Some paper have given the formula between the rotation angle and the index. How to get it really confuses me ?please give me some explanation. Thank you
You rotate one lattice point, that can be represented as r=m a1 + n a2, (m,n) to an equivalent position t1=n a1 + m a2 (n,m)(Where a1 and a2 are the lattice vectors of graphene (for instance (Sqrt(3)/2,-0.5)*a and (Sqrt(3)/2,-0.5)*a ) ) , it is called a a commensurate rotation.
It is then not very difficult to calculate the rotational angle between these two vectors r and t1 (using dot product definition), You obtain then an analytical result of the angle between the two vectors.