Frequency attempt

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]... 

Equation (3.32) demonstrates that the decay rate for a metastable state is equal to the inverse period of classical vibrations in the well ( attempt frequency ) times the barrier transparency. 

The term Oq now contains n and ze as well as the information on attempt frequency and jump distance. This expression accounts for the fact that ionic conductivity increases with temperature. 

It only remains to specify the time constant, r0 [Eqs. (5.14) and (5.15)], related inversely to the attempt frequency with which the monomers attempt to cross barriers in the torsional potential (Fig. 1.2b). We have not attempted to calculate this time constant from first principles, but rather fixed it by comparison to experiment on chain self-diffusion at T = 450K [178]. This yields r0 1/50 picoseconds. This small number can be understood from the fact that because of the potentials, Eqs. (5.12) and (5.13), at T = 450 K only a few percent of the attempted hops of the effective monomers are successful the time constant for successful hops is of the order of 1 ps. These considerations... 

The pre-exponential factor is assumed to be temperature independent and proportional to the inverse of the attempting frequency, a-1. 

Here N0 is the initial concentration of centers, v is the attempt frequency, and Ea the binding energy. For As—H and Sb—H samples it was observed that the concentration of centers initially increased upon annealing. It was supposed that the excess hydrogen that is often observed in SIMS profiles at the surface of n-type samples might be causing this anomalous increase. 

When the samples were etched mildly, the anomalous increase upon annealing was not observed. In an isothermal annealing experiment performed at 423 K for As—H complexes, the exponential decay given by Eq. (3) was verified for a 50 times reduction in concentration. In Fig. 11 the results of a series of 30 min isochronal anneals are shown for each of the donor-H complexes. The curves are given by Eq. (3) with an assumed attempt frequency of 1013 s-1 and binding energies of 1.32 eV for P—H and 1.43 eV for As—H and Sb—H. 

The frequency vx is interpreted as an attempt frequency that describes the number of times per second that the atom tries to escape. The rest of the expression represents the probability of escape at any given attempt. The term a /ay specifies the ratio of the window sizes at A and X through which the atom has to pass. The final term is a temperature-dependent Boltzmann factor that specifies the ratio of equilibrium number densities at A and X. 

The atoms in a crystal are vibrating continually with frequency v, which is usually taken to have a value of about 1013 Hz at room temperature. It is reasonable to suppose that the number of attempts at a jump, sometimes called the attempt frequency, will be equal to the frequency with which the atom is vibrating. The number of successful jumps that an atom will make per second, the jump frequency T, will be equal to the attempt frequency v, multiplied by the probability of a successful move, that is,... 

Henceforth we take the primitive path co-ordinate s=L-z from the free end inwards to the branch point so that t(s) is an increasing function of s. The prefactor Tq is an inverse attempt frequency for explorations of the potential by the free end, and may be expected to scale as the Rouse time for the star arm (in fact this is not quite true - the actual scaling is as [25,26]). The relaxation mod-... 

Eqn (6.7) may be expressed in a number of slightly different forms which depend on the model and assumptions made in the original derivation. If ionic diffusion is considered to be an activated process as, for example, in the case of glasses and ceramics, then included in the preexponential term of Eqn (6.7) is the attempt frequency, Vq, for ion mobility. Several... 

The experimental studies described in this chapter certainly led to a better understanding of the coalescence phenomena in concentrated emulsions. Despite the complexity and variety of the destruction scenarios, different methods for measuring the coalescence frequency, ty, have been proposed. It should be within the reach of future work to measure ty for a large variety of systems in order to establish a comparative stability scale. This is a necessary step to determine the microscopic parameters that control the activation energy Ea and the attempt frequency coo. 

In obtaining the last expression we have used the substitutions after Eqs. (7) and (10). Eec is the activation energy for detachment from a step edge into the vapor , and Vec is the attempt frequency for that process, (see also [8] and case A of [13]). As can be seen from Eq. (15), in this case, G,(r) is independent of q at short times. 

A common approach for the study of activated barrier crossing reactions is the transition state theory (TST), in which the transfer rate over the activation barrier V is given by (0)R/2jt)e where 0)r (the oscillation frequency of the reaction coordinate at the reactant well) is an attempt frequency to overcome the activation barrier. For reactions in solution a multi-dimensional version of TST is used, in which the transfer rate is given by... 

Since in eq. (23) there is one frequency more in the numerator than in the denominator it is often interpreted as an attempt frequency of the reactant system multiplied by a Boltzmann factor corresponding to the energy barrier between the initial state and the saddle point in the transition state. The transition state theory result is often written in the form (see page 110 of [3]) ... 

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