I'm writing a program and need a way to identify and enumerate the number of possible distinct crystallographic domains when depositing one 2D lattice on top of another.

I've posed this as a question in Math SE: Number of possible rotational domains of one 2D lattice on top of another? but possibly because I lack the proper "math words" it might not be receiving views by the users who can answer it.

I'm sure that this is a solved problem for mathematicians, and for crystallographers!


  1. Would this question be on-topic here if suitably rewritten for this site?
  2. If so, might it be likely to receive an answer?

Snippet from the linked question:

Square and hexagonal lattices shown at 0, 30 and 60 degrees:

square and hexagonal lattices shown at 0, 30 and 60 degrees

Square and hexagonal lattices shown at +/-10 degrees:

square and hexagonal lattices shown at +/-10 degrees

  • 2
    $\begingroup$ Given the number of degrees of freedom of just rotating/translating various patterns of dots over various other patterns of dots, I'm not sure the question is even answerable. And that is before actual physics or chemistry is applied to the possible bonding states, reconstructions, etc. Or am I missing something? $\endgroup$
    – Jon Custer
    Mar 3, 2020 at 13:52
  • 1
    $\begingroup$ @JonCuster Check the question in Math SE itself, you'll find "Ignoring translation..." right up there near the top. I haven't specified that the lattice pairs are or are not commensurate in any way. The numbers are mostly going to be single digits, probably between one and six; I don't think anything higher ever comes up. $\endgroup$
    – uhoh
    Mar 3, 2020 at 14:10
  • 1
    $\begingroup$ Well, surface science is quite rich in the variety of patterns formed on homo-surfaces, much less hetero-surfaces. so the math folks may have their viewpoint but Mother Nature begs to differ. $\endgroup$
    – Jon Custer
    Mar 3, 2020 at 14:14
  • $\begingroup$ No, for the question in Math SE, as currently written, I don't think that that is the case. If you are sure, then please show an example proving me incorrect. $\endgroup$
    – uhoh
    Mar 3, 2020 at 14:15
  • $\begingroup$ @JonCuster What I'm after is in Chapter 6 of Crystallography and Surface Structure by Klaus Hermann, 2nd edition. It's too much for me to digest all at once and I want to start with only the simplest symmetry groups rather than do everything at once. $\endgroup$
    – uhoh
    Mar 4, 2020 at 10:24
  • 1
    $\begingroup$ I've been musing about this for a bit. OK, you have two 2-D lattices each with a specific rotational symmetry and a rotational symmetry axis. You place one atop the other and align the symmetry axes. Now you want to know what rotation(s) produce a pattern exhibiting some other symmetry of interest for some values of the two lattice types and parameters? No buckling, no surface reconstruction, nothing but rotation around the aligned symmetry axes? (I'll note that I am suspicious that your examples above do not go out enough lattice parameters to see the pattern decay away.) $\endgroup$
    – Jon Custer
    Mar 4, 2020 at 14:10
  • $\begingroup$ @JonCuster for now, I just want to consider only the most general cases; hexagonal, square, rectangular and oblique latices as defined by their two lattice vectors only. I can't take in the whole problem at once. So for example hexagonal graphene on a simple fcc(111) crystal surface (also hexagonal) will either form one or two domains; one if it's aligned at 0, 30 60... degrees, and two if its minimum energy is at some angle other than those (i.e. $\mod(|\theta|, 30) \ne 0$). Never mind if they are in some way commensurate in some supercell or not, there'll be some angles with minimum energy. $\endgroup$
    – uhoh
    Mar 4, 2020 at 14:18


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