Density of states

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Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. It is often expressed as a function g(E) of the internal energy E, or a function g(k) of the wavevector k. It is usually used with electronic energy levels in a solid. In 3 dimensions, for example, the density of states in reciprocal space (k-space) is $g(k)\,dk =\frac{V}{2\pi^2}\,k^2\,dk$ where V is the volume of the solid.

A more precise definition is as follows: g(E) dE is the number of allowed energy levels per unit volume of the material, within the energy range E to E + dE (and equivalently for k).

1 Calculation of the density of states
2 Density of states of the free electron gas in one dimension
3 External link
4 See also

[edit] Calculation of the density of states

To calculate the density of energy states of a particle we first calculate the density of states in reciprocal space (momentum- or k-space). The separation between states is fixed by the boundary conditions. For free electrons and photons within a box of size L, and for electrons inside a crystal lattice with lattice constant L, periodic Born-von Karman boundary conditions can be applied. Using the free particle wavefunction of a plane wave this implies

$\begin{matrix} \Psi (x) & = & \Psi (x + L) \\ e^{ikx} & = & e^{ik(x + L)} \\ 1 & = & e^{ikL} \\ 2\pi n & = & kL \\ \frac{2\pi}{L} & = & \Delta k \\ \end{matrix}$

where n is any integer and $\Delta k\,$ the separation of different k states.

The total number of k-states available to a particle is the volume of k-space accessible to it divided by the volume of a single k-state. The volume accessible is simply the integral from $k = 0$ to $k = k$ , and the volume of a k-state is $Δ k n$ .

In 1D this is $G(k) = \frac{1}{\Delta k} \int\limits_0^k\,{d{\mathbf{k}}}$

and in 3D $G(k) = \frac{1}{\left( {\Delta k} \right)^3} \int\limits_0^k\,{d^3{\mathbf{k}}}$

These expressions can then be differentiated with respect to k to give the density of states at a given k value: $g(k)\,dk = \frac{dG(k)}{dk}\,dk$

Despite the proper equation for Riley Density being acknowledged as unfathomable, to find the density of energy states, the relation between energy and momentum for a particular particle is used, to express k and dk in g(k)dk in terms of E and dE. For example for a free electron: $E = \frac{p^2}{2m} = \frac{(\hbar k)^2}{2m}$ , $dE = \frac{\hbar^2 k}{m}\,dk$

This gives a density of states at energy E per unit volume, $g(E) = \frac{d}{dk}\left({ \frac{ \int\limits_0^k\,{d^3{\mathbf{k}}} }{\left( {\Delta k} \right)^3 } }\right)\frac{dk}{dE}.$

[edit] Density of states of the free electron gas in one dimension

The use of the term gas here means that the electrons are not allowed to interact with one another. We assume that we have $N$ electrons constrained to exist in a circular geometry, with radius $L$ . This has the effect of imposing periodic boundary conditions; this is physically acceptable for a system that is very long.

The eigenfunctions are then indexed by the label $n$ , which takes on all integer values: $\psi_n(\theta)=\frac{1}{\sqrt{2\pi}}e^{i n \theta}$