How to determine the degree of how high a symmetry of high-symmetry points in the first Brillouin zone?
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For exmple, we have a hexagonal lattice with hexagonal Brillouin zone, shown in the picture
The points ΓΓ, K, M and ΛΛ are high symmetry points. Now, ΓΓ point is the highest-symmetry point, while others are high-symmetry points with less number of symmetries. How to determine, what symmetries correspond to each of the symmetry points (and symmetry lines connecting them) exactly on an intuitive level? The way I imagine it is the following. ΓΓ point is zero-frequency point, i.e. particle doesn't move in the crystall structure. So all 12 symmetry operations will not change local environment that a particle feels. At M point, the particle feels an oscillating at frequency M environment, and oscillations are only along a certain direction (from ΓΓ to M). So, now there is no 3-fold proper or 6-fold improper rotations... Is that the right intuitive way to look at it?
Is there a table of symmetries that make a correspondence of high-symmetry points (and lines) and number of symmetries that those symmetry points/lines capture?
What are the point groups corresponding to these symmetry points and connecting them symmetry lines for 2D hexagonal lattice?
Note: ΛΛ should be treated as a symmetry line rather than a symmetry point (my sketch is misleading in this regard).
The points ΓΓ, K, M and ΛΛ are high symmetry points. Now, ΓΓ point is the highest-symmetry point, while others are high-symmetry points with less number of symmetries. How to determine, what symmetries correspond to each of the symmetry points (and symmetry lines connecting them) exactly on an intuitive level? The way I imagine it is the following. ΓΓ point is zero-frequency point, i.e. particle doesn't move in the crystall structure. So all 12 symmetry operations will not change local environment that a particle feels. At M point, the particle feels an oscillating at frequency M environment, and oscillations are only along a certain direction (from ΓΓ to M). So, now there is no 3-fold proper or 6-fold improper rotations... Is that the right intuitive way to look at it?
Is there a table of symmetries that make a correspondence of high-symmetry points (and lines) and number of symmetries that those symmetry points/lines capture?
What are the point groups corresponding to these symmetry points and connecting them symmetry lines for 2D hexagonal lattice?
Note: ΛΛ should be treated as a symmetry line rather than a symmetry point (my sketch is misleading in this regard).
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