Categories: Physics, Statistics.

Density of states

The density of states g(E)g(E) of a physical system is defined such that g(E)dEg(E) \dd{E} is the number of states which could be occupied with an energy in the interval [E,E+dE][E, E + \dd{E}]. In fact, EE need not be an energy; it should just be something that effectively identifies the state.

In its simplest form, the density of states is as follows, where Γ(E)\Gamma(E) is the number of states with energy less than or equal to the argument EE:

g(E)=dΓdE\begin{aligned} g(E) = \dv{\Gamma}{E} \end{aligned}

If the states can be treated as waves, which is often the case, then we can calculate the density of states g(k)g(k) in kk-space, i.e. as a function of the wavenumber k=kk = |\vb{k}|. Once we have g(k)g(k), we use the dispersion relation E(k)E(k) to find g(E)g(E), by demanding that:

g(k)dk=g(E)dE    g(E)=g(k)dkdE\begin{aligned} g(k) \dd{k} = g(E) \dd{E} \quad \implies \quad g(E) = g(k) \dv{k}{E} \end{aligned}

Inverting the dispersion relation E(k)E(k) to get k(E)k(E) might be difficult, in which case the left-hand equation can be satisfied numerically.

Define Ωn(k)\Omega_n(k) as the number of states with a kk-value less than or equal to the argument, or in other words, the volume of a hypersphere with radius kk. Then the nn-dimensional density of states gn(k)g_n(k) has the following general form:

gn(k)=D2nkminndΩndk\begin{aligned} \boxed{ g_n(k) = \frac{D}{2^n k_{\mathrm{min}}^n} \: \dv{\Omega_n}{k} } \end{aligned}

Where DD is each state’s degeneracy (e.g. due to spin), and kmink_{\mathrm{min}} is the smallest allowed kk-value, according to the characteristic length LL of the system. We divide by 2n2^n to limit ourselves to the sector where all axes are positive, because we are only considering the magnitude of kk.

In one dimension n=1n = 1, the number of states within a distance kk from the origin is the distance from kk to k-k (we let it run negative, since its meaning does not matter here), given by:

Ω1(k)=2k\begin{aligned} \Omega_1(k) = 2 k \end{aligned}

To get kmink_{\mathrm{min}}, we choose to look at a rod of length LL, across which the function is a standing wave, meaning that the allowed values of kk must be as follows, where mNm \in \mathbb{N}:

λ=2Lm    k=2πλ=mπL\begin{aligned} \lambda = \frac{2 L}{m} \quad \implies \quad k = \frac{2 \pi}{\lambda} = \frac{m \pi}{L} \end{aligned}

Take the smallest option m=1m = 1, such that kmin=π/Lk_{\mathrm{min}} = \pi / L, the 1D density of states g1(k)g_1(k) is:

g1(k)=DL2π2=DLπ\begin{aligned} \boxed{ g_1(k) = \frac{D L}{2 \pi} \: 2 = \frac{D L}{\pi} } \end{aligned}

In 2D, the number of states within a range kk of the origin is the area of a circle with radius kk:

Ω2(k)=πk2\begin{aligned} \Omega_2(k) = \pi k^2 \end{aligned}

Analogously to the 1D case, we take the system to be a square of side LL, so kmin=π/Lk_{\mathrm{min}} = \pi / L again. The density of states then becomes:

g2(k)=DL24π22πk=DL2k2π\begin{aligned} \boxed{ g_2(k) = \frac{D L^2}{4 \pi^2} \:2 \pi k = \frac{D L^2 k}{2 \pi} } \end{aligned}

In 3D, the number of states is the volume of a sphere with radius kk:

Ω3(k)=4π3k3\begin{aligned} \Omega_3(k) = \frac{4 \pi}{3} k^3 \end{aligned}

For a cube with side LL, we once again find kmin=π/Lk_{\mathrm{min}} = \pi / L. We thus get:

g3(k)=DL38π34πk2=DL3k22π2\begin{aligned} \boxed{ g_3(k) = \frac{D L^3}{8 \pi^3} \:4 \pi k^2 = \frac{D L^3 k^2}{2 \pi^2} } \end{aligned}

All these expressions contain the characteristic length/area/volume LnL^n, and therefore give the number of states in that region only. Keep in mind that LL is free to choose; it need not be the physical size of the system. In fact, we typically want the density of states per unit length/area/volume, so we can just set L=1L = 1 in our preferred unit of distance.

If the system is infinitely large, or if it has periodic boundaries, then kk becomes a continuous variable and kmin0k_\mathrm{min} \to 0. But again, LL is arbitrary, so a finite value can be chosen.


  1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.
  2. B. Van Zeghbroeck, Principles of semiconductor devices, 2011, University of Colorado.