For a free electron gas the procedure for determining the density of states is as follows. Apply periodic boundary conditions to the free electron waves over a cube of side L. This gives us that there is one state per volume 2[itex]\pi[/itex]/L^{3}=2[itex]\pi[/itex]/V And from there we can find the number of states at a given energy E by multiplying by the volume of a sphere at E in k space. One big problem with this is however: Why do we assume that material is necessarily a cube? What if we worked with a ball of metal?
There is a theorem, I think by Sommerfeld, that the boundary becomes unimportant in determining e.g. the DOS in the limit V->infinity
I thought that the proof is in an article by a guy named W. Ledermann, published in 1944. http://rspa.royalsocietypublishing.org/content/182/991/362.full.pdf I actualy found it mentioned in a review paper from 1993: http://www.jstor.org/discover/10.2307/52288?uid=3739728&uid=2&uid=4&uid=3739256&sid=21102824926163 Do you know something about Sommerfeld writing something along the same lines?