Solid state physics energy, translation

[usn7564]

Hey,

I'm having some trouble understanding why the energy changes here between the blue and red paths (see attached image), it's representing state of an electron in a square lattice crystal. Like mathematically I can see it because obviously plug and chugging the different values k in E(k) give different energies but the physics I'm completely lost on. My thinking was that the energy curves would be equal as everything should be.. periodic, I mean we're just in another '1st' Brillouin zone with the other point as a reference.

Obviously it's not the case though unless maths started lying. What am I actually looking at with E(k - G), like what does the translation actually mean physically? We have all the possible states represented in the first Brillouin zone with the allowed k's there, so what am I looking at with k vectors outside the first Brillouin zone?

Thanks in advance, having a hard time wrapping my head around reciprocal space. Not using the template as the question was "find the energies" which I have, I just can't interpret them.

 

[nilesh_pat]

The image you’ve provided is a representation of the reciprocal space of a square lattice crystal. In order to understand why the energy changes between the blue and red paths, it is important to understand what the translations from one point to another represent in terms of energy. The E(k - G) term represents a shift of the wave vector k by a reciprocal lattice vector G. This shift in wave vector changes the momentum of the electron, which in turn affects its energy. In a reciprocal lattice, the size of the reciprocal lattice vector G and the magnitude of the wave vector k determine the energy of the electron. As such, the red path would have a greater energy than the blue because the red path has a larger wave vector than the blue path.In short, the energy of an electron in a reciprocal lattice is determined by the size of the reciprocal lattice vector G and the magnitude of the wave vector k. When these values are shifted, the energy of the electron changes accordingly.