Density of states in 3d

by aaaa202
Tags: density, states
 P: 1,005 My book gives a treatment of this problem for crystal vibrations, but I don't really understand it. It says: There is one allowed value of K per volume (2$\pi$/L)3. But at the same time it has just shown that Kx,Ky,Kz can take values ±2$\pi$/L which would certainly lead to more combinations of Kx,Ky,Kz within the volume confined by (2$\pi$/L)3. What am I misunderstanding. Also: applying periodic boundary conditions yields the condition that Kx,Ky,Kz=±n2$\pi$/L, while fixed ends yielded K=n$\pi$/L, but my book says the two approaches yield identical results. How is that?? Attached Thumbnails