@@ -113,14 +113,13 @@ The key conclusion is that the lattice potential couples plane-wave states that
Hint: The Fourier series of $V(x)$ is $V(x)=A(e^{2\pi/ax}+e^{-2\pi/ax})/2$, so the only non-zero Fourier components are $V_1=V_{-1} = A/2$.
The different models can be organized as a function of the strength of the lattice potential $V(x)$:
## General description of a band structure in a crystal - Bloch theorem
The different models considered thus far can be organized as a function of the strength of the lattice potential $V(x)$:
## Bloch theorem
How is it possible to describe an electron that can scatter off all the atoms in a solid by something that even remotely looks like a plane wave? The answer lies in that the periodic potential created by the atoms can only scatter an electron between momentum states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$ *if these momenta differ by a reciprocal lattice vector*. This condition is very similar to the Laue condition of X-ray scattering. In this lecture we will explicitly analyze it in the context of the nearly-free electron model. The condition is known as the **conservation of crystal momentum** and is central to Bloch's theorem, which provides a general framework for computing band structures in crystals.
We have seen that in the nearly-free electron model, the electrons behave as plane waves that are only slightly perurbed by the lattice potential. How is it possible that an electron that can scatter off all the atoms in a solid can even remotely look like a plane wave? The answer lies in that the periodic potential of the atoms can only scatter an electron between momentum states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$ *if these momenta differ by a reciprocal lattice vector*. This condition is very similar to the Laue condition of X-ray scattering. In this lecture we have explicitly analyzed it in the context of the nearly-free electron model. The condition is known as the **conservation of crystal momentum** and is central to Bloch's theorem, which provides a general framework for computing band structures in crystals.
