# Claim that DeBroglie relation doesn't work in crystal

In this Wikipedia article on Position and Momentum Space, https://en.wikipedia.org/wiki/Position_and_momentum_space

there is a claim that "the de Broglie relation is not true in a crystal" in the sentence before the content box.

Is this claim valid? If so, why? What is the implications for quasi-particles (e.g. plasmons and polaritons) in materials?

• Since particles and quasi-particles are not "free" in a periodic crystal potential, the de Broglie relation does not apply. You need to use Bloch functions to describe the particles. – Jon Custer May 14 at 17:04

In a crystal, $$\vec p$$ does not necessarily have the same direction than $$\vec k$$. So, I suppose that it's indeed true that the de Broglie relation ($$\vec p = \hbar \vec k$$) does not always hold in a crystal.
If we take a perfect crystal, then the wavefunction of an electron can be written as the Bloch electron wavefunction $$\Psi = u(\vec r) e^{i\vec k \cdot \vec r}$$ where $$u(\vec r)$$ is a periodic function whose periodicity matches the lattice's. By applying the momentum operator $$\hat p =-i\hbar \nabla_\vec r$$ to that wavefunction, one finds that it's equal to $$\hbar \vec k \Psi + \text{something not proportional to } \Psi$$ (nor to $$\vec k$$ for that matter.). Here, $$\hbar \vec k$$ is called the crystal momentum and does not match the electron's momentum. See Ashcroft and Mermin pages 139 and 219 for a detailed discussion about that.
• Then what is the tensor in units of $\hbar$ that gives you a different momentum direction from the wavevector? – wcc May 14 at 22:54