How does ensemble distance work?

[oliverkjohnson]

I ran a test to see how the VFZ changes when the reference point (oref) changes to understand how the ensemble distance method works. We had two hypotheses about it: (i) the VFZ borders change as oref changes, or (ii) the borders of the VFZ never change, but the place that points get mapped into it changes (so they rotate through being in the interior where distance calculations are accurate and being near the exterior where distance overestimation occurs).

Here are VFZs for [100] tilt GBs (produced by the same algorithm as the general VFZ but simplified for this subspace) using 3 different reference points (the white square). The marker color is the original location index with $[\omega,\beta]=[0,0]$ corresponding to 1 (dark blue) and $[\omega,\beta]=[\pi/4,\pi/4]$ corresponding to 2116 (dark red) which was the total number of points. It is apparent from this illustration, that the VFZ borders are changing as oref changes (i.e. hypothesis 1 is correct). This suggests that which GBs inhabit the perimeter of the VFZ depends on the choice of oref.

[oliverkjohnson]

Note that the mirror borders are fixed, it's only the interior border that intersects the 2-fold symmetry axis that changes. This is true in the example on the sphere as well if you use a point-group symmetry that has rotational symmetry axes through which no mirrors pass (the m-3m point group has mirrors passing through all rotational symmetry axes, which is why its FZ is bordered entirely by mirrors, and in this case changing the reference point has no impact--the FZ is unique). Here is an example with 23 point group symmetry, there is a 3-fold rotation axis along the <111> directions through which no mirrors pass, and the borders of the VFZ that intersect this rotation axis are not unique (whereas the mirrors are), so if you change the reference point you get a different VFZ:

[oliverkjohnson]

So this is fundamentally why the ensemble methods work, some of the borders of the full GBFZ are not unique even for Oh point group symmetry, because there must be symmetry axes through which no mirrors pass. The mirror borders of the full GBFZ are fixed, but the other borders change when you change oref.

[sgbaird]

@oliverkjohnson wow! This is great. A longstanding conundrum resolved.