Ionic conductivity (${\sigma }_{\mathrm{ion}}$)

Electrical conductivity arising from motion of ions. In a dilute, uncorrelated vacancy model, a mobile oxygen vacancy has effective charge $+2e$ and moves opposite to the oxygen ions that fill it.

With the explicitly stated convention $H_R=D_{\mathrm{tr}}/D_{\sigma }$, the tracer and collective diffusion coefficients are related by $D_{\sigma }=D_{\mathrm{tr}}/H_R$. The Nernst-Einstein relation then gives $\sigma =n{q}^2{D}_{\mathrm{tr}}/(H_R{k}_{B}T)$. State the convention, since reciprocal definitions can otherwise cause confusion.

The activation free-energy barrier for an ion to hop between sites. ionic conductivity also depends on the concentration of mobile defects and correlations between successive hops.

For dilute, independently migrating carriers of number density $n$, charge $q$ and diffusion coefficient $D$, balancing drift with diffusion gives mobility $D/(k_{B}T)$ per applied force. Multiplying drift velocity by charge density yields $\sigma =n{q}^{2}D/(k_{B}T)$. State which species defines $n$ and $D$: a vacancy diffusivity must be combined with vacancy density. Correlated motion requires a correction rather than automatic use of a tracer diffusivity.

The density and diffusivity in $\sigma =n{q}^{2}D/(k_{B}T)$ must refer to the same carrier species and correlation convention. Oxygen vacancy diffusivity belongs with vacancy density; tracer diffusion of oxygen atoms is not automatically the vacancy diffusivity. A transport inversion may determine an effective density without uniquely determining a chemical defect concentration.