Van Hove singularities

Van Hove Singularities are divergences in the Density of states (that tend to infinity). i.e. $g(E)=A/((E-E_{0})^{n})$ as the energy tends to $E_{0}$.

Typically, they require the group velocity of the particle goes to zero.

It can be viewed as coming from the DOS

$g(E)dE=dN=$ $g(E){\frac {dE}{dk}}dk=dN$

So $g(E)={\frac {dN/dK}{dE/dK}}$

Note, this should be done in 3D form with $\nabla$ notation, but this is the simple version for easy learning.

A full version is presented on Wikipedia.