Hopping frequency

A more recent study performed on nC6 and hydrogenated fullerenes presents more detailed effects on hopping frequency, heart rate, appendage movement, and post-abdominal claw curling. The results show intoxication effects, which lead an invertebrate population to be more easily plundered (Lovem et al., 2007). 

In addition to structural information, Li MAS NMR Tz relaxation measurements and analysis of Li line shapes have been used to probe the dynamics of the lithium ions. Holland et al. identified two different species with different mobilities (interfacial Li (longer Tz, rapid dynamics) and intercalated lithium (shorter Tz, slower dynamics)) in the elec-trochemically lithiated V2O5 xerogel matrix. Li hopping frequencies were extracted from an analysis of the Li line widths and the appearance of a quadru-polar splitting as the temperature decreased in a related system. ... 

An observation of motion of single atoms and single atomic clusters with STEM was reported by Isaacson et al,192 They observed atomic jumps of single uranium atoms on a very thin carbon film of —15 A thickness or less. Coupled motion of two to three atoms could also be seen. As the temperature of the thin film could not be controlled, no Arrhenius plot could be obtained. Instead, the Debye frequency , kTIh, was used to calculate the activation energy of surface diffusion, as is also sometimes done in field ion microscopy. That the atomic jumps were not induced by electron bombardment was checked by observing the atomic hopping frequencies as a function of the electron beam intensity. 

A characteristic property of surface migration is that ((Ar)2 varies linearly with time. Note that the very definition of a hopping frequency Th tacitly implies statistic averaging over many hopping events. The time difference between the individual jumps of a specific particle varies stochastically. The corresponding tracer diffusion coefficient is defined as ... 

As already mentioned before, if one could localize a core hole on a given nucleus it would oscillate with a hopping frequency determined by the level splitting Wa, and the n-charge would have a relax in the presence of a moving hole, i.e. dynamic relaxation. However, for a deep core hole the hopping frequency would be practically zero and we would then have static relaxation (cf. Sect. 3). We shall now discuss these two cases in some detail. 

When the hopping frequency of the a-hole is equal to the resonance frequency of the jr-bond there are two equally important normal modes of the total system o-hole and 7r-bond, one where the hole and screening charge oscillate together in phase between the... 

Conceptually, the process is thought to occur as a random walk where adparticles hop between adjacent sites, i.e. from an occupied to an adjacent empty site. The hopping frequency depends then exponentially on the temperature of the system which leads to the following form of the diffusion coefficient ... 

In the lattice gas model, the hopping frequency of a carrier decreases exponentially with the hopping distance p as v = vQ exp(-2p/p0). Here, pQ is a wavefunction decay constant and vQ a frequency factor. From the Einstein relationship, the zero-field mobility is... 

On the other hand, for a non-adiabatic reaction, k i 1, /CeiVn = Vgi and the rate constant is given by Eq. 23 where Vei is the electron hopping frequency in the activated complex. The Landau-Zener treatment yields Eq. 24 for Vei [16, 17]. 

The above treatment is based upon the traditional Born-Oppenheimer approximation which states that, when nuclei move, the electrons can almost instantaneously adjust to their new positions. Another relevant time frame is the time required to establish the electronic polarization of the medium. To characterize this time frame, Kim and Hynes consider the ratio of Vei, the electron hopping frequency, to Vep, the frequency characteristic of the solvent electronic polarization. The Bom-Oppenheimer-based treatment is valid provided that this ratio is much less than unity, i.e., the time scale for the adjustment of the electronic polarization is much shorter than that for the transferring electron [22-26]. 

The electron hopping frequency may be estimated from time-dependent perturbation theory. If Hab is treated as a constant perturbation, the system will start to oscillate between the two diabatic states once the perturbation is turned on. In a bimolecular reaction, for example, the perturbation is turned on upon formation of the precursor complex, while in a covalently attached (bridged) binuclear system it can be turned on upon reduction (oxidation) of one end of the fully oxidized (reduced) system by an external reagent or by photoexcitation. If the system is in the diabatic reactant state at / = 0, then the probability of it being in the product state at some later time t is given by the Rabi formula [27]. 

The rates of the elementary reactions have been chosen in accordance with experimental findings, whenever this was possible. For a comparison with experimental data, see for example reference [76]. In total, the model contains 51 reactions, fourteen of which involve four sites or more. The values for the reaction rates chosen in our model are shown in Table 1. Note that the rate constant for diffusion is in fact a hopping frequency, because we have modeled diffusion as a hopping process. Compared to realistic values, the diffusion rate is very low realistic rates are about five orders of magnitude faster. High diffusion rates can only be simulated with much simpler models and smaller simulation grids than we have used in our simulations. 

Figure 2. Left AB production rate as a function of temperature. Oscillations occur in the hatched area, with rates alternating between between the lower and the upper boundary of the hatched area. Right amplitudes of the oscillations in the simulations (solid line) and in experiments on Pt(lOO) (dashed line). Experimental values taken from reference [67]. Grid size used in these MC simulations 256x 256. Increasing the grid size did not significantly alter the results. In the absence of diffusion, the AB production was absent at temperatures below 400 K. The hopping frequency for diffusion was 30 s , Pa = 2 mPa, Pb2 = 40 mPa. In the experiments, an oxygen pressure of 55 mPa was employed and the CO pressure was varied between 0.13 and 13 mPa [67].

Figure 5. Grid size dependence of oscillation amplitudes at 490 K. L is the linear dimension of the grid. The solid line shows the grid size dependence in absence of diffusion, the dashed line for a hopping frequency of 30 s . The lines are only guides to the eye.

Figure 1. Density of states g(t) of a probe scattered by bosonic atoms in a ID optical lattice. Solid, dotted and dashed curves stand for A(t) and the thick curve stands for A(e) (dispersion), see text. All curves are numerically computed from G(t) and correspond to average random couplings (IT2) = 0.4, 2 and 10 respectively. The hopping frequency J = 1, and for all curves f deg(t) = 1. Inset (a) a probe weakly scattered by a randomly occupied lattice. Inset (b) a probe multiply scattered by a regular atomic distribution.

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