The density of states g(E) for a typical three-dimensional solid showing van Hove singularities indicated by black arrows. The function g(E) tends to have square-root singularities since ${\displaystyle g(E)\approx \int {\frac {dE}{{\vec {\nabla }}E}}}$ and since for a spherical Fermi surface ${\displaystyle E=\hbar ^{2}k^{2}/2m}$ so that ${\displaystyle {\vec {\nabla }}E=\hbar ^{2}k/m=\hbar {\sqrt {\frac {2E}{m}}}}$.