On a Stochastic Theory of Ion Transfer at Intermediate Damping

Abstract

The stochastic method, which was formulated earlier by Gurevich and Kharkats on the basis of a classic solution of the Fokker-Planck equation in a phase space of coordinates and momenta, is developed to extend it to the case of intermediate damping. In contradistinction to the Gurevich-Kharkats model, the new theory is based on the Fokker-Planck equation in the energy-action coordinates. Analytical expressions for the exchange current and conductance are derived. The expressions are valid in a broad damping regime. As compared with the Gurevich-Kharkats model, the expression for the exchange current contains an additional factor, which depends on the energy losses sustained by the ion upon its collision with the cage of a crystal lattice. An explanation for the isotope effect in the high-temperature protonic conductors is suggested. It is demonstrated that the friction coefficient for these solid electrolytes may correspond to the underdamped-to-intermediatedamping regime. This fact makes it possible to put forth an explanation for the existing contradictions between their properties and simple hopping models, which ignore the interaction of ions with the crystal lattice.

Simulation of prefactor anomalies in superionic conductors in the case of arbitrary energy losses. II. Energy distributions in the barrier region

“…It is possible also to use the close relation [29] δ=(Γ/2)E (a) where Γ is a dimensionless friction. Some experimental data agree with Γ =2 $ 0:02 À 0:04 [10], yielding the range Δ $ 0:34 À 0:8. Line 2 in Fig.…”

“…In particular, in surface diffusion studies [27], γ is small but Δ may be of the order of unity. By using the values [4] U = 0.72 eV and U =0.83 eV for Ba 2 (Ga 1.1 Nb 0.9 )O 6 and Ba 3 (Sr 1.18 Nb 1.82 )O 9 and assuming γ/ω ∼ 0.01 for HTPCs [10], together with the expression [11,28] Δ=4(γ/ω)(U/kT), the damping constants are Δ ∼0.7 and Δ∼0.8, respectively (at T=500 K). It is possible also to use the close relation [29] δ=(Γ/2)E (a) where Γ is a dimensionless friction.…”

“…Since f(ɛ) is directly expressed [8] in terms of the probability of trapping, P(ɛ), our line of attack is thus to reduce the calculation problem to a correct discrete WH equation for f(ɛ). Many of the kinetic properties of different solid electrolytes can be understood based on the assumption that the range of Δ is essentially different from that of ART and exhibits rather intermediate values [2,5,9,10]. In order to examine such a possibility, we used an optimized version of the algorithm implemented in the prefactor study.…”

Simulation of prefactor anomalies in superionic conductors in the case of arbitrary energy losses

“…When the above simulation results are compared with the existing findings [2,6] for proton conductors, such as SrZrO 3 (10%Sc) and Sr 3 Ca 1.18 Nb 1.82 O 9-δ , we obtain formal appropriate evaluation of the prefactor at P =−0.58. This provides excellent agreement with our key hypothesis that the average energy transfer per collision may be comparable to the thermal energy (or somewhat below kT) in several ion-conducting systems for which prefactor anomalies are seen due to a role of collective degrees of freedom.…”

“…1, the kernel W(ɛ′, ɛ) defines the probability that an ion will be transferred from one energy level in the well to another. Recent works [3,5,6] indicate that the intermediate friction regime may be most likely for many superionic conductors. On the other hand, wide prefactor anomalies have been established in recent years for a lot of materials, especially in studies of kinetic isotope effects in proton conductors [2] for which δ is likely to be greater than in the case [1] of Li 2 Ti 3 O 7 .…”