# Understanding electrons in a weak periodic potential fourier analysis

I have been trying to understand Ashcroft's take on electrons in a weak periodic potential, and his approach by Fourier analysis is proving to be extremely challenging. I understand how to get to the master equation:

and then he assumes this (which by itself is odd, but I assume it is to isolate a band and study it isolated from the others):

But this is what I can't wrap my head around:

I can't understand the behaviour of $$O(U^2)$$. How does he get to that conclusion. I have read the text several times, but I can't make sense of it. Can someone help me? I have looked into parseval's theorem to see if it would help relate the coefficients $$c_{k-K}$$ but that theorem relates the square of the coefficients to the energy, so it didn't quite help.

Well, this is easy. Simple math, I suppose.

We have $$\mathbf{c_{k-K} = \frac{U_{K_1-K}\, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K}} + \sum_{K'\ne K_1} \frac{U_{K'-K}\, c_{k-K'}}{\epsilon-\epsilon^{\circ}_{k-K}}}$$

Now in place of $$\mathbf{c_{k-K'}}$$ in the numerator of the 2nd expression in RHS, you plug in the above expression for $$\mathbf{c_{k-K}}$$.

Then you get the following expression:

$$\mathbf{c_{k-K} = \frac{U_{K_1-K}\, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K}} + \sum_{K'\ne K_1} \frac{U_{K'-K}}{\epsilon-\epsilon^{\circ}_{k-K}}} \left(\mathbf{\frac{U_{K_1-K'} \, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K'}} + \sum_{K''\ne K_1} \frac{U_{K''-K'}\, c_{k-K''}}{\epsilon-\epsilon^{\circ}_{k-K'}}}\right)$$

which on further simplification gives:

$$\mathbf{c_{k-K} = \frac{U_{K_1-K}\, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K}} + \sum_{K'\ne K_1} \frac{U_{K'-K}\, U_{K_1-K'} \, c_{k-K_1}}{(\epsilon-\epsilon^{\circ}_{k-K})(\epsilon-\epsilon^{\circ}_{k-K'})} + \sum_{K'\ne K_1} \sum_{K''\ne K_1} \frac{U_{K'-K}\, U_{K''-K'}\, c_{k-K''}}{(\epsilon-\epsilon^{\circ}_{k-K})(\epsilon-\epsilon^{\circ}_{k-K'})}}$$

Now, if you look at the second and third terms in the RHS, it involves quadratic terms of $$U_k$$. In other words, in big O notation, $$\mathcal{O}(U^2)$$.

In this way, the author arrives at:

$$\mathbf{c_{k-K} = \frac{U_{K_1-K}\, c_{k-K_1}}{\epsilon-\epsilon^{\circ}_{k-K}} + \mathcal{O}(U^2) }$$

Since he is talking about weak potentials here, the author later neglects quadratic terms in U to quantify the effect of weak periodic potentials to create Wannier functions, leading to the formation of the s-band, p-band so on, for different overlap integrals.

Keep on reading Ashcroft and Mermin, it is a very good book for understanding the basics of solid-state physics and condensed matter physics.

Hope this helps you.

It is because of this passage before Eq. (9.10):

Since we are examining that solution for which $$c_{k-K}$$ vanishes when $$K\ne K_1$$ in the limit of vanishing $$U$$

In other words, in the sum in Eq. (9.10) the coefficient $$c_{k-K'}$$ are also of order $$\mathcal{O}(U)$$, which multiplied by $$U$$ (and assuming no degeneracy, so that the denominator cannot be small) gives $$\mathcal{O}(U^2)$$.