On an application of the causality principle to the theory of ion transport processes
Introduction
The theory of solid electrolytes faces problems that cannot be explained by means of the ordinary absolute rate theory (ART). Among them are isotope effects in simple and complex perovskite-like oxides. It was stated in my recent paper [1] that the treatment of such effects in terms of the Kramers problem [2] with arbitrary damping correlates well with other known approaches (for example, with the atomistic description [3]). Probably, the many-particle interaction in the lattice should be taken into account for the most adequate description of ion transport processes. The expression for the diffusion coefficient D in doped SrCeO3-type high-temperature proton conductors (HTPC's) [1] (or the stochastic expression for the ionic conductivity σ [4]) includes an additional factor A1 (δ/kT), where δ is the average energy loss of the ion per one oscillation. The aforementioned correlation generates interest in the search for other theoretical derivations of this correction to the reaction rate τ−1 in solids without resorting to the Kramers problem and details of the transport mechanism.
Modern methods for describing the ion transport are based on the theory of frequency factors proposed by Vineyard in 1957 [5] who considered the ART with a special attention to many-body aspects. The expressionis used for HTPC's [6], where
Here ν is equal to a typical frequency (for instance, we denote by the frequency of oscillation at the bottom of the well), z is the number of jump directions, e is the charge, d is the jump distance, v0 is the molecular volume, k is Boltzmann's constant, Ceff is the effective concentration, and E(a) is the activation energy. Traditionally, there is no fundamental distinction between ν and ν0. For most experimental devices, it is important to know the actual value of ν. However, the calculation of ν is a complicated problem.
The purpose of the present paper is to discuss several reasons for thinking about frequency factors in the framework of the principle of physical causality. A feature of this strategy is the use of analytic methods similar those used in quantum field theories and optics.
Section snippets
Attempt frequency from integral equations
The correction to the pre-factor A for coupling to a bath is of the form μA1, where μ is the Kramers coefficient [2]. In the following, we assume that μ=1. At small Δ=δ/kt, the damping factor is given by A1∼Δ. From the Wiener–Hopf (WH) equation for the distribution function of particles over their energies, f(ε), we havewhich was derived [7] by solution of the Fokker–Planck (FP) equation in the energy-position variables with the boundary condition
S-matrix formalism of transfer rate in solid electrolytes
For studying the kinetic properties of superionic conductors, we use techniques from S-matrix theories. With this formalism it is easier to understand how ionic oscillations in wells are incorporated in the transfer rate scheme and to explain why some frequency factors have a quite remarkable dependence on the many-particle interaction.
We begin by briefly recounting the conceptual grounds of the S-matrix theory according to Refs. [9], [10], [11]. The state vector Φ=Φ(t) obeys the Schrödinger
Discussion
It follows from Eq. (11) that to escape and, hence, contribute to the conductivity, ions should perform all cycles of n oscillations in a well. During these cycles, they accumulate energy for the final jump. This is just the manifestation of the causality principle in the ion transport. The principle of detailed balance is also significant in the S-matrix theory. As distinct from [8], where this principle was used as an additional condition for asymptotic properties of the kernel given in Eq.
Conclusion
Calculations of kinetic characteristics of solid electrolytes were discussed in terms of the causality principle. A special emphasis was placed on the correlation between the obtained results and findings known in the Kramers model. This approach permits to obtain further insight into the appropriate contribution from the phonon effects.
A general expression for the ion attempt frequency without invoking details of the transport mechanism was suggested. It was shown that frequency factors are
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