Nearest-neighbor site probability

Fig. 4. Stability of carbon on different sites (A-D) on a pure nickel(l 11) surface (below) and a gold-alloyed nickel(l 11) surface (above). The probability of nucleation of graphite is determined by the stability of the adsorbed carbon atoms. The less stable the adsorbed carbon, the larger the tendency to react with adsorbed oxygen to form CO and the lower the coverage. On the pure nickel) 111) surface, the most stable adsorption configuration of carbon is in the threefold (hep) site (lower curve). The upper graph shows that carbon adsorption in threefold sites next to a gold atom is completely unstable (sites B and C), and even the threefold sites that are next-nearest neighbors (sites A and D) to the gold atoms are led to a substantial destabilization of the carbon. From Reference (79).

The quantities K0 and Kt thus define the solution. As indicated in Appendix A, the result, Eqs. (5)-(9), is identical with the familiar statistical mechanical solution for the case of nearest-neighbor interactions, summarized for example, by Schwarz.2 We note the ease with which the results have been obtained here. The procedure could be extended to other cases, for example, a copolymer (i.e., a linear lattice with two types of sites) distributed in a prescribed manner and undergoing a transition to two other types of sites. For the finite chain, however, the use of nearest-neighbor conditional probabilities and detailed balancing will not yield the complete solution.3... 

The diffusion of vacancies is uncorrelated for the same reasons given above for diffusion of the interstitial atoms. After each jump, a vacancy will have the possibility of jumping into any one of its 12 nearest-neighbor sites with equal probability. 

A rough estimate for f can be obtained based on the number of nearest-neighbors and the probability that a tracer atom which has just jumped and vacated a site will return to the vacant site on the vacancy s next jump. A vacancy jumps randomly into its nearest-neighbor sites, and the probability that the return will occur is 1 jz. This event will then occur on average once during every 2 jumps of an atom. For each return jump, two atom jumps are effectively eliminated by cancellation, and the overall number of tracer-atom jumps that contribute to diffusion is reduced by the fraction 2/z. According to Eq. 8.3, D is proportional to the product Tf, and since the number of effective jumps is reduced by 2/z, f can be assigned the value f 1 - 2/z = 0.83 for f.c.c. crystals. More accurate calculations (see below) show that f = 0.78. 

Solution. There are six first-neighbor sites in the primitive cubic lattice, and the first-neighbor jump distance, r, is equal to the lattice constant, a. Once an atom has jumped into a given site, the probability that it will next jump into any of its nearest-neighbor sites (with the exception of the site from which it just jumped) is (1 —p)/5. Therefore,... 

Thus we are considering the case of a random walker moving in an infinitely extended lattice, and making jumps from one site of this lattice to one of the two nearest neighbor sites with equal probability. Thanks to Eq. (68), the matrix K of Eq. (59) becomes... 

Isotropic translational diffusion has been simulated by a simple random walk process in which each spin — representing one or more nematic molecules — jumps to one of its nearest neighbor sites with equal probability [11]. After the diffusion jump has been performed, the spin acquires the orientation of the local director at the new coordinates. Calculating G t) we have, like in the diffusion-less case, updated from the MC data the spin configuration inside the droplet 8 times per NMR cycle. Now additional diffusion steps have been added in between these structural updates, with their number A ranging from 1 to 32. In this last case the spectra are completely motionally averaged due to dififiision effects since for A = 32 each of the spins exhibits a total of 256 jumps within the duration of one NMR cycle. This already corresponds to the fast diffusion limit with C to-... 

The random walk is not limited to rectangnlar lattices. In the nom-ectangular lattices such as a triangular lattice and a diamond lattice with lattice unit = b, we let the random walker choose one of the Z nearest-neighbor sites with an equal probability irrespective of its past (Markoffian). Then, the same statistics holds for Ar, as the one in the rectangular lattices ... 

The sites are independent. In general this assumption requires that the adsorption of A on one site does not affect the probability of adsorption on any other site. Dependence between sites can occur either because of interactions between adsorbed molecules on different (usually nearest-neighbors) sites or by indirect interaction transmitted by the sites. Since we have already made assumption 4, it is sufficient to state here that there are no direct interactions between adsorbed molecules on different sites. Note that assumption 1 is equivalent to infinite repulsive interaction between two ligands brought to the same site. 

Note that substitution of equations 7.2 (including KcPc) and 7.10 into equation 7.11 does NOT result in a second-order dependence on L (i.e., ), as mistakenly stated in various texts and papers, because nearest neighbor sites are required in step 7.8 above, and the probability for site pairs is 1/2 Z/L[2], where Z is the number of nearest neighbor sites around a site on the surface. Consequently, the term must be replaced by 1/2 ZL, and the rate is always proportional to the site density L in the regime of kinetic control. 

Finally, substitution of equations 2, 5 and 7 into equation 1 yields the following rate law, after including the probability of nearest neighbor sites and dividing r by the Fcs concentration (per g) from Table 7.7 to convert... 

This is so because for an electron with a given spin to acquire kinetic energy, such an electron must have available a nearest neighbor site not already occupied by an electron of the same spin. The probability of encountering such an electron is n/2 and the same for the appropriate empty configuration is (1 — n/2). This gets multiplied by the total band energy. The associated probability of a random double occupancy is... 

We know that the stochastic motion of a particle exhibits a range of dynamics (anomalous diffusion, diffusion, and drift) from short to long asymptotic time regime. Let us consider the motion of a random walk of a particle on a cubic lattice in which a particle hops to one of its nearest-neighbor sites (six on a cubic lattice) randomly at each time step. The probability that the random walker reaches a distance R from the origin (the starting point) in f hops (or time steps) is P(t,R) =... 

In a recent computer simulation Zumofen and Blumen treated the situation where hopping to all possible sites occurs with a distance dependence corresponding to the Forster mechanism and for the case of no emission or trapping [81]. Their results are shown in Table 3. It is apparent from a comparison of Tables 2 and 3 that when hopping is not limited to nearest neighbors the probability that a site is resampled decreases. In addition, there is little site resampling in short walks. 

Now, let us consider an approximation to obtain the coherent medium. Suppose that in a coherent (average) medium all the transition probabilities associated with a given pair of nearest-neighbor sites (say 1 and 2) are given their... 

The percolation theory (or more precisely, random bond percolation on nearest-neighbor lattices) assumes that each bond between two nearest-neighbor sites on an infinite periodic lattice is formed randomly with probability p. Detailed reviews, which are partially outdated, are available , and even a movie about percolation has been made. Its possible applications range fi-om quark matter in high-energy physics and the extraction of crude oil from porous media to, perhaps, gelation ... 

Figure 5. The solid line shows the probability for findin the charge on a Cu site between q and q+dq in a fee disordered alloy with any concentration. The dotted lines show the conditional probabilities corresponding to sites with a concentration of Cu atoms on the nearest-neighbor shell of 100%, 75%, 50%, 25%, and 0%.

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