# density of states in 3d

by aaaa202
Tags: density, states
 P: 957 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
 P: 37 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
 P: 957 Well in this context, what does the word branch refer to? Different combinations of Kx,Ky,Kz?
P: 1,757

## density of states in 3d

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).

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