Number of states and density of states

• The number of states $S(E)$ is the total number of allowed electron states (including spin) with energy from 0 up to $E$ (usually per unit volume).
• The density of states $g(E)$ is the number of states per unit energy per unit volume at energy $E$: $g(E) = dS_v/dE$. So $g(E)\,dE$ is the number of states between $E$ and $E + dE$ per unit volume.

Electron in a three-dimensional box

Treat the free electrons of a metal as particles in a cubic box of side $L$ (infinite well in 3D). The energies are

$$E = \frac{h^2}{8mL^2}\left(n_1^2 + n_2^2 + n_3^2\right), \qquad n_1, n_2, n_3 = 1, 2, 3, \ldots$$

Each set $(n_1, n_2, n_3)$ is one orbital state, i.e. one point in "n-space" with unit volume per point. Define

$$n^2 = n_1^2 + n_2^2 + n_3^2 = \frac{8mL^2E}{h^2}$$

All states with energy $\le E$ lie inside a sphere of radius $n$. Since $n_1, n_2, n_3$ are positive, only one octant (1/8) of the sphere counts.

   n2
   ^
   | . . . .
   | . . . . .    one octant of a sphere
   | . . . . . .  of radius n; each dot
   | . . . . . .  = one orbital state
   +--------------> n1

Number of states

Each orbital state holds 2 electrons (spin), so

$$\begin{aligned} S(E) &= 2\times\frac{1}{8}\times\frac{4}{3}\pi n^3 = \frac{\pi}{3}n^3 \\ &= \frac{\pi}{3}\left(\frac{8mL^2E}{h^2}\right)^{3/2} \end{aligned}$$

Dividing by the volume $L^3$:

$$S_v(E) = \frac{\pi}{3}\left(\frac{8m}{h^2}\right)^{3/2}E^{3/2}$$

Density of states

$$\begin{aligned} g(E) &= \frac{dS_v}{dE} = \frac{\pi}{3}\left(\frac{8m}{h^2}\right)^{3/2}\cdot\frac{3}{2}E^{1/2} \\ &= \frac{\pi}{2}\left(\frac{8m}{h^2}\right)^{3/2}E^{1/2} = \frac{8\sqrt{2}\,\pi\,m^{3/2}}{h^3}E^{1/2} \end{aligned}$$

So $g(E) \propto \sqrt{E}$.

 g(E)
  ^                 __..--
  |           _.--''
  |       _.-'      g(E) ~ sqrt(E)
  |    .-'
  |  /
  | /
  +-------------------------> E

Use

Multiplying $g(E)$ by the Fermi–Dirac function $f(E)$ gives the number of electrons per unit energy, $n_E = g(E)f(E)$. Integrating up to $E_F$ at 0 K gives the Fermi energy $E_F = \frac{h^2}{8m}\left(\frac{3n}{\pi}\right)^{2/3}$.