The Linear Two-State Landau-Zener Model

Abstract

The two-state nonadiabatic-coupling model discussed in this chapter was the first attempt to describe nonadiabatic coupling over the small range of interatomic separations near to the pseudocrossing or crossing of adiabatic terms. Since the time when Landau obtained analytical expressions for the transition probabilities in the weak [8.1] and near-adiabatic [8.2] coupling cases, and Zener [8.3] and Stueckelberg [8.4] found solutions for strong coupling in the semiclassical approximation, this model has been widely used for the interpretation of various experiments on atomic and ionic collisions [8.5–9]. Moreover, the Landau-Zener formula has often been used beyond the validity range both of the model as such and of the approximations adopted in the solution of nonadiabatic coupling equations. In particular, the solutions found by Landau, Zener and Stueckelberg were obtained making the assumption that the motion of the system in the nonadiabaticity region was quasi-classical. This assumption imposes a lower limit on the nuclear velocity, and the condition that the extent of the interaction region be small imposes an upper limit. However, the range of velocity variations satisfying both the model and the solution constraints appeared to be insufficiently wide for the application of the theory to certain problems. This stimulated the publishing of many papers attempting to clarify the range of applicability of the model, to widen it as much as possible, or even to remove the restrictions imposed by the initial formulation [8.10–12].