Transition probabilities nearest-neighbor

In this discussion of the transition elements we have considered only the orbitals (n— )d ns np. It seems probable that in some metals use is made also of the nd orbitals in bond formation. In gray tin, with the diamond structure, the four orbitals 5s5p3 are used with four outer electrons in the formation of tetrahedral bonds, the 4d shell being filled with ten electrons. The structure of white tin, in which each atom has six nearest neighbors (four at 3.016A and two at 3.17.5A), becomes reasonable if it is assumed that one of the 4d electrons is promoted to the 5d shell, and that six bonds are formed with use of the orbitals 4dSs5p35d. 

We now introduce nearest-neighbor interactions in the restricted chain models. In our particular model these interactions are nonzero only for those nearest-neighbor contacts which can lead to excluded configurations upon addition of a single step. Thus for example, in a square lattice, nonzero interactions involve only contacts created by the three-step chain configuration denoted by q(i The new transition probabilities are... 

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... 

Next, we plot transition diagrams separately by transition probabilities, for more detailed investigation, as shown in Figs. 15 and 16 for b = 0.040 and b = 0.100, respectively. We put circles on each end of transitions in addition to directional arrows, so that the region that contains only circles without an arrow means there is no transition from there, even to the nearest-neighbor regions. 

If the matrix contains only the nearest-neighbor transition probabilities, the corresponding eigenvectors and eigenvalues are related by the equation ... 

The transport equation (1) reduces to that proposed by E uing and co-workers > > at a steady state with only nearest-neighbor transition probabilities. By setting e = 0o no a nd dCJt)jdt= 0 at the steady state, a transport equation of the following form is obtained from (1) ... 

We shall present here some potential profiles which give analytic expressions of ip s and s for any N, and shall generalize these results to include broader multi-barrier problems. Nearest-neighbor transition probabilities only are involved throughout the following considerations except for the perturbation theory treatment the inclusion of the next-nearest neighbor may be introduced by the perturbation method as discussed later. 

A. A Potential Profile with Equal Forward-Nearest-Neighbor and Equal Backward-Nearest-Neighbor Transition Probabilities... 

For the sake of completeness of our discussion, the perturbation on matrix A will next be briefly consid< red. The perturbation may be due to the deviation from the perfect absorption at the N - - I)th well. It may also be due to the fact that A has non-zero next-nearest-neighbor transition probabilities, or the transition probabilities may be dependent on concentration and time in such a way that the additional elements in a perturbed matrix are small compared with the corresponding unperturbed matrix. Only the stationary perturbation will be considered here. The analogous case for time-dependent perturbation follows in a manner similar to that used in quantum mechanics. 

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