Landau Zener model

The motivation comes from the early work of Landau [208], Zener [209], and Stueckelberg [210]. The Landau-Zener model is for a classical particle moving on two coupled ID PES. If the diabatic states cross so that the energy gap is linear with time, and the velocity of the particle is constant through the non-adiabatic region, then the probability of changing adiabatic states is... 

This is, beyond all doubt, the most important process and the only one which has been already tackled with theoretically. Nevertheless, the prediction given by the classical overbarrier transition model is not correct for this collision [9] and the modified multichaimel Landau-Zener model developed by Boudjema et al. [34] caimot explain the experimental results for collision velocities higher than 0.2 a.u.. With regard to the collision energy range, we have thus performed a semi-classical [35] collisional treatment... 

The probability P in equations (61) and (62) may be related to the electronic coupling matrix element through equation (63) by application of the Landau-Zener model ... 

In the framework of the Landau-Zener model, P is related to H by means of equation (75). These equations are also valid when both the stretching and solvent reorganization coordinates are taken into account as in the case of dissociative electron transfer. 

Section 6.5.1), two avoided crossings arise in Figure 6.10c at t = +10 fs between states /) and 1 3), the first of which is marked by a gray circle. Due to the highly nonadiabatic time evolution, diabatic transitions between these dressed states are likely to occur. The Landau-Zener model [48, 104, 105] estimates the probability for a diabatic transition at the avoided crossings as... 

Various models to compute the probability of hopping exist. One of the simplest is the Landau-Zener model for avoided crossings in a single coordinate. The probability of the hop is determined as... 

Let a crossing of diabatic surfaces of potential energy occur in a certain point R0. Taking into account only the linear expansion term of the difference between the energies of the diabatic states near the crossing point (the Landau-Zener model)... 

Fig. 7. Cross-section for the charge exchange in H + H1 collisions [20]. The points are the experimental data. The line has been calculated using the Landau Zener model.

The probability of transition from one diabatic term to another when passing the point q, can be calculated using the Landau-Zener model [10]... 

Having considered the connection between the multiphoton resonances and the microwave threshold field for the K (n + 2)s —> (n,k) transitions, it is now interesting to return to the analogous n — n + 1 transitions which are responsible for microwave ionization and consider them from this point of view. We start with a two level description based on the extreme n and n + 1 m = 0 Stark states, a description which is the multiphoton resonance counterpart to the single cycle Landau-Zener model presented earlier. The problem is identical to the problem... 

Pillet et al. observed that adding small static fields dramatically reduces the microwave fields required for the ionization of Li.19 For example the application of a static field of 1 V/cm lowers the 15 GHz ionization threshold of the Li 42d state from 200 V/cm, to a broad threshold centered at 20 V/cm, a field only slightly in excess of E = 1/3n5 = 13 V/cm. The threshold field 200 V/cm corresponds to the hydrogenic threshold field of 1/9 n4, which will be described shortly. A small field has virtually no effect in a single cycle Landau-Zener model, but its dramatic... 

A (>0) is the electronic factor. 1-P is the probability for continuing on the lower PES, which corresponds to ET. If the barrier disappears, the Landau-Zener model should not be used and it may be necessary to include the nuclear coordinates in a wave packet model. 

We consider next the so-called Landau-Zener model that provides insight into non-adiabatic dynamics. The Landau-Zener model concerns the transition probability between two one-dimensional linear intersecting diabatic potentials... 

In the Landau-Zener model, dynamics is described by a single trajectory which due to the constant force undergoes an accelerated motion in the crossing region. The probability of a transition from diabatic state 1 to state 2 is denoted by P 2l which is also the probability of remaining in the lower adiabatic state, and the transition probability from the lower to the upper adiabatic state is then Pnonadia. = 1 — -P12, which is given by [16,17]... 

Fig. 4.2.2 Potential energy diagram for the Landau-Zener model. Adiabatic potentials (solid lines) and diabatic potentials (dashed lines), with /3i < 0 and (32 > 0. The arrow illustrates the dynamics on the lower adiabatic (ground-state) potential.

E. E. Nikitin, Theory of Non-Adiabatic Transitions. Recent Development on the Landau-Zener Model, in Chemische Elementazprozesse edited by H. Hartmann, Springer-Verlag, Berlin, 1968. 

The recently available spectroscopic data and the RKR potentials of the alkali hydrides allow us to determine the "experimental" values of the parameters relevant to the transition probability of the charge transfer processes. In the Landau-Zener model these parameters are the energy gap between the and X S adiabatic potentials at the avoided crossing distance and the coupling matrix elements. In this paper the coupling matrix elements are evaluated in a two-state ionic-covalent interaction model. The systema-tic trends found in the alkali hydride series for their X e potentials are presented. This leads to a simple model for the ionic potentials. 

We consider the Landau-Zener model (H.) for calculating the charge transfer cross section for high collision energies ... 

In the avoided crossing region the experimental RKR potential of the state and the "essentially experimental" potential of the state are used to determine the crossing distance R(- and the coupling matrix element T -. in the two state approximation. These quantities are relevant to the evaluation of the total charge transfer cross section at high energy (e.g. in the Landau-Zener model). 

For cases of electron transfer between relatively weakly coupled reactants, the 2-state Landau-Zener model leads to the following expression for the electronic transmission factor, (as in... 

Fig. 8. Relative velocity dependence of integral cross sections calculated for Na + O collisions for the indicated exit channels. The solid curve is the charge transfer cross section calculated using a multichannel Landau-Zener formalism (see text). The dashed curve is the two-state Landau-Zener cross section. Charge transfer calculations by van den Bos are indicated by triangles. Full circles and squares are the respective excitation channels as determined using the multichannel Landau-Zener model.

E.E.Nikitin, Theory of nonadiabatic transitions. Recent development of Landau-Zener model, in Chemische Elementarprozesse, ed. H.Hartmann, Berlin-Heidelberg, Springer, 1968, p.43... 

E.E.Nikitin, The Landau-Zener model and its region of applieability. Comm. At Mol. Phys. 1, 166 (1970)... 

E.E.Nikitin and A.I.Reznikov, Calculation of transition probabilities using the Landau-Zener model, Phys. Rev. A6, 522 (1972)... 

E.E.Nikitin, The Landau-Zener model in the theory of atomic colhsions, in Physics of Ionized Gases, ed. M.V.Kurepa, Belgrade, Institute of Physics, 1972, p.117... 

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