the diffusion coefficient

If diffusion is atoms wandering through a solid, the diffusion coefficient D is the single number that says how good they are at wandering. A large D means atoms move readily, so a firing reaches completion quickly; a tiny D means they are nearly frozen in place, so even days at temperature achieve little. It is the 'mobility rating' of a species in a particular material, and almost every practical diffusion question comes down to knowing its value.

D carries units of area per time — square metres per second (m^2/s) — which is exactly what you need to feed the sqrt(D times t) rule for how far atoms travel. Its size is set by microscopic events: how often an atom attempts a jump, how far each jump is, and what fraction of attempts succeed. Roughly, D is about (jump distance)^2 times (jump frequency), so anything that makes jumps longer or more frequent raises D. Typical numbers span an enormous range: a fast interstitial in a hot oxide might have D near 10^-10 m^2/s, while a sluggish substitutional cation near room temperature can sit below 10^-25 m^2/s — fifteen orders of magnitude apart. Because D depends so violently on temperature, quoting a diffusion coefficient without its temperature is meaningless.

In ceramics you rarely deal with just one D. There is a different coefficient for each species (oxygen and the cation diffuse at wildly different rates), a different one for each path (lattice, grain boundary, surface), and the measured 'chemical' or interdiffusion coefficient for a couple can differ from the 'tracer' coefficient of a single labelled atom. The practical payoff is direct: D, plugged into the diffusion equation, tells the ceramist how hot and how long a part must be fired to dope it, react it, or densify it. Choose too low a temperature and D is so small the process never finishes; that is why firing temperatures are high.