potential curve is determined using the classical rainbow angle: = 1.8eV
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
from the curve of A05-m5aii-F1, the rainbow
parameter q defined by
Taking into account the supernumerary spacings and convolution effects, the classical rainbow can be expected at 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:
The repulsive steepness coefficient is determined via a doubtful classical fit: = 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.|
The differential cross sections of Fig. A05-m4bi1-F1
indicate equal total cross sections for chemi-ionization of
|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:
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 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 .