Diffusion trajectory

If the diffusion trajectory of an ion is divided into sections of length e and the number of divisions is denoted as N(e),... 

The fractal behavior of diffusion trajectories of ions has been studied in the molten phase of Agl as well as in the a-phase. The Devalues for an MD system with 250 Ag and 250 I" at 900 K were calculated from Fig. 21 to be 2 and 2.17, respectively. The mean-square displacements are shown in Fig. 22 in comparison with those of the a-phase at 670 K. As results of supplementary MD simulations, these authors obtained Dj = 1 for Ag and D = 2.17 at 1000 K and Df = 2 for both ions at 2000 K. Thus, they have concluded that (1) at an extremely high temperature above the melting point, the system is in a completely liquid state, which leads to a... 

Figure21. The division number A(e) for the trajectories of Ag and I" at 900 K. (Reprinted from M. Kubayashi andF. Shimojo, Molecular Dynamics Studies of Molten Agl.II. Fractal Behavior of Diffusion Trajectory, J. Phys. Soc. Jpn. 60 4076-4080, 1991, Fig. 6, with permission of the Physical Society of Japan.

The independent reaction time (1RT) model was introduced as a shortcut Monte Carlo simulation of pairwise reaction times without explicit reference to diffusive trajectories (Clifford et al, 1982b). At first, the initial positions of the reactive species (any number and kind) are simulated by convolving from a given (usually gaussian) distribution using random numbers. These are examined for immediate reaction—that is, whether any interparticle separation is within the respective reaction radius. If so, such particles are removed and the reactions are recorded as static reactions. 

Figure 16. A double-well potential for reaction IV X) along a one-dimensional reaction coordinate X in the Kramers model, and a reactive diffusive trajectory represented by a zigzag line surmounting a reaction barrier from the reactant to the product well.

This correspondence between equipotential and tangential field surfaces leads to a simple construction for the likely diffusion trajectories of ions within fast-ion conductors, also called "solid electrolytes". These solids (typically binary salts) are electrically highly conducting (the electrical conductivity of... 

Dicthyl/.inc addition to bcn/aldeliydc 142 Diffusion ctTects 22S Diffusion-reaction equations 197 Diffusion trajectory I9S Di-Grignard reagents 288-9. 367-85 9. tlM)ihydroanthraecne 409... 

In fact, single molecule spectroscopy (SMS) experiments have recently become a reality. The first experiments were performed on pentacene (the chromophore) in a p-terphenyl crystal [8-10]. I will focus here on the experiments of Ambrose, Basche, and Moemer [9, 10], which involved repeated fluorescence excitation spectrum scans of the same chromophore. For each chromophore molecule they found an identical (except for its center frequency) Lorentzian line shape whose line width is determined by fast phonon-induced fluctuations (and by the excited state lifetime), as discussed above. However, for each of a number of different chromophore molecules Moemer and coworkers found that the chromophore s center frequency changed from scan to scan, reflecting spectral dynamics on the time scale of many seconds The transition frequencies of each of the chromophores seemed to sample a nearly infinite number of possible values. Plotting the transition frequency as a function of time produces what has been called a spectral diffusion trajectory (although the frequency fluctuations are not necessarily diffusive ). These fascinating and totally... 

In the second type of experiment that measures single molecule spectral dynamics one performs repeated fluorescence excitation scans of the same molecule. In each scan the line shape is described as above, but now there is the possibility that the center frequency of the line will change from scan to scan because of slow fluctuations. Thus one can measure the center frequency as a function of time, producing what has been called a spectral diffusion trajectory. This trajectory can, in principle, be characterized completely by the spectral diffusion kernel of Eqs. (16) and (19), but of course it must be understood that only the slow Kj < 1 /t) TLSs contribute. In fact, the experimental trajectories are really too short to be analyzed with this spectral diffusion kernel. Instead, it is useful [11, 12] to consider three simpler characterizations of the spectral diffusion trajectories the frequency-frequency correlation function in Eq. (14), the distribution of frequencies from Eq. (15), and the distribution of spectral jumps from Eq. (21). For this application of the theoretical results, in all three of these formulas j should be replaced by s, the labels for the slow TLSs. 

Spectral diffusion trajectories due to spontaneous (rather than light-induced) fluctuations have been measured for Tr in PE [14] and for TBT in PIB [15,16]. As in the crystalline case these trajectories reflect dynamics of the slow TLSs. The three published trajectories show that in two cases the chromophore visits a large number of frequencies, and in one case, only four. In this latter case the chromophore is presumably strongly coupled to two TLSs. A correlation function analysis was applied to the PIB system, but for neither the PIB nor the PE system was a temperature-dependent study reported. 

Extremely exciting experimental data for glasses are now beginning to emerge. It has been shown that line shape measurements, fluorescence intensity fluctuations, and spectral diffusion trajectories can all be used to probe TLS dynamics on different time scales. Furthermore, as has been emphasized already, these experiments on individual molecules will provide information complementary to that obtained from more traditional echo and hole burning experiments. At this point what we need is more data. In an ideal world all three of the above experiments would be performed on the same individual molecule at a variety of temperatures, and then would be repeated on many molecules, and all of the above would be repeated for several different systems. Although the basic theoretical apparatus is in place for analyzing these experiments, more refined theoretical results will surely be needed. 

Quantum-chemical analysis of the electronic states in the MX-H systems was carried out by Kupryazhkin et al (1984) making use of 19-atom clusters [V(C,N)5DVi2 + H], Along with the initial (octahedral or tetrahedral) position of H, the clusters containing H at the centre of an elementary diffusion trajectory were considered. These were the clusters which are met when in the diffusion process H moves from one tetrahedral position in the crystal lattice to another. In the cases of both highly symmetric and displaced positions of H, H atoms were found to interact more strongly with their surroundings when placed near the X vacancy. 

In a melt of long chains, each chain is entangled with its naghbom and the characteristic relaxation time of movement of the whole chain is given by the time of the chain to diffuse trajectory equal to its length. According to deGeiines [20], the relaxation time of the chain reptation motion, can be expressed as ... 

The IRT model [1,2, 12, 15] is essentially a Monte Carlo algorithm which assumes the independence of reaction times (i.e. each reaction is independent of other such reactions and that the covariance of these reaction times is zero). Unlike the random flights simulation, the diffusive trajectories are not tracked but instead encounter times are generated by sampling from an appropriate probability density function conditioned on the initial separation of the pair. The first encounter takes place at the minimum of the key times generated min(h t2, fs...) and all subsequent reactions occur based on the minimum of surviving reaction times. Unlike random flights... 

Accurate treatment of products which are capable of further reactions is another challenging problem in the IRT algorithm, since again the diffusive trajectories are not traced. The necessity to generate the correct spatial distribution of reactive products is of paramount importance, as this ultimately affects the subsequent kinetics that follow. In the literature, three approximations have been discussed by Clifford et al. [12] which aim to calculate either the new interparticle separation or a new reaction time directly. A brief discussion is now presented. 

Consider particles i, j,k which can all react with each other at geminate times tij, tik and tjk- If particles i and j react to produce I (which replaces i), then the diffusive trajectory for Ik will be the same as ik had reaction not occurred but it will be followed with the new relative diffusion coefficient. Hence, the new species I follows the same trajectory path as species i would have, with a new relative diffusion coefficient up to the encounter distance point The new scaled reaction time for the geminate recombination time Ik is then given by... 

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