# Density of levels in Fermi-Dirac statistics

We know that $$u=\int_{-\infty}^{+\infty} D(\mathcal{E})\mathcal{E}f(\mathcal{E})d\mathcal{E}$$ where $$D(\mathcal{E})$$ is called density of levels per unit volume. My textbook (Kittel) says that:

• $$D(\mathcal{E})$$ is the density of single-particle states as function of energy
• $$f(\mathcal{E})D(\mathcal{E})$$ is the density of filled orbitals

I don't understand the difference. And orbital and single.particle state are the same thing? Each orbital is defined by all the quantum numbers and only one electron can occupy it. So the density of energy state D(E) is the number of suitable orbitals in that range of energy?

• $D$ is the density of levels, whether or not they are occupied. $f$ is the probability that a level is occupied. Does that help? – garyp Jul 10 at 20:16

A part for a factor $$2$$, due to the spin degeneracy, yes they are the same thing.
The difference between $$D(\mathcal{E})$$ and $$f(\mathcal{E})D(\mathcal{E})$$ is entirely in the adjective filled.