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I made a binary special quasi-random structure (SQS) using mcsqs (distributed as part of the alloy theoretic automated toolkit (ATAT)). The structure was generated using their own example FCC input file - rndstr.in:

1 1 1 90 90 90
0   0.5 0.5
0.5 0   0.5
0.5 0.5 0
0 0 0 A,B

The space group for FCC is 225, but the SQS generated are random. One had a space group of 8, another 12, yet another 160.

As far as I understand, SQS maintain symmetry. They are random, so perhaps an $L1_2$ ordered structure with space group 221 would not be an SQS, but 225 FCC is what any FCC SQS should be. Why then is mcsqs giving such haphazard results?

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    $\begingroup$ I am no expert in SQS, but I think your problem may be the following: when one says that an alloy has FCC structure, what that means is that the ensemble of all possible random configurations of the atoms has FCC symmetry. But any individual configuration will not have FCC symmetry but a much lower symmetry because it is a particular realisation of a disordered configuration. In this view, I think your results make perfect sense... $\endgroup$ – ProfM Sep 3 '20 at 14:56
  • $\begingroup$ @ProfM sorry, I don't follow. I understand the part about the ensemble of all possibilities showing an FCC structure, but why would any individual configuration be of a lesser symmetry? $\endgroup$ – Hitanshu Sachania Sep 3 '20 at 16:59
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These are some general comments about modelling disorder which I think address your question, but note I cannot provide specifics about the codes you are using.

Imagine a simple configuration with only two sites for the discussion:

enter image description here

Both atoms are of the same type (blue) so that this configuration has a mirror plane down the middle of the two atoms (dashed black line). This is a symmetry of the pure crystal (think your FCC symmetry). Now imagine that we can alloy this system with another type of atom (red) in a 50-50 composition, such that the average alloy composition looks like this:

enter image description here

In this case, the mirror plane down the middle is still a symmetry of the alloy. This is because we are building this abstract concept of a partial occupancy of a site (half blue half red), and this is precisely what we mean when we say that an alloy has a certain symmetry.

However, the real material is not as shown in the schematic above because you cannot have partial occupancies of a given site, you either have one atom or the other. These are two examples of real configurations of the system:

enter image description here

You will notice that none of these obey the mirror symmetry down the middle. This is equivalent to your particular realizations of disordered supercells: none of those will obey the nominal FCC symmetry of the crystal.

So again, what do we mean by an alloy having a particular symmetry? If you consider all possible realization of disorder (none of which will in general obey a particular symmetry), this forms your ensemble of disorder realizations. If you now calculate the average over all these realizations, then you will get the "partial occupancy" picture (half blue half red sites above), which does obey the symmetry of interest.

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    $\begingroup$ Thank you, I understand now. It's similar to how we write a resonance structure for Benzene with a pi bond spread across all C-C bonds, even though at any instant, only alternating C-C bonds have a pi bond. But this also makes me think about the relevance of partial density of states in the case of alloys. I guess I'll post this as a follow-up question. $\endgroup$ – Hitanshu Sachania Sep 9 '20 at 17:48
  • $\begingroup$ Follow-up question: mattermodeling.stackexchange.com/questions/2231/… $\endgroup$ – Hitanshu Sachania Sep 9 '20 at 18:49

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