# Band structure and Density of states (DOS)

Can someone explain how these two plots are related?

How are the peaks in the right are associated with the left figure?

• Peaks in the DoS occur when dispersion is near zero in a large area of $k$-space. But do you understand what Gamma, X, S, and Y mean? – Pieter May 8 '18 at 11:26
• Peaks of DoS always stands for van-Hove singularities – FangXie Apr 11 '19 at 19:06

If you have dispersions of the form $$\epsilon_\mu(k)$$ where $$\mu$$ is the band index and $$k$$ is momentum, the DOS is given by:
$$\rho(\omega) = \sum_\mu \int \frac{dk}{(2\pi)^d} \delta ( \omega- \epsilon_\mu(k))$$
Where $$d$$ is the spatial dimension. It should be pretty self-explanatory. For every value of $$\omega$$ you "sum" over all possible states that contribute to that value. In a translational invariant systems you count states by counting momenta (with a certain factor of $$2\pi$$).