The Density of States

Going through the algebra to calculate the translational partition function we turned a sum over the integers $n_x$, $n_y$ and $n_z$ which count the number of half wavelengths along the three sides, to an integral over $\bf k$. Since the energy depends only on $k=\vert\bf k\vert$, we could do the integral over the direction of $\bf k$ leaving only the integral over $k$; in this process we collected a number of factors and called them the density of state: $D(k)=V k^2/2\pi^2$, so that

$$Z_1=\int_0^\infty \!D(k) e^{-\varepsilon(k)\beta}{\rm d}k$$

We see that $D(k)$ is acting as a ``degeneracy factor'', which we first met in the context of the rotor. If there is more than one energy level with the same energy, and we replace the sum over individual states with a sum over allowed energies, we need to include a factor in front of the Boltzmann factor for degenerate levels so that they are counted often enough.

The picture above shows a graphical representation of the allowed states in $k$-space. Since

$${\bf k}=\left({\pi n_x\over L_x},{\pi n_y\over L_y},{\pi n_z\over L_z}\right),$$

with $n_x$ etc positive, the allowed values of k form a three-dimensional lattice. The density of states is the number of states within an infinitesimal range of $k$, and hence of energy. This is just the volume of an octant of a spherical shell, $(1/8)\times 4\pi k^2\times {\rm d}k$, divided by the volume of $k$-space per state, $\pi^3/V$, giving

$$D(k) {\rm d}k={V k^2\over 2\pi^2}{\rm d}k.$$