Isotopic exchange and SIMS diffusion profile measurement

[Dario56]

Suppose there is a cylindrical (pellet) sample in the oxygen atmosphere as shown on the photo attached. Oxygen diffuses from the outside to the sample interior everywhere on the outer surface of the sample. From the photo, it can be seen that diffusion profile of oxygen is measured in the axial direction of the cylinder. In another words, diffusion is regarded as one-dimensional even though the sample is three-dimensional. I'm not sure how is it possible that one-dimensional solution of the diffusion equation can be applied here?

Oxygen can move in all three dimensions and will do so if partial derivative of concentration is non-zero in the other two Cartesian coordinates (cylindrical coordinates are more appropriate here, but that's not super important for the question).

 

[jrmichler]

I'm not sure how is it possible that one-dimensional solution of the diffusion equation can be applied here?

IF the sample thickness is small compared to the diameter, the one-dimensional solution applies to all but the outer perimeter. The region around the outer perimeter to a depth of about one sample thickness would require a two-dimensional solution, but a one-dimensional solution will be accurate farther toward the center.

A three-dimensional solution would be necessary if the sample diameter is of similar dimension to the sample thickness.

 

[Astronuc]

Suppose there is a cylindrical (pellet) sample in the oxygen atmosphere as shown on the photo attached. Oxygen diffuses from the outside to the sample interior everywhere on the outer surface of the sample. From the photo, it can be seen that diffusion profile of oxygen is measured in the axial direction of the cylinder. In another words, diffusion is regarded as one-dimensional even though the sample is three-dimensional. I'm not sure how is it possible that one-dimensional solution of the diffusion equation can be applied here?

Oxygen can move in all three dimensions and will do so if partial derivative of concentration is non-zero in the other two Cartesian coordinates (cylindrical coordinates are more appropriate here, but that's not super important for the question).

So this is about isotopic exchange (¹⁸O for ¹⁶O) in a metal oxide pellet? As in ZrO₂, Al₂O₃ or MgO?

It will also depend in whether the oxide is an n-type vs p-type oxide, and how the specimen is mounted, e.g., precluding diffusion from the circumference.

But, as jrmichler indicated, the problem can be treated in 1-D if the t << D.