@@ -135,7 +135,7 @@ where we have used that $k'-k =2\pi/a$ because we are analyzing the first crossi
Everything we did can also be applied to the higher-energy crossings seen in the figure above. We note that all crossings occur between parabola's that are shifted by integer multiples of reciprocal lattice vectors $n 2\pi/a$. The first crossing corresponds to $n=1$, and we found that the magnitude of the gap is given by $V_1$. Similarly, $V_2$ determines the gap between the second and third bands, $V_3$ for the crossing between third and fourth, etc.
The key conclusion is that the Fourier components of the lattice potential couple plane-wave states that differ by integer multiples of reciprocal lattice vectors. The coupling alters the band structure most strongly where the free-electron eigenenergies cross.
The key conclusion is that the lattice potential couples plane-wave states that differ by integer multiples of reciprocal lattice vectors. This coupling alters the band structure most strongly where the free-electron eigenenergies cross, opening up gaps determined by the Fourier components of the lattice potential.
### Repeated vs reduced vs extended Brillouin zone
