Hopping processes

On the other hand, the rope of SWCNTs gave an increase of resistivity with increasing temperature which suggests the existence of metallic CNTs [21], This is accounted for by the small effect of inter-chain barrier and the absence of interlayer hopping processes. 

More recently, D. Emin [24] developed an alternative analysis of activated hopping by introducing the concept of coincidence. The tunneling of an electron from one site to the next occurs when the energy state of the second site coincides with that of the first one. Such a coincidence is insured by the thermal deformations of the lattice. By comparing the lifetime of such a coincidence and the electron transit time, one can identify two classes of hopping processes. If the coincidence lime is much laigcr than the transit lime, the jump is adiabatic the electron has lime to follow the lattice deformations. In the reverse case, the jump is non-adia-batic. 

Electronic conductivity arises from delocalized electrons, tunneling or hopping processes. 

Apart from the problems of low electrocatalytic activity of the methanol electrode and poisoning of the electrocatalyst by adsorbed intermediates, an overwhelming problem is the migration of the methanol from the anode to the cathode via the proton-conducting membrane. The perfluoro-sulfonic acid membrane contains about 30% of water by weight, which is essential for achieving the desired conductivity. The proton conduction occurs by a mechanism (proton hopping process) similar to what occurs... 

The occurrence of intramolecular electron-hopping processes will now be discussed in detail, considering on an empirical basis systematic variation of relevant structural factors in the substrates. First, however, a few theoretical and experimental aspects will be summarized. 

A number of different techniques have been applied to test the distance and orientation dependence of ET reactions (Closs and Miller, 1988 Closs et al, 1989 Liang et al., 1990 Reimers and Hush, 1990 Fox and Chanon, 1988 Wasielewski, 1989 Paddon Row and Jordan, 1988 Joachim et al, 1990 McConnell, 1961). Our method of analysing the mode of charge distribution in charged species is esr spectroscopy, which defines the timescale of the detectable dynamic species (Gerson, 1967 Kurreck et al, 1988 Wertz and Bolton, 1972). If an electron transfer is slow relative to the esr timescale (<10 7s) the spectrum corresponds to that of monomeric model compounds with a single electrophore. If the hopping process is rapid on the esr timescale, one will detect an effective delocalization. 

Thus, in the present approach, the major focus is on the question of how we can influence the external parameters like solvent and counterion and the intrinsic structural parameters within the systems A-l-A to force the electron-hopping process into the timescale of the experiment, or at least to establish clearly the borderline cases. That we are still looking at an electron-hopping process in the case of effective charge delocalization over the entire molecule and not at a pure resonance phenomenon may be reassured by VIS/NIR spectroscopy of the neutral and charged species the absorption of a single chromophore should be detected unless a very fast process > 1012 Hz is taking place. 

In the radical anions of the di(2-anthryl) system [9] and of the ort/io-cyclophane [13] a spin-localized situation prevails independently of temperature and ion pairing. On the other hand, for the corresponding di(9-anthryl) species and even for the di(l-anthryl) species a rapid hopping process can be detected if suitable ion pairing is established (Becker et al., 1991). Thus, according to both cyclic voltammetric and esr spectroscopic... 

The hopping process in the cancrinite cage is only of local nature, and long-range conductivity has been observed only for the sodium cations in the onedimensional 12-ring channels, where the chromate ions are also located [283]. 

This distance is far beyond the equilibrium region of primary chemical interest, so we do not consider the long-range-hopping process further. 

As indicated, the power law approximations to the fS-correlator described above are only valid asymptotically for a —> 0, but corrections to these predictions have been worked out.102,103 More important, however, is the assumption of the idealized MCT that density fluctuations are the only slow variables. This assumption breaks down close to Tc. The MCT has been augmented by coupling to mass currents, which are sometimes termed inclusion of hopping processes, but the extension of the theory to temperatures below Tc or even down to Tg has not yet been successful.101 Also, the theory is often not applied to experimental density fluctuations directly (observed by neutron scattering) but instead to dielectric relaxation or to NMR experiments. These latter techniques probe reorientational motion of anisotropic molecules, whereas the MCT equation describes a scalar quantity. Using MCT results to compare with dielectric or NMR experiments thus forces one to assume a direct coupling of orientational correlations with density fluctuations exists. The different orientational correlation functions and the question to what extent they directly couple to the density fluctuations have been considered in extensions to the standard MCT picture.104-108... 

The second-order character of the electron hopping process is responsible for the nonlinear expression of the term that depicts the field effect. 

As discussed above, this discrepancy may be caused by classically forbidden electronic transitions—that is, cases in which a proposed hopping process is rejected due to a lack of nuclear kinetic energy. Figure 11c supports this idea by showing the absolute numbers of successful (thick fine) and rejected (thin line) surface hops. In accordance with the initial decay of the adiabatic population, the number of successful surface hops is largest during the first 20 fs. For larger times, the number of rejected hops exceeds the number of successful surface hops. This behavior clearly coincides with the onset of the deviations between the two classically evaluated curves Nk t) and P t). We therefore conclude that the observed breakdown of the consistency relation (42) is indeed caused by classically forbidden electronic transitions. 

In activated ion hopping processes such as occur in solid electrolytes, there is an inverse correlation between the magnitude of the activation energy and the frequency of successful ion hops this leads us to the concept of ion hopping rates. 

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