1 Introduction

[page 176, §1]
[176.1.1] A large variety of dynamic phenomena in condensed systems can be described by a lattice gas model. [176.1.2] Examples include transport in superionic solids, phase separation in binary alloys, kinetics of spin models, diffusion in metal-hydrogen systems or order-disorder phenomena in chemisorbed monolayers at surfaces[1, 2, 3, 4]. [176.1.3] Moreover, the popular model of hopping transport on a percolating network can be viewed as a special case. [176.1.4] This is seen by considering a lattice gas with two species of particles, A and B, for the case where the B-particles are frozen into some random configuration. [176.1.5] If, as usual, the A-particles are allowed to hop only into vacant lattice sites then the problem is that of diffusion in a frozen disordered environment. [176.1.6] Our interest in this problem stems from a case in which the B-particles are not completely immobile, but very slow compared to the A-particles. [176.1.7] The general objective in this paper will be to calculate the transport coefficients for the A-particles.

[176.2.1] More specifically, we are motivated by the problem of calculating the frequency dependent conductivity of $\beta^{{\prime\prime}}$ -alumina, a superionic conductor, well known for its ability to transport a variety of cations[5]. [176.2.2] $\mathrm{Na}^{+}$ /Bapp- $\beta^{{\prime\prime}}$ -alumina is a layered compound where $\mathrm{Na}^{+}$ -ions hop between the sites of a hexagonal lattice forming a stack of two dimensional conduction planes[6]. [176.2.3] The activation energy for $\mathrm{Na}^{+}$ is roughly $0.35\,\mathrm{eV}$ while that for Bapp is approximately $0.58\,\mathrm{eV}$ [5]. [176.2.4] Thus at sufficiently low temperatures the Bapp-ions are essentially frozen. [176.2.5] They play the role of the B-particles in the lattice gas described above. [176.2.6] At higher temperatures the Bapp become mobile. [176.2.7] Now the $\mathrm{Na}^{+}$ -ions (A-particles) experience a dynamic instead of a frozen disordered environment. [176.2.8] The fundamental parameter characterizing the different time scales in the problem is the ratio $\tau=\tau _{{\mathrm{B}}}/\tau _{{\mathrm{A}}}$ between the charcteristic hopping time $\tau _{{\mathrm{B}}}$ of the Bapp and $\tau _{{\mathrm{A}}}$ of the $\mathrm{Na}^{+}$ -ions.

[176.3.1] Despite its conceptual simplicity the model contains many interesting features, even if we neglect for the moment all interactions between particles, except, of course, the hard core repulsion. [page 177, §0] [177.0.1] The hard core repulsion prevents double occupancy of lattice sites. [177.0.2] In the limit $\tau\to 1$ the tracer diffusion coefficient for the A-particle shows pronounced correlation effects[7]. [177.0.3] On the other hand in the limit $\tau\to\infty$ the system will exhibit a percolation transition[8]. [177.0.4] For low concentrations of B-particles the Aâparticles can diffuse along an infinite network of vacancies while at high concentrations they are confined to finite clusters, and there will be no long range transport.

[177.1.1] Generally the system discussed above can be described by a many particle master equation[1]. [177.1.2] This remains true even if interactions between particles are included. [177.1.3] In the case of $\mathrm{Na}^{+}$ /Bapp- $\beta^{{\prime\prime}}$ âalumina the Coulomb repulsion between the mobile ions will be important. [177.1.4] Additional correlations can arise from lattice relaxation effects, or from simultaneous hops of groups of ions. [177.1.5] The latter is a well known phenomenon for the related $\beta$ -alumina. [177.1.6] Clearly the problem has to be simplified even if one resorts to a computer simulation. [177.1.7] A careful Monte-Carlo-study of the tracer diffusion problem was carried out by Kehr, Kutner and Binder[7, 9, 10, 11, 12]. [177.1.8] They focussed on the case $\tau=1$ on a face centered cubic (fcc) lattice, and considered systems with and without short range attractive/repulsive interactions. [177.1.9] Theoretical attempts[13, 14, 15] have also concentrated on the case $\tau=1$ and were therefore unable to reproduce the percolation transition for $\tau\to\infty$ . [177.1.10] In this paper we present a theory for general $\tau$ which allows to incorporate correlation effects arising from the blocker dynamics or from other sources such as particle interactions.

