I have question about graphene.

When you have the graphene lattice two types of atoms can be distinguished, let's call them type A and B.You can draw a unit cell that has the shape of a parallelogram. It contains exactly one of each type of atoms. There is then an inversion symmetry center between atom of type A and atom of type B.

I thought that I understood what an inversion symmetry center was. I thought it was a point and if you move the particles in straight lines through those points you can get the original structure back.

But why does this work for graphene. If you exchange A and B, then you have a different unit cell, right?

  • $\begingroup$ But aren't A and B both carbon atoms? Yes, they form a 2-atom basis, but why does that rule out an inversion center? Looking at the hexagonal network, it should be visually obvious that the center of the hexagon is an inversion center, so an inversion center must exist for the Bravais lattice + basis representation. Now, for a hexagonal boron nitride sheet - then A and B are actually different. $\endgroup$ – Jon Custer Apr 27 '16 at 14:07

You can only distinguish the sublattices in this case because you've tagged them A,B. The process of inversion only exchanges identical carbon with carbon, leaving the crystal physically unchanged. If you gave me a crystal with one orientation and I then returned it to you without telling you whether or not it's been inverted, you'd have no way of knowing.

You can compare to hexagonal boron nitride (h-BN), which is identical to graphene except one sublattice is composed of boron and the other of nitrogen. If you create the same unit cell and invert, you exchange chemically distinct atoms with different onsite energies. As an example, you could measure the atomic lattice with STM and see two distinct sublattices which trade places upon inversion.

  • $\begingroup$ yes, good answer. Or shorter: the honeycomb lattice is inversion symmetric, boron nitride is not (though its atoms are on the lattice) $\endgroup$ – Ilja Apr 27 '16 at 14:28

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