Hopping models

By using this approach, it is possible to calculate vibrational state-selected cross-sections from minimal END trajectories obtained with a classical description of the nuclei. We have studied vibrationally excited H2(v) molecules produced in collisions with 30-eV protons [42,43]. The relevant experiments were performed by Toennies et al. [46] with comparisons to theoretical studies using the trajectory surface hopping model [11,47] fTSHM). This system has also stimulated a quantum mechanical study [48] using diatomics-in-molecule (DIM) surfaces [49] and invoicing the infinite-onler sudden approximation (lOSA). 

A celebrated derivation of the temperature dependence of the mobility within the hopping model was made by Miller and Abrahams 22. They first evaluated the hopping rate y,y, that is the probability that an electron at site i jumps to site j. Their evaluation was made in the case of a lightly doped semiconductor at a very low temperature. The localized states are shallow impurity levels their energy stands in a narrow range, so that even at low temperatures, an electron at one site can easily find a phonon to jump to the nearest site. The hopping rate is given by... 

Another famous hopping model is Mott s variable range hopping [23], in which it is assumed that the localized sites are spread over the entire gap. At low temperatures, the probability to find a phonon of sufficient energy to induce a jump to the nearest neighbor is low, and hops over larger distances may be more favorable. In that case, the conductivity is given by... 

We note a temperature dependence of the zero field mobility as exp[—( F()/F)2], a behavior which is indeed encountered in real organic semiconductors, and differs from both Millers-Abrahams fixed range and Moll s variable range hopping models. 

Point defects in solids make it possible for ions to move through the structure. Ionic conductivity represents ion transport under the influence of an external electric field. The movement of ions through a lattice can be explained by two possible mechanisms. Figure 25.3 shows their schematic representation. The first, called the vacancy mechanism, represents an ion that hops or jumps from its normal position on the lattice to a neighboring equivalent but vacant site or the movement of a vacancy in the opposite direction. The second one is an interstitial mechanism where an interstitial ion jumps or hops to an adjacent equivalent site. These simple pictures of movement in an ionic lattice, known as the hopping model, ignore more complicated cooperative motions. 

The electronic band structure of a neutral polyacetylene is characterized by an empty band gap, like in other intrinsic semiconductors. Defect sites (solitons, polarons, bipolarons) can be regarded as electronic states within the band gap. The conduction in low-doped poly acetylene is attributed mainly to the transport of solitons within and between chains, as described by the intersoliton-hopping model (IHM) . Polarons and bipolarons are important charge carriers at higher doping levels and with polymers other than polyacetylene. 

Hopping Models Hole-Resting-Site and Phonon-Assisted Polaron Transport... 

In a second possibility, the polaron-like hopping model, a structural distortion of the DNA stabilizes and delocalizes the radical cation over several bases. Migration of the charge occurs by thermal motions of the DNA and its environment when bases are added to or removed from the polaron [23]. 

The hole-resting-site and polaron-like hopping models can be distinguished by the distance and sequence behavior of radical cation migration. Analysis of the hole-resting-site model leads to the prediction that the efficiency of radical cation migration will drop ca. ten-fold for each A/T base pair that separates the G resting sites [33]. 

The phonon-assisted polaron-like hopping model is unique because it is built upon an understanding of the dynamical nature of DNA in solution. The fundamental assumption of this model is that the introduction of a base radical cation into DNA will be accompanied by a consequent structural change that lowers the energy for the system. 

In the hopping model the electrochemical potential, p, of electrons is expressed conventionally in the same way as that of ions as shown in Eqn. 2-26 ... 

We now consider the relationship which connects the electrochemical potential of electrons in the hopping model with that in the band model. The total concentration, N, of electron sites for the hopping model may be replaced by the effective state density, JVc, for the band model. For the two models thereby we obtain from Eqn. 2-27 the following equation ... 

The hopping model This model assumes that molecules can hop over the surface. The surface flux is calculated by the mean hopping distance and the velocity, with which the molecules leave their site. Weaver and Metzner (1966) developed a detailed model to calculate the mean hopping distance. Ponzi et al. (1977) developed a simpler way of estimating the mean hopping distance. 

An adsorbed phase can exhibit monolayer adsorption as well as multilayer adsorption. Surface flow in the presence of multilayer adsorption can be accounted for in the models described in the previous section. For example Okazaki and Tamon (1981) describe multilayer diffusion in their random hopping model. 

Fig. 4.27 Q-dependence of the amplitude of the relative quasi-elastic contribution of the -process to the coherent scattering function S (Q)/S(Q) obtained for PB from the hopping model solid line) with dp=l.5 A. The static structure factor S(Q) at 160 K [123] is shown for comparison dashed-dotted line)...

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