Landau-Zener formula transition

In order to apply quantitatively the Landau-Zener formula Eq. (2), we first saturated the crystal of Fes clusters in a field of Hz = -1.4 T, yielding an initial magnetization Min = -Ms. Then, we swept the applied field at a constant rate over one of the resonance transitions and measured the fraction of molecules which reversed their spin. This procedure yields the tunneling rate P 1010 and thus the tunnel splitting A 1010 , Eq. (2), with n = 0, 1, 2,. 

Coulson derived a new method for computing P(t) that gives the Landau-Zener formula as a special case. He considered three cases (1) transition between two discrete states [approximate internal conversion when the molecule passes from one discrete (bound) state to another], (2) transition from a bound state to the continuum (predissociation), and (3) transitions between two states of the continuum (corresponding to scattering problems). Coulson omitted in his article the computational details for P(t) for case (1). For case (2) he gave... 

This result is due to Dykhne [48] and Davis and Pechukas [49] and has been extended to A-level systems [50]. The conditions of validity of the so-called Dykhne-Davis-Pechukas (DDP) formula has been established in [51,52], This formula allows us to calculate in the adiabatic asymptotic limit the probability of the nonadiabatic transitions. This formula captures, for example, the result of the Landau-Zener formula, which we study below. 

The results obtained in our laboratory as well as by other experimentalists [3, 4] have inspired a considerable amount of theoretical work on this system [2, 5-8], Archirel and Levy [7] have calculated a set of potential energy surfaces for the states N2 (X) + Ar, N2(A) + Ar, and N2 + Ar+(2P) as well as the couplings between these surfaces using a novel computational technique. From their results they developed a set of diabatic vibronic potential energy curves, and they assumed that transitions could occur when two curves crossed. Cross sections were computed using either the Demkov or Landau-Zener formula, as appropriate, and good agreement was obtained with the experimental values in most cases. Nikitin et al. [8] have taken a somewhat similar approach to this system. They estimated the adiabatic vibronic interaction curves for this system, and they assumed that transitions... 

If we further assume that the diabatic coupling V(t) is constant, then Eq. (12) can be solved exactly in terms of the Weber function. Then the final transition probability is exactly equal to Eq. (10). The linearity in time t is very much different from the linearity in coordinate R and the effects of turning points are completely neglected in the former approximation. In Landau s treatment this corresponds to the assumption of the common straightline classical trajectory with constant velocity. Thus, the Landau-Zener formula Eq. (10) is valid only at collision energies much higher than the crossing point. 

The Landau-Zener transition probability pZN is as simple as the famous Landau-Zener formula of Eq. (10) but much better. Note that p is expressed in terms of a2 and b2 as... 

Another practical method is TSH (18), in which ordinary classical trajectories are run until they come close to the surface crossing region where the trajectories are jumped to the other surface with probability given by the Landau-Zener formula. This method is simple and convenient, but suffers from the following drawbacks all phases are completely neglected and only the probabilities (not the probability amplitudes) are handled the detailed balance is not necessarily satisfied and nonadiabatic transitions at energies... 

The details of the nonadiabatic transition can be handled in a variety of The simplest approach is to use the Landau-Zener formula for the probability of crossing. " For a single crossing the probability of a transition from the zero-order curve corresponding to X -FHY(0) to that corresponding to X-FHY(n) is given by... 

N-N zero vibrations M is the nitrogen atomic mass and co is the N2 vibration frequency. To finalize determination of the transition probability between relevant electronic n -> n ) and vibrational (vi V2) states (6-17), the electronic transition probability p"" can be fonnd based on the Landau-Zener formula (Landau Lifshitz, 1981a,b) ... 

The situation is radically different in the inverted region, as well as in certain cases of nonequihbrium back transfer (see below), which are always nonadiabatic whatever the coupling strength is. For large V, the ET rate is no longer controlled by transport to the transition region but rather by nonadiabatic transitions between adiabatic states (see Fig. 9.1). Therefore, one should expect a decrease of the ET rate with increasing V to follow the solvent-controlled plateau. Usually, the Landau-Zener formula is used for the description of nonadiabatic transitions in the classical limit [162,163]. 

The Landau Zener formula is a scattering solution to the problem of crossing between two diabatic cmves, derived by assmning that the nuclei follow a classical trajectory. The diabatic curves are linearized around the crossing point, which translates into an electronic energy gap varjung linearly with time. The transition probability P is ... 

The probability of the Pj transition depends essentially on the form of the potential surfaces , and E. Its correct calculation for real multidimensional PES s constitutes a complex mathematical problem, which is why it is a common practice to perform in this case one-dimensional approximation (the reaction coordinate is approximated by one parameter) and make use of the classical expression for the probability of a transition between the PES s ( and known as the Landau-Zener formula. 

VIII. Appendix Zhu-Nakamura Theory and Summary of the Formulas A. Landau—Zener Type of Transition... 

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

Electron-jump in reactions of alkali atoms is another example of non-adiabatic transitions. Several aspects of this mechanism have been explored in connection with experimental measurements (Herschbach, 1966 Kinsey, 1971). The role of vibrational motion in the electron-jump model has been investigated (Kendall and Grice, 1972) for alkali-dihalide reactions. It was assumed that the transition is sudden, and that reaction probabilities are proportional to the overlap (Franck-Condon) integral between vibrational wavefunctions of the dihalide X2 and vibrational or continuum wave-functions of the negative ion X2. Related calculations have been carried out by Grice and Herschbach (1973). Further developments on the electron-jump mechanism may be expected from analytical extensions of the Landau-Zener-Stueckelberg formula (Nikitin and Ovchinnikova, 1972 Delos and Thorson, 1972), and from computational studies with realistic atom-atom potentials (Evans and Lane, 1973 Redmon and Micha, 1974). 

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