Linear response formulae for the phenomenological coefficients LAA, LAB and LBB for matter transport by single vacancies in a very dilute isotropic alloy of solute B in solvent A are studied by the matrix method of random walk theory familiar in the calculation of impurity correlation factors. A symmetry classification of vacancy jumps based on symmetry with respect to the diffusion axis is introduced and applied to the linear response formulae in order to reduce them to compact forms. For typical models these forms contain a small number of functions Kij which are linear combinations of vacancy migration Green functions for paths between sites of symmetry types i and j that do not pass through the impurity. The Kij are calculated by inversion of a certain matrix of jump probabilities. Detailed calculations are made for models with first- and second-nearest-neighbour vacancy-impurity interactions for BCC, FCC and diamond lattices. The calculations of LAB and LAA are seen to be straightforward extensions of those already familiar in the calculation of LBB and the impurity correlation factor. This suggests that extensions to other crystal structures and migration mechanisms can be made.
A model polymer is allowed to grow in the same direction as that of the previous unit with probability α or turn at right angles with probability β but not to reverse. The mean square end-to-end distance <R2≳ of a polymer of N units, is calculated to be 〈R2〉/a02=[(1+α)/(1−α)]N−2α (1−αN)/(1−α)], where a0 is the unit length. An exact expression for the probability that the Nth unit points in the same direction as that of the zeroth unit is also obtained. A general way of interpreting the conformation of simple polymers in terms of the present model and the connection with other theories of the polymer conformation are discussed.
Based on the idea that different temperatures generate different carrier densities and the resulting diffusion causes the thermal emf, a new formula for the Seebeck coefficient S is obtained: [Formula: see text], where q, n, εF, [Formula: see text]. and [Formula: see text]. are respectively charge, carrier density, Fermi energy, density of states at ∊F and volume. Ohmic and Seebeck currents are fundamentally different in nature. This difference can cause significantly different transport behaviors. For a multi-carrier metal the Einstein relation between the conductivity and the diffusion coefficient does not hold in general. Seebeck (S) and Hall (RH) coefficients in noble metals have opposite signs. This is shown to arise from the Fermi surface having "necks" at the Brillouin boundary.
A walker is allowed to move on the simple cubic lattice with the following rules: If it should arrive at any site, it may move in the same direction as that of the previous step with probability a, turn at right angles with probability y, reverse with probability P, or remain with probability o. , normalized such that a + 4y +P + o. = l. If it were at rest, it may move in any direction with the same probability p or remain with probability o', normalized such that 6p + o' = 1. An exact expression for the mean-square displacement (r') after N units of time is derived. From this expression, the diffusion coefficient D is obtained as follows: D =(1/6)(l + & -P)(1 -& +P) ' XI1+0. /(1o')) 'aor ', where ao is the step length and v. the unit of time. Similar results are obtained for the face-centered and body-centered cubic lattices. These results are used to discuss the atomic diffusion in cubic crystals with impurities, which act as traps. Comparison with previous experimental and theoretical results is made and discussed.
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