[177.2.1] Let us first discuss the general framework of our approach. [177.2.2] Instead of focussing on the case $\tau=1$ the basic idea is to start from the frozen problem, i. e. $\tau=\infty$ . [177.2.3] We discuss the diffusion of a single A-particle in the percolation geometry produced by the immobile B-particles. [177.2.4] Interactions between the particles lead to correlations for the random walk of the A-particle. [177.2.5] In general the effect of such correlations is to change the transition probabilities to nearest neighbours of the particle[16]. [177.2.6] In particular let us assume that the primary effect is to change the transition rate to the site that was occupied before the last step. [177.2.7] Thus we assume that the A-particle has a memory of its previous position, and consequently its random walk no longer has the Markov property. [177.2.8] The corresponding problem on a regular lattice is well known[17, 18, 19, 20] and can be solved exactly. [page 178, §0] [178.0.1] This allows us to formulate a generalized effective medium theory for the correlated hopping of an A-particle on the frozen disordered network.

[178.1.1] Finally the B-particles are allowed to move, i. e. $\tau<\infty$ , and the A-particle now experiences a dynamic disordered environment. [178.1.2] The solution to this dynamic percolation problem can be expressed in terms of the solution for the frozen problem. [178.1.3] The result is valid for arbitrary $\tau$ and allows to incorporate correlation effects via the special transition rate for transitions to the previously occupied site. [178.1.4] The main advantage of this approach is its simplicity. [178.1.5] The three main parameters are the concentration of B-particles, $p$ , the ratio of attempt frequencies, $\tau$ , and the ratio between the hopping rates for return to previously occupied sites and transitions to other neighbours, $b$ . [178.1.6] The first two are essentially fixed from the experiment, while $b$ can be determined from a measurement of the conductivity at any single frequency, e. g. at $\omega=0$ .

[178.2.1] The objective of this paper, as mentioned in the beginning, is the calculation of the frequency dependent conductivity for the A-particles. [178.2.2] In addition we wish to evaluate the correlation factor for the selfâdiffusion constant in the d. c. limit. [178.2.3] We assume here that the B-particles do not contribute to the conductivity. [178.2.4] We will present results for the hexagonal and the fcc-lattice. [178.2.5] The first because of its low coordination number and its relevance for $\mathrm{Na}^{+}$ /Bapp- $\beta^{{\prime\prime}}$ -alumina. [178.2.6] The second because of the possibility to compare with d. c. results from extensive Monte-Carlo simulations in the literature.

[178.3.1] For the a.c. response we find a crossover between a low frequency regime dominated by the effects of blocker motion, and at high frequency regime in which the blockers, or equivalently vacancies, appear to form a frozen network. [178.3.2] In addition we predict the appearance of novel features in the real and imaginary part as a result of correlations. [178.3.3] In the d. c. limit we calculate the correlation factor for the self diffusion coefficient. [178.3.4] It interpolates smoothly between the case $\tau=0$ and $\tau\to\infty$ . [178.3.5] Although our theory does not contain adjustable parameters for the uncorrelated case ( $b=1$ ) we find good agreement with Monte-Carlo simulations by Kehr, Kutner and Binder[10, 11]. Our results will be presented in Section 5 and are then discussed in Section 6. [page 179, §0] [179.0.1] In the next section we formulate our model. [179.0.2] In Section 3 we treat the correlated random walk in a frozen disordered environment, and in Section 4 we use the result from Section 3 as input for the dynamic problem.

Author:	R. Hilfer and R. Orbach
originally published in:	Dynamical Processes in Condensed Molecular Systems, J. Klafter et al. (eds.), World Scientific, Singapore, 1989, pages 175-202, ISBN 981-02-3457-0
originally published:	1989