# Diffusion coefficient of a crystal

I've been trying to work this out so I can give a hand waving argument for one of the effects I'm observing on the fly and I find myself going down a rabbit hole that seems way too complicated for what I need. Thanks for any help.

I have a TiO2 rutile (so the unit cell is a body centered tetragonal cell) material. The unit cell parameters are a=b=4.5 and c=3 angstroms. I know the diffusion coefficient for oxygen along the a direction (call it Da) and the coefficient along the c direction (call it Dc). My question is: Given that I know Da and Dc, what is the diffusion coefficient in the (111) direction?

While I'm at it, once I have the diffusion coefficient, how can I calculate the percentage loss of oxygen at a given distance from the surface. That is to say, say I find out that there is sufficient time for the oxygen a nanometer in to diffuse to the surface, that obviously doesn't mean that all of the oxygen a nanometer from the surface will migrate to the surface. So how do I calculate what percentage will migrate for a given time?

If it turns out that this can't be conceptualized simply, then I suppose I'll brush off my condensed matter textbooks, but it seems to me I worked this type of problem really early on in my studies. So I hope someone can help. Thanks!

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