Influence of electron interactions in a band metal.

Let's make it a goal to finish by Friday our problem involving the possible influence of electron repulsion in a band metal. Specifically, we are considering N electrons in a band that has a total of 2N states and which has a constant DOS. So \(D_o B = 2N\), where B is the bandwidth and N is the number of sites in the crystal.

We know how to calculate the energy of the system (at T=0), by integrating over the occupied states. (Can someone post that integral here?) We spoke about imagining separating the DOS into two equal halves, one for spin up and one for down. With that in mind one can then calculate the total energy of the system as a function of N_up and N_down. Can someone do that and post it here. Can you post a graph of the energy of the system as a function of \(N_{up}\)? Then we can consider and discuss what is the equilibrium value and what might influence the extent of fluctuations away from equilibrium. We assume \(N_{up} + N_{down} =N\). Does that make sense? What does that plot look like? What is the equilibrium value?

Now let's suppose every electron interacts with every other electron with an interaction expectation value \(e^2/L\)  (where L=Na). How many electron pairs are there if there are 100 atoms and 100 electrons in this band? How many are there if there are N electrons? What is the total electron-electron interaction energy?

Then let's make a slightly different assumption. Let's suppose that every spin up electron interacts with every other spin up electron with a smaller energy, \(e^2/2L\). Same for down- down pairs. The interactions between up and down pairs we leave at the larger value \(e^2/L\). (Does that make sense? Perhaps someone an explain that more lucidly?) How does this new assumption change and influence things? What is the total electron electron repulsion energy with this assumption. Graph it as a function of \(N_{up}\). Think about it and discuss.