The typical shape of the A-B intermolecular potential is shown in Fig. MESO-m3c-mod-F1. The chemi-ionization process is seen to be endothermic with an energy
The crossing distance Rc is approximately given by
The groundstates of the system are described by two curves.
There exist many analytical expressions to describe the ionic alkali-halide potential curve.
For the ionic-potential curve we have chosen a Rittner potential
of the form:
species and the multiplicity of the ionic and covalent ground states are the same. At the crossing point the transitions between the covalent and the ionic configurations are induced by a strong coupling between the state . The Landau-Zener theory provides a simple expression for this transition as a function of the radial velocity at the crossing point. At a collision, ionization of the neutral particles takes place via such a non-adiabatic transition by electron transfer.
The potential curves of Fig. MESO-m3c-mod-F1 clearly show the two collision trajectories leading to chemi-ionization. Incoming on the covalent curve and outgoing on the ionic one, the particles follow either the ionic potential curve inside the crossing distance with a covalent-ionic> transition at the first passing of Rc followed by a covalent-covalent crossing, or the covalent potential curve inside the crossing distance Rc (with a covalent-covalent transition followed by a covalent-ionic one). The former process will be indicated by ``ionic'' scattering, the latter one by ``covalent'' scattering. In terms of an electron jump between the two interactants, ionic scattering means that the electron has jumped from A to B when these first approached to the crossing distance Rc of each other and that it remains there during the transition that takes place the second time the particles are at Rc. In covalent scattering there is no charge transfer in a diabatic transition at the first passing of the crossing distance, but the electron jumps when the separating interactants are again at Rc.
In he two-dimensional case the transition probability between different configurations has been given firstly by Landau, Zener and Stueckelberg (commonly called the Landau-Zener theory and abbreviated here as L-Z theory). This theory has been criticized and extended by different authors . Transitions between multi-dimensional surfaces have been treated by Teller . Further work has been done during the last decade by Nikitin .
The probability Pb
for a covalent-ionic transition at a single passage of the crossing at Rc
is given by the Landau-Zener formula:
At thermal energies the probability of the electron jump will be close to unity, except for the largest Rc values. At higher energies, however, the probability for a transition from covalent to ionic at the crossing point decreases, while the covalent-covalent transition probability increases: at centre of mass energies larger than the probability that only one electron jump takes place is non-negligible, such that ion pair formation can occur due to an electron jump at the first crossing while passing the second time no change occurs or vice versa. .
From the potential-energy curves and the transition probability, the total and differential cross sections for ion-pair formation via the different trajectories can be calculated via the deflection function.
In the simplified atom-atom model, it is assumed that the system has an isotropic, static two-body potential; vibrational and rotational degrees of freedom are neglected. The potential associated to harpoon reaction
Calculations on three-atomic systems are very difficult because they require a knowledge of the complete energy surfaces involved. Moreover, the situation is complicated very much because multiple passing of the crossing region is possible in contrast with the two-atom case, where the crossing region is passed only twice during the collision. An attempt has been made by Bjerre and Nikitin to calculate cross sections for quenching of Na* by N2. This quenching is also supposed to take place via crossings of potential surfaces. They made trajectory calculations on three-dimensional crossing surfaces, determining the transition probabilities at the crossings by an extended L-Z theory. Such calculations are very computer-time consuming and, because the knowledge of the potential energy surfaces is very poor, one may not expect results with more than only qualitative meaning.
It is known now that the quantitative validity of the L-Z theory is very limited. Moreover, this theory in its simple form is applicable neither to three-particle collisions nor to endothermic processes. Indeed it is found to be impossible to fit the energy dependence of the measured cross section with that predicted by the L-Z theory. However, one can try to use the results of the theory as a kind of ``adiabatic criterion'' to determine the position and also the order of magnitude of the maximum cross section of this type of inelastic processes.