---
title: "Density of states"
sort_title: "Density of states"
date: 2021-05-08
categories:
- Physics
- Statistics
layout: "concept"
---

The **density of states** $$g(E)$$ of a physical system is defined such that
$$g(E) \dd{E}$$ is the number of states which could be occupied
with an energy in the interval $$[E, E + \dd{E}]$$.
In fact, $$E$$ 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 $$\Gamma(E)$$ is the number of states with energy
less than or equal to the argument $$E$$:

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

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

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

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


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

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

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

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

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

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

$$\begin{aligned}
    \lambda = \frac{2 L}{m}
    \quad \implies \quad
    k = \frac{2 \pi}{\lambda} = \frac{m \pi}{L}
\end{aligned}$$

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

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

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

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

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

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

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

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

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

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

All these expressions contain the characteristic length/area/volume $$L^n$$,
and therefore give the number of states in that region only.
Keep in mind that $$L$$ 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 $$L = 1$$ in our preferred unit of distance.

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



## References
1.  H. Gould, J. Tobochnik,
    *Statistical and thermal physics*, 2nd edition,
    Princeton.
2.  B. Van Zeghbroeck,
    [Principles of semiconductor devices](https://ecee.colorado.edu/~bart/book/book/chapter2/ch2_4.htm), 2011,
    University of Colorado.