Prefactor

The only modification of equation (Al.6.90) for spontaneous Raman scattering is the multiplication by the density of states of the cavity, equation (Al.6.24). leading to a prefactor of the fonn cojCOg. ... 

Note the presence of the ra prefactor in the absorption spectrum, as in equation (Al.6.87) again its origm is essentially the faster rate of the change of the phase of higher frequency light, which in turn is related to a higher rate of energy absorption. The equivalence between the other factors in equation (Al.6.110) and equation (Al.6.87) under linear response will now be established. 

Within this general framework there have been many different systems modelled and the dynamical, statistical prefactors have been calculated. These are detailed in [42]. For a binary mixture, phase separating from an initially metastable state, the work of Langer and Schwartz [48] using die Langer theory [47] gives the micleation rate as... 

Non-parabolic barrier tops cause the prefactor to become temperahire dependent [48]. In the Smoluchowski... 

The second integral in Eq. (155) seemed to be singular when n + / + q = 0. However, in this case, (i must be zero, and consequently this term will never contribute to the final result for being suppressed by the prefactor. With the definition in Eq. (132), we can write... 

The prefactor M(T), also called a frequency factor, has units of inverse seconds. It may have a weak dependence on temperature. Some theoretical models predict a variation with, but such variation is frequently ignored and M is taken as constant over limited temperature ranges. The prefactor M is often... 

Figure 6 shows the field dependence of hole mobiUty for TAPC-doped bisphenol A polycarbonate at various temperatures (37). The mobilities decrease with increasing field at low fields. At high fields, a log oc relationship is observed. The experimental results can be reproduced by Monte Carlo simulation, shown by soHd lines in Figure 6. The model predicts that the high field mobiUty follows the following equation (37) where d = a/kT (p is the width of the Gaussian distribution density of states), Z is a parameter that characterizes the degree of positional disorder, E is the electric field, is a prefactor mobihty, and Cis an empirical constant given as 2.9 X lO " (cm/V). ...

According to (2.6), when the temperature is decreased, both the apparent activation energy 3 and the apparent prefactor = ko exp[ — 2S E )/h ] decrease. The rate constant k given by (2.6)... 

Exploration of the region 0 < T < requires numerical calculations using eqs. (2.5)-(2.7). Since the change in /cq is small compared to that in the leading exponential term [cf. (2.14) and (2.18)], the Arrhenius plot k(P) is often drawn simply by setting ko = coo/ln (fig. 5). Typical behavior of the prefactor k and activation energy E versus temperature is presented in fig. 6. The narrow intermediate region between the Arrhenius behavior and the low-temperature limit has width... 

Fig. 5. Arrhenius plot of k T) for one-dimensional barrier with to Iota = 1. 0.5, 0.25 for the curves 1-3, respectively 2nVa/hota = 10, prefactor is taken constant.

Fig. 6. Apparent activation energy (1,2) and logarithm of apparent prefactor Ink, (1, 2 ) versus temperature. The value 2nVo/hco is taken 40 and 20 for the curves 1, 1 and 2, 2, respectively.

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

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]... 

This formula, aside from the prefactor, is simply a one-dimensional Gamov factor for tunneling in the barrier shown in fig. 12. The temperature dependence of k, being Arrhenius at high temperatures, levels off to near the cross-over temperature which, for A = 0, is equal to ... 

The origin of the isotope effect is the dependence of coq and co on the reacting particle mass. Classically, this dependence comes about only via the prefactor coq [see (2.14)], and the ratio of the rate constants of transfer of isotopes with masses mj and m2 m2 > mj) is temperature-independent and equal to... 

The uncertainty principle necessitates that any extremal trajectory should be spread , and the next step in our calculation is to find the prefactor by incorporating the small fluctuations around... 

This formula resembles (3.32) and, as we shall show in due course, this similarity is not accidental. Note that at n = 0 the short action 1 2 ( q) taken at the ground state energy Eq is not equal to the kink action (3.68). Since in the harmonic approximation for the well Tq = 2n/o)o, this difference should be compensated by the prefactor in (3.74), but, generally speaking, expressions (3.74) and (3.79) are not identical because eq. (3.79) uses the semiclassical approximation for the ground state, while (3.74) does not. 

Once the instanton trajectory has been numerically found, one proceeds to the calculation of prefactor, which amounts to finding determinants of differential operators. The direct two-dimensional generalization of (3.46) is... 

It is readily seen that when p is large enough and the hyperbolic sines in (4.18) can be replaced by exponents, the effect of the prefactor B, is to replace the potential V s) by the vibrationally adiabatic... 

At temperatures above there is no instanton, and escape out of the initial well is accounted for by the static solution Q = Q with the action S ff = PVo (where Vq is the adiabatic barrier height here) which does not depend on friction. This follows from the fact that the zero Fourier component of K x) equals zero and hence the dissipative term in (5.38) vanishes if Q = constant. The dissipative effects come about only through the prefactor which arises from small fluctuations around the static solution. Decomposing the trajectory into Fourier series. 

The situation changes when moving on to low temperature. Friction affects not only the prefactor but also the instanton action itself, and the rate constant depends strongly on rj. In what follows we restrict ourselves to the action alone, and for the calculation of the prefactor we refer the reader to the original papers cited. For the cusp-shaped harmonic potential... 

As the temperature drops, (5.80) starts to incorporate quantum corrections. When friction increases, T u decreases and the prefactor in (5.80) increases. This means that the reaction becomes more adiabatic. However, the rise of the prefactor is suppressed by the strong decrease in the leading exponent itself The result (5.80) may be recast in a TST-like form. If the transition were classical, the rate constant could be calculated as the average flux towards the product valley... 

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