I have a question about insulator. The story starts from the fact that, taking a 1-D model, there are N possible values of wave vectors k within the first Brillouin zone [becasue number of possible k's = (2*Pi/a)/(2*Pi/L) = L/a = N where L is the length of the crystal, a is the distance between two adjacent cells]. Also, we know that each k state could have up or down spin. Hence there are in total 2N levels in each band, which is always an even number. This far is fine. However, how can we proceed from this point to argue that "each cell contributing an even number of electrons" is a necessary condition for a material to be an insulator? To be specific, the confusion is this: It seems to me that EVERY cell in the crystal contribute when talking about band-filling. Hence isn't it true that if we have an even number of cells, we can fill all 2N states despite each cell contributing an odd number???