Inversion symmetry points of graphene

Question

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?

Answer 1

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.