# The nearly free electron model and states in a band?

1. Nov 3, 2009

### jeebs

hi,
I have this problem where I am supposed to show, using periodic boundary conditions, that there are 2N states in an energy band in the "nearly free electron" model of solids, where N is the number of atoms.

I have been looking through my course notes and my textbook, and my textbook goes through the "free electron" model first. This is where the electrons are assumed to be moving in zero potential, ie. the lattice ions are completely ignored. It shows that the energy of an electron is given by
$$\epsilon = \frac{\hbar^2k^2}{2m}$$

and shows a plot of $$\ \epsilon$$ vs. wavenumber k, which as you hopefully know, looks like a sort of steep bowl when +ve and -ve k values are plotted.

Then I go on to the next chapter, about the "nearly free electrons", where the potential is no longer a uniform zero throughout the solid, but has a periodic negative peak corresponding to the positions of the lattice ions (which are separated by a distance a). The book goes on to show a plot of $$\epsilon$$ vs. k, but this time it results in the graph of the "Brillouin zones", which show the regions of energy bands and the band gaps that occur at every value of k = $$n\pi$$/a. This plot is superimposed on top of the curve for the free electron model, and the two curves coincide until k gets close to the Brillouin zone boundaries, where the nearly free electron curve tapers off.
This is what I assume my question is talking about with "periodic boundary conditions".

However, neither my book nor my notes are clear about where this plot comes from. Can anyone show me?
All I can guess is that the equation for $$\epsilon$$ as a function of k must have some sine or cosine part in it. I'm hoping that once I know where this plot comes from, it will help me understand why there are 2N states per band.

Basically my problem is that my book says there are N allowed values of k in the range $$-\frac{\pi}{a} < k < \frac{\pi}{a}$$ and and hence 2N electrons per band (allowing for 2 different electron spins), but I have no idea why.

Can anyone enlighten me?

Thanks.