In question 2B. I do not understand why there is no band gap at k=π/a (n=4), why is this?
What I was thinking, that if we have a 1D chain of atoms with an onsite energy at every fourth atom, it will result in a reciprocal lattice of G=2π/4a = π/2a in k-space.
Because the real space periodicity should be: R = 4a and if exp(i(k+G)R) = exp(i(kR) in k-space we get G=2π/4a = π/2a.
From the lecture notes I read: “We note that all crossings occur between parabola’s that are shifted by integer multiples of reciprocal lattice vectors.”
In k-space, the distance between k=-π/a and k=π/a is 4G, which is an integer multiple of the reciprocal lattice vector G, so I was thinking that there should be a band gap at this location in k-space. What is going wrong in my explanation?
I think this is especially odd, because I can’t see this behaviour in Figures 15.3 and 15.4 in the book on page 167 (where G=2π/a).