6.19 DOS for a Periodic Box

The density of states is the number of single-particle states per unit energy range. It turns out that the formulae for the density of states given in section 6.3 may be used for the periodic box as well as for the closed box. A box can hold about the same number of particles per unit volume whether the boundary conditions are periodic or not.

It is not that hard to verify. For a periodic box, the wave numbers can be both positive and negative, not just positive like for a closed box. On the other hand, a comparison of (6.3) and (6.28) shows that the wave number spacing for a periodic box is twice as large as for a corresponding closed box. That cancels the effect of the additional negative wave numbers and the total number of wave number vectors in a given energy range remains the same. Therefore the density of states is the same.

For the periodic box it is often convenient to have the density of states on a linear momentum basis. It can be found by substituting $k$ $\vphantom0\raisebox{1.5pt}{$=$}$ $p$$\raisebox{.5pt}{$/$}$$\hbar$ into (6.5). That gives the number of single-particle states ${\rm d}{N}$ in a momentum range of size ${\rm d}{p}$ as:

{\rm d}N = {\cal V}{\cal D}_p {\,\r...
...p \qquad
{\cal D}_p = \frac{2s+1}{2\pi^2\hbar^3} p^2
$} %
\end{displaymath} (6.29)

Here ${\cal D}_p$ is the density of states per unit momentum range and unit volume. Also, $s$ is again the particle spin. Recall that $2s+1$ becomes $2s$ for photons.

The staging behavior due to confinement gets somewhat modified compared to section 6.12, since zero wave numbers are now included. The analysis is however essentially unchanged.

Key Points
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The density of states is essentially the same for a periodic box as for a closed one.