Theory of one-dimensional hopping conductivity and diffusion

Hopping motion is considered in a linear chain containing an arbitrary density of particles that are not allowed to hop to occupied sites. The general case of two inequivalent lattice sites $A$ and $B$ is treated. Transition-rate equations are solved by Monte Carlo simulation and, where possible, by analytic techniques. Only for the case of equivalent sites do the results here agree with those recently obtained by Huber from nonlinear differential equations for site occupancy which neglect certain correlations. The conductivity is mean-field-like for equivalent sites, but shows sizeable departure for inequivalent sites, having an activation energy increased over the mean-field value. Here "mean-field" behavior is one where the only effect of forbidden hops to occupied sites is to reduce effective transition rates by a factor $1-n$, where $n$ is the average occupation number of the site to which a jump occurs. The velocity correlation function is shown to consist of a $\delta$-function part which reproduces the mean-field conductivity and a function $\beta \mathrm{}\left(t\right)\mathrm{}$ which is negative for times $t>\mathrm{}0\mathrm{}$. This is qualitatively quite different from the picture given by Huber's equations. Motion of a single distinguishable particle shows an anomalous $x^2\propto t^{\frac{1}{2}}$ dependence of the mean square displacement upon time $t$, but the displacement $X$ of all the particles does obey a diffusion relation $X^2\propto t$. This difference is explained in terms of the number of particles which have to be pushed aside in order for a particle to move a distance $x$, and in terms of the ensuing density fluctuation. Differences between time dependences and attempt fequencies as measured by bulk conductivity and microscopic probes such as NMR are noted and discussed in light of data on the one-dimensional superionic conductor $\beta$ eucryptite (LiAlSi${\mathrm{O}}_4$). Reinterpretation of NMR relaxation data on some of the organic charge-transfer salts is also suggested.