lattice diffusion
Lattice diffusion is atoms moving through the crystal interior — hopping from site to site across the orderly bulk of a grain, far from any boundary or surface. It is the 'main road' of diffusion: the route every textbook draws first, the one the Arrhenius diffusion coefficient usually describes, and the standard against which the faster shortcut paths are compared. When people say 'the diffusion coefficient of oxygen in alumina', they almost always mean the lattice value.
In the crystal interior an atom is fully surrounded by its neighbours in the regular pattern, so moving requires either an adjacent vacancy (the vacancy mechanism) or an open interstice (the interstitial mechanism) — the atomic mechanisms in their purest form. Because the bulk lattice is dense and well-ordered, the activation energy for a jump is high, so lattice diffusion is the slowest of the three paths but also the one with the most room: it happens throughout the entire volume of every grain, not just on thin interfaces. Its rate follows the Arrhenius law with a large Q, which is why lattice diffusion is negligible at low temperature and only becomes significant when the ceramic is fired truly hot.
Whether lattice diffusion dominates depends on the competition with the short-circuit paths. At high temperature, where the exponential has swelled the bulk D, lattice diffusion carries most of the flux simply because it has the whole grain volume to work with. At lower temperature its high Q shuts it down faster than the low-Q boundary and surface paths, so those shortcuts take over. This crossover — lattice-controlled when hot, boundary- or surface-controlled when cool — is exactly the kink often seen in an Arrhenius plot, and it is central to predicting how a ceramic densifies and reacts across its firing range.
In a coarse-grained, high-purity alumina fired near 1600 degrees C, most cation transport goes by lattice diffusion through the grain interiors; drop the temperature or refine the grains and grain-boundary diffusion, with its lower activation energy, takes the lead instead.
Lattice (bulk) diffusion is atoms hopping through the crystal interior — the slowest path but the one with the whole grain volume, and it dominates at high temperature.
Lattice diffusion has a higher activation energy than the grain-boundary and surface paths, so it is only the dominant path when the ceramic is hot and the grains are coarse. In fine-grained ceramics at moderate temperature the boundaries usually win.