Solid state - Energy of electron in Brillouin zone

[Confundo]

Homework Statement

Using geometrical arguments or otherwise, derive how the energy of an electron in the second Brillouin zone may be less than the energy of an electron in the first zone. [3]

Homework Equations

The Attempt at a Solution

I'm thinking this has something to do with overlapping bands, I think divalent metals show these characteristics. I'm not really sure where to start off from with a derivation though.

 

[weejee]

Suppose we have a simple cubic lattice, whose 1st Brillouin zone (1BZ) is a cube. Let's consider a vertex and the center of a face on this cube.

We know the the distance of the vertex from the origin is sqrt(3) times that of the center of the face. Therefore, in the (nearly) free electron model, the energy at the vertex is bigger.

Now consider a point near the vertex but still inside 1BZ and a point near the center of a face but a little bit outside 1BZ. Which one has the bigger energy?

This kind of thing happens in any shape other than sphere, which is impossible to be the shape of the Brillouin zone.