Landau-Zener crossing

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.

For vanishing angle 6 this expression is identified as the Landau-Zener formula/ Small values of B imply that at J < J ,. the adiabatic states merge into the diabatic states, similarly as for R > R. A typical example for a Landau-Zener crossing is given in the inset of Figure 9. 

The Landau-Zener transition probability is derived from an approximation to the frill two-state impact-parameter treatment of the collision. The single passage probability for a transition between the diabatic surfaces H, (/ ) and R AR) which cross at is the Landau-Zener transition probability... 

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... 

In the study of (electronic) curve crossing problems, one distinguishes between a situation where two electronic curves, Ej R), j — 1,2, approach each other at a point R = Rq so that the difference AE[R = Rq) = E iR = Rq) — Fi is relatively small and a situation where the two electronic curves interact so that AE R) Const is relatively large. The first case is usually treated by the Landau-Zener fonnula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

The transition described by (2.62) is classical and it is characterized by an activation energy equal to the potential at the crossing point. The prefactor is the attempt frequency co/27c times the Landau-Zener transmission coefficient B for nonadiabatic transition [Landau and Lifshitz 1981]... 

In the above numerical examples the held parameter F is taken to be the laser frequency and the nonadiabatic transition used is the Landau-Zener type of curve-crossing. The periodic chirping method, however, can actually be more... 

On the basis of a Landau-Zener curve crossing formalism, Borgis and Hynes derived the nonadiabatic rate constant k, which is similar to that expressed by the DKL model but where the tunneling term Cnm found in Eq. (4) is significantly modified due to the influence of the low-frequency promoting mode Q, with a frequency 0)q, on the tunneling rate. The dependence of Cnm on Q is given by [13]... 

K can be estimated by assuming that the system passes through the crossing region with a uniform velocity, so that the quantum one-dimensional problem can be solved to obtain the Landau-Zener form - ... 

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... 

Herein a is the rate of change of the lower dressed state energy i(t) (black dashed line in Figure 6.10c) evaluated at the inflection points at t = +15 fs, and the Rabi frequency H22 is evaluated at the crossing times. For symmetry reasons, the Landau-Zener probability is the same for both avoided crossings. Now the second requirement concerning the field amplitude is to tailor the Rabi frequency of the main pulse such that = 0.5. Then 50% of the population is transferred... 

Figure 15.7 Near approach (or avoided crossing) of two electronic states as a function of nuclear coordinate Q. The inset expands the region of the avoided crossing to facilitate the definition of quantities appearing in the Landau-Zener surface-hopping-probability model...

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... 

Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e., coherence) in the two different wells via the area shown in Fig. 2. A variety of applications to time-independent problems may be found in the literature [1]. 

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 process may then be described by classical trajectories and by transition probabilities for changing the potential surfaces when these trajectories come close to a crossing. The transition probability for changing the diabatic curves at a crossing line R, is given by a Landau-Zener... 

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