Zener approach

Within the Zener approach, and its nuclear spin variant (Frohlich and Nabarro 1940), the degree of spin ordering, Mq, at given temperature T is found by minimizing the total free energy of the spin and carrier subsystems, F[Mq], Here Mq denotes the Fourier... 

In equation (30), Vei is the frequency of electronic motion across the reaction coordinate near the transition state. Other expressions for /Cei have been proposed, for example, based on Landau-Zener approaches. ... 

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 Snoek pendulum, which in the most general sense is a device to measure relaxations, has also been used to measure relaxation caused by tangential displacements at grain boundaries. This application has been the central concern of a distinguished Chinese physicist, Tingsui Ke, for all of the past 55 years. He was stimulated to this study by Clarence Zener, in 1945, and pursued the approach, first in Chicago and then in China. This exceptional fidelity to a powerful quantitative technique was recognised by a medal and an invitation to deliver an overview lecture in America, recently published shortly before his death (Ke 1999). 

For k(r) we shall assume at first, as in (19), that the reaction is adiabatic at the distance of closest approach, r = a, and that it is joined there to the nonadiabatic solution which varies as exp(-ar). The adiabatic and nonadiabatic solutions can be joined smoothly. For example, one could try to generalize to the present multi-dimensional potential energy surfaces, a Landau-Zener type treatment (41). For simplicity, however, we will join the adiabatic and nonadiabatic expressions at r = a. We subsequently consider another approximation in which the reaction is treated as being nonadiabatic even at r = a. 

Meyer and coworkers (18). In this case, we see from Figure 6 that the Weiner method predicts faster rates than the perturbation approach, the difference being about a factor of 5 at room temperature and more than an order of magnitude at low temperature. As might be expected, the two methods continue to diverge as increases. If gets so large that the transfer rate becomes comparable to v, the Landau-Zener correction ((5), eq 155) may be applied. c... 

Relevant data extracted from activity measurements in the Fe-C phase were later included in the assessment by Zener (1946) and Darken and Smith (1951). All this work followed a traditional approach solely based on thermodynamic data derived under equilibrium conditions. By contrast, Kaufman and Cohen (1958) showed that the data of Johannson (1937) were more consistent with information derived from Fe-based martensite transformations than the interpretation used by Darken and Smith (1951). Together with the work of Svechnikov and Lesnik (1956) this was a notable attempt to combine thermodynamic information derived from low-temperature metastable transformations with those from more traditional sources. 

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...

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]. 

In the case of semiconductors, the idea of electron tunneling has been used by Zener [42] to describe the so-called interband tunneling. Such tunneling represents one of the possible mechanisms of semiconductor breakdown. To understand the nature of interband tunneling, we shall first follow Ziman [43] and consider the one-dimensional motion of an electron in a separate band under the influence of an electric field. If we use the scheme of repeated bands, then the electron motion in momentum space is an up and down motion along the OABC periodic curve (Fig. 16). In the coordinate space, the electron, starting from point O, accelerates then slows down as it approaches point A here, the direction of the motion is changed to the... 

First, we shall approach the problem as a transition driven by the time variation of a quasi-static field, in other words, in the Landau-Zener terms we used to describe microwave ionization. We can write the Schroedinger equation for this problem as... 

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