Dimensionless coupling parameters

Using the somewhat smaller transition temperatures Td,2 of van Dijk (1994) with ATc = 0.054 K (fig. 29) one obtains for the dimensionless coupling parameter of SC and AF order... 

At last, the Fermi coupling potential may be expressed beyond the exchange approximation by introducing the dimensionless Fermi coupling parameter A ... 

First let us restrict the comparison to very weak H bonds because in such a case we can expect that the adiabatic approximation is fulfilled. We made some sample calculations shown in Figs. 8(c) and 8(d), with a small dimensionless anharmonic coupling parameter, ac = 0.6. The figures displays /sf ex [dashed line (d)] and 7Sf [dashed line (c)] at 300 K. Each one is compared with the adiabatic spectra 7f (superimposed full lines). Note that the adiabatic spectra (c)... 

As with the other reaction schemes involving the coupling of electron transfer with a follow-up homogeneous reaction, the kinetics of electron transfer may interfere in the rate control of the overall process, similar to what was described earlier for the EC mechanism. Under these conditions a convenient way of obtaining the rate constant for the follow-up reaction with no interference from the electron transfer kinetics is to use double potential chronoamperometry in place of cyclic voltammetry. The variations of normalized anodic-to-cathodic current ratio with the dimensionless rate parameter are summarized in Figure 2.15 for all four electrodimerization mechanisms. 

The response of a reversible reaction (2.146) depends on two dimensionless adsorption parameters, Pr and po. When pR = po the adsorbed species accomplish instantaneously a redox equilibrium after application of each potential pulse, thus no current remains to be sampled at the end of the potential pulses. The only current measured is due to the flux of the dissolved forms of both reactant and product of the reaction. For these reasons, the response of a reversible reaction of an adsorbed redox couple is identical to the response of the simple reaction of a dissolved redox couple (2.157), provided Pr = po- As a consequence, the real net peak current depends linearly on /J, and the peak potential is independent of the frequency. If the adsorption strength of the product decreases, i.e., the ratio increases, the net peak current starts to increase (Fig. 2.73). Under these conditions, the establishment of equilibrium between the adsorbed redox forms is prevented by the mass transfer of the product from the electrode surface. Thus, the redox reaction of adsorbed species contributes to the overall response, causing an increase of the current. In the hmiting case, when ]8o —0, the reaction (2.146) simplifies to reaction (2.144). 

Voltammetric features of adsorption coupled EC mechanisms (2.177) [128] and (2.178) [129] are rather unpredictable and deviate strongly from the EQ mechanism of a dissolved redox couple. Their voltammetric behaviour is mainly controlled by the adsorption parameter p, and the dimensionless chemical parameters k"s = j and... 

In order to obtain a formula for the energy difference between the complete ground state energy E and the noninteracting energy Eks, and thus for E c, the interaction Hamiltonian H = H- is supplemented by a dimensionless coupling strength parameter g in such a way,... 

The relativistic adiabatic connection formula is based on a modified Hamiltonian H g) in which not only the electron-photon coupling strength is multiplied by the dimensionless scaling parameter g but also a g-dependent, multiplicative, external potential is introduced. 

The coordinates, coupling parameter C and frequency fl in (4.29), are dimensionless, and time is measured in dimensionless units o>0t, where 0 is the frequency of small vibrations in the well for the adiabatic potential... 

With the substrate biased at a potential slightly more positive than E° of A/B couple, B is oxidized to form A for both DISP1 and ECE mechanisms. However, in the latter case the reduction of C also occurs at the substrate. The numerical solution of corresponding diffusion problems (see Ref. [85] for problem formulations) yielded several families of working curves shown in Fig. 12 (DISP1 pathway) and Fig. 13 (ECE pathway). In both cases, the tip and the substrate currents are functions of the dimensionless kinetic parameter, K = ka2/D. 

Within these nonsymmetrized coordinates, the indirect damping parameter is y and not y, whereas the dimensionless anharmonic coupling parameter a° remains to be given by... 

Suppose that the lowest cavity mode is resonant. Then one can evaluate the dimensionless coupling constant asK (e2/In mc2L) (here we return to the dimensional variables). The maximum value of parameter is (see the discussion in Section X) max dmaxvs/2nc, where 5max 0.01 is the maximal possible relative deformation in the material of the wall, and vs 5 103 m/s is the sound velocity inside the wall. Then the ratio e/k cannot exceed the value 5max (mirL/Hne2) 0.05 for L 1 cm and m the mass of electron (for these parameters k 2 10 7). Consequently, one may believe that in the real conditions e/k [Pg.369]

This spectral density has a characteristic low-frequency behavior J((o) — rjo), where rj is the usual ohmic viscosity. The system-bath coupling strength can then be measured in terms of the dimensionless Kondo parameter K, and time scale of bath motions is described by a cutoff frequency (o. For many problems in low-temperature physics, this cutoff frequency is taken to be the largest frequency scale in the problem. In the case of electron transfer, the same spectral density with some intermediate value for is most appropriate for a realistic description of... 

Rg. 9l6 Dynamic mechanical moduli of the nonlinear chain A = 10 with tension-dissociation coupling g =0.2. The dimensionless diffusion parameter e/j is changed from 0 (affine network, broken fine) to 0.20 (solid lines). (Reprinted with permission from Ref. [23].)... 

Table 2. Dimensionless kinetic parameters for various methods and chemical steps coupled with reversible electron transfer (adapted from [3])...

The problems discussed in this section have been restricted to reversible electron transfer processes coupled with first-order chemical reactions (for the most part). The current responses are usually expressed as functions of the dimensionless kinetic parameters (cf. Table 2) involving the life-time of mercury drop, For the estimation of the chemical rate constants of reversible reactions the equilibrium constants K should be known. As in other voltammetric methods (see below), the experimental data are transformed into normalized quantities. Kinetic... 

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