# Solid solution hardening, concentration dependence

When you add impurity atoms to a material, the yield strength often increases by a process known as solid solution hardening. This is because the impurity atoms create a barrier to dislocation motion. The literature describing this phenomenon dates back to the 1960s with some famous papers by R.L. Fleischer.

Anyway, this hardening is supposedly dependent on the spacing between the impurity atoms. Since the spacing between impurities should be proportional to the impurity concentration in some way (i.e., more impurities, closer spacing between them), increase in yield strength is often plotted against impurity concentration. Experimentally, this increase has been shown to vary with the square-root of impurity concentration.

*My question is this: Why is it a square-root dependence, and not a cube-root dependence? Atomic concentration is per unit volume, so if you increase impurity concentration, the spacing between these impurity atoms should scale with the cube root of the concentration, right? Every resource I have found states that impurity spacing varies with the square-root of impurity concentration but I have yet to find a good explanation for this. Am I missing something? Can someone explain?

mfb
Mentor
You could argue that atoms in a material rarely move along a line. You can have different parts moving against each other, where the whole area is affected. The number of impurity atoms in a specific area goes with ##n^{2/3}##. The true dependence could be between those effects.

CarlJose
You could argue that atoms in a material rarely move along a line. You can have different parts moving against each other, where the whole area is affected. The number of impurity atoms in a specific area goes with ##n^{2/3}##. The true dependence could be between those effects.

Why would the number of impurity atoms in a specific area go with ##n^{2/3}##? (and I'm assuming you are using '##n##' to represent atomic concentration?) Is there some sort of geometrical justification for this?

Also, I guess I'd be willing to accept that the ##n^{1/2}## dependence is just an empirical observation (and between two functional limits), but I've found a few sources that state it as if it has some sort of theoretical justification. I'm suspicious that it has something to do with the impurities necessarily being within a crystalline lattice, but I can't figure out why that would make a difference...

mfb
Mentor
Sorry, my previous post did not make sense. A surface cutting through a volume of N atoms will have N2/3 atoms (neglecting prefactors), but the number of impurity atoms within that surface will scale linearly with the total number of impurity atoms in the material.

The mean area without impurities (as part of a larger area we consider) should scale with the inverse square of the average linear distance, which gives n-2/3, but that is harder to translate to material properties.

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