moire1.nb

(*Define a function to generate a 2D lattice.*)
createLattice[n_, latticeVectors_] := 
  Flatten[Table[
    i latticeVectors[[1]] + j latticeVectors[[2]], {i, -n, n}, {j, -n,
      n}], 1];

(*Define a function to rotate the lattice by a given angle.*)

rotateLattice[lattice_, angle_] := 
  RotationTransform[angle][#] & /@ lattice;

(*Define a function to plot the interference pattern of two lattices.*)

plotInterference[lattice1_, lattice2_] := 
  Show[ListPlot[lattice1, AspectRatio -> Automatic, 
    PlotStyle -> PointSize[Large]], 
   ListPlot[lattice2, AspectRatio -> Automatic, 
    PlotStyle -> {PointSize[Large], Red}]];

(*Define the lattice vectors for the two materials.*)

latticeVectors1 = {{3/2, Sqrt[3]/2}, {3/2, -Sqrt[3]/2}};
latticeVectors2 = {{2, 0}, {2, Sqrt[3]}};

(*Generate the lattices for a given range.*)

lattice1 = createLattice[10, latticeVectors1];
lattice2 = createLattice[10, latticeVectors2];

(*Define a Manipulate function to allow interactive adjustment of the \
angle and spacing.*)

Manipulate[lattice2Rotated = rotateLattice[lattice2, angle];
 plotInterference[lattice1, lattice2Rotated], {angle, 0, 
  2 Pi}, {spacing, 1, 10}]