The transmission coefficient

The transmission coefficient Cl (Qj,t), considering transient (broadband) sources, is time-dependent and therefore accounts for the possible pulse deformation in the refraction process. It also takes account of the quantity actually computed in the solid (displacement, velocity potential,...) and the possible mode-conversion into shear waves and is given by... 

The use of air-bome ultrasound for the excitation and reception of surface or bulk waves introduces a number of problems. The acoustic impedance mismatch which exists at the transducer/air and the air/sample interfaces is the dominant factor to be overcome in this system. Typical values for these three media are about 35 MRayls for a piezo-ceramic (PZT) element and 45 MRayls for steel, compared with just 0.0004 MRayls for air. The transmission coefficient T for energy from a medium 1 into a medium 2 is given by... 

Voth G A 1990 Analytic expression for the transmission coefficient in quantum mechanical transition state theory Chem. Phys. Lett. 170 289... 

There is still some debate regarding the form of a dynamical equation for the time evolution of the density distribution in the 9 / 1 regime. Fortunately, to evaluate the rate constant in the transition state theory approximation, we need only know the form of the equilibrium distribution. It is only when we wish to obtain a more accurate estimate of the rate constant, including an estimate of the transmission coefficient, that we need to define the system s dynamics. 

Now we can compute the transmission coefficient [17,24]. It will be the difference between the positive and negative contributions, or... 

This discussion of geometric effects ignored the attenuation of radiation by material through which the radiation must travel to reach the receptor. The number of particles, dN, penetrating material, equals the number of particles incident N times a small penetration distance, dx, divided by the mean free path length of the type of particle in the type of material (equation 8.3-8). Integrating gives the transmission coefficient for the radiation (equation 8.3-9). 

For gas-phase reactions, Eq. (5-40) offers a route to the calculation of rate constants from nonkinetic data (such as spectroscopic measurements). There is evidence, from such calculations, that in some reactions not every transition state species proceeds on to product some fraction of transition state molecules may return to the initial state. In such a case the calculated rate will be greater than the observed rate, and it is customaiy to insert a correction factor k, called the transmission coefficient, in the expression. We will not make use of the transmission coefficient. 

Many computational studies in heterocyclic chemistry deal with proton transfer reactions between different tautomeric structures. Activation energies of these reactions obtained from quantum chemical calculations need further corrections, since tunneling effects may lower the effective barriers considerably. These effects can either be estimated by simple models or computed more precisely via the determination of the transmission coefficients within the framework of variational transition state calculations [92CPC235, 93JA2408]. 

Measure up the area of each type of surface and compute the loss through each surface by multiplying the transmission coefficient by the measured area by the difference between the inside and the outside temperatures. 

Some gas phase data suggest that a certain fraction of the transition states for some reactions are reflected back to products. One can multiply the right side of Eq. (7-55) by k, the transmission coefficient, to account for this, in which case k < 1. We shall ignore this factor k, taking it as unity. Indeed, we shall ignore a large body of experimental research on gas phase reactions and the theoretical calculations on them. 

Phenomenological evidence for the participation of ionic precursors in radiolytic product formation and the applicability of mass spectral information on fragmentation patterns and ion-molecule reactions to radiolysis conditions are reviewed. Specific application of the methods in the ethylene system indicates the formation of the primary ions, C2H4+, C2i/3+, and C2H2+, with yields of ca. 1.5, 1.0, and 0.8 ions/100 e.v., respectively. The primary ions form intermediate collision complexes with ethylene. Intermediates [C4iZ8 + ] and [CJH7 + ] are stable (<dissociation rate constants <107 sec.-1) and form C6 intermediates which dissociate rate constants <109 sec. l). The transmission coefficient for the third-order ion-molecule reactions appears to be less than 0.02, and such inefficient steps are held responsible for the absence of ionic polymerization. 

In the case where they represent quantum vibrational modes, this leads to the appearance of a small tunnel factor in the transmission coefficient k. ... 

The assessment of k is of some importance since it relates to the question as to how much if any of the free energy of activation barrier is due to the spin-forbidden character of the transition. From the experimental point of view, Eq. (49) shows that the transmission coefficient k and the activation entropy AS appear in the temperature-independent part of the rate constant and thus cannot be separated without additional assumptions. Possible approaches to the partition of — TAS have been discussed in Sect. 4 for spin transition complexes of iron(II) and iron(III). If the assumption is made that the entropy of activation is completely due to k, minimum values between 10 and 10 are obtained for iron(II) and values between 10 and 10 for iron(III). There is an increase of entropy for the transition LS -+ HS and thus the above assumption implies that the transition state resembles the HS state. On the other hand, volumes of activation indicate that the transition state should be about midway between the LS and HS state. This appears indeed more reasonable and has the... 

The term exp[iax] in equations (2.47) indicates travel in the positive x-direction, while exp[—iax] refers to travel in the opposite direction. The coefficient A is, then, the amplitude of the incident wave, B is the amplitude of the reflected wave, and F is the amplitude of the transmitted wave. In region III, the particle moves in the positive x-direction, so that G is zero. The relative probability of tunneling is given by the transmission coefficient T... 

The transmission coefficient T in equation (2.58) is the relative probability that a particle impinging on the potential barrier tunnels through the barrier. The reflection coefficient R in equation (2.59) is the relative probability that the particle bounces off the barrier and moves in the negative v-direction. Since the particle must do one or the other of these two possibilities, the sum of T and R should equal unity... 

In the limit as a oo, as Vo oo, as m oc, or any combination, the transmission coefficient T approaches zero and the refiection coefficient R approaches unity, which are the classical-mechanical values. We also note that in the limit h 0, the classical values for Tand R are obtained. 

Find the expression for the transmission coefficient T for Problem 2.4 when the energy E of the particle is equal to the potential barrier height Fq. 

As these functions cannot be normalized, it is sufficient here to pose A[2 = 1 and calculate the relative probability densities in each succeeding step. Then, R = 1512 represents the reflection coefficient and T = F 2 the transmission coefficient Assuming that the particle cannot remain trapped within the hairier, the relation... 

Equation (SI) can be verified by calculation of R = B 2 from the simultaneous equations for the coefficients and substitution in Eq. (80). The result represented by Eq. (81) shows that the transmission coefficient decreases as the height V or the thickness t of the barrier increases. 

The use of Eq. (5-10) to evaluate the reaction rate is characterised by the calculation of Hessians for a large number of points along the MEP which are required to locate the free energy maximum and also to evaluate the curvature required for evaluation of the transmission coefficient. In view of the associated computational expense, high-level electronic structure calculations are not feasible and alternative strategies, one of which is to use a semi-empirical method, are usually employed [81]. 

If the probability for the system to jump to the upper PES is small, the reaction is an adiabatic one. The advantage of the adiabatic approach consists in the fact that its application does not lead to difficulties of fundamental character, e.g., to those related to the detailed balance principle. The activation factor is determined here by the energy (or, to be more precise, by the free energy) corresponding to the top of the potential barrier, and the transmission coefficient, k, characterizing the probability of the rearrangement of the electron state is determined by the minimum separation AE of the lower and upper PES. The quantity AE is the same for the forward and reverse transitions. 

The activation energy for the nonadiabatic reaction, "ad, is determined by the point of minimum energy on the intersection surface of PES Ut and Uf9 and the transmission coefficient k is determined by the electron resonance integral... 

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