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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'').


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 \
rac{2\pi}{L} & = & \partial k \
\end{matrix}

where ''n'' is any integer and ''dk'' 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 \partial k^n.

:In 1D this is G(k) = rac{ \int\limits_0^k\,{d{\mathbf{k}}} }{\partial k}

:and in 3D G(k) = rac{ \int\limits_0^k\,{d^3{\mathbf{k}}} }{\left( {\partial k} ight)^3}

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

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 = rac{p^2}{2m} = rac{(\hbar k)^2}{2m}, dE = rac{\hbar^2 k}{m}\,dk

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


DENSITY OF STATES OF THE FREE ELECTRON GAS IN ONE DIMENSION

The use of the term ''gas'' here means that it is not allowed for the electrons 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( heta)= rac{1}{\sqrt{2\pi}}e^{i n heta}

This eigenfunction has energy eigenvalue
E_n = rac{\hbar^2 n^2}{2mr^2}

It should be remembered that is a gas of electrons, not just a single one, and one starts to fill the states.


EXTERNAL LINK



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