Understanding Density of States in 3d Crystals: A Comprehensive Explanation

[aaaa202]

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??

 

[Hyo X]

if you read the sentence without pausing, the meaning is slightly different.
"There is one allowed value of K per volume , for each polarization and for each branch."
so there are multiple K values allowed per volume

 

[aaaa202]

Well in this context, what does the word branch refer to? Different combinations of Kx,Ky,Kz?

 

[nasu]

No, same combination of kx,ky,kz may correspond to different states with different energies.
Look at the acoustic and optic branch in a 1D chain with two types of atoms. This is the simplest example of "branches".
Here for each k there are two energies (or frequencies).