I Density of States: 1-Dim Linear Phonons & Electrons Differences

[cozycoz]

I'm to get the density of states of 1-dim linear phonons, with N atoms. I think it's a lot similar to that of 1-dim electrons, except that two electrons are allowed to be in one state by Pauli exclusion principle. To elaborate,

$dN=\frac{dk}{\frac{2π}{a}}=\frac{a}{2π}dk$ for phonons,

$dN=2⋅\frac{dk}{\frac{2π}{a}}=\frac{a}{π}dk$ for electrons.

But in Kittel's solid state physics, the latter is described as a phonon case. What's wrong with my procedure?

 

[Mentz114]

I'm to get the density of states of 1-dim linear phonons, with N atoms. I think it's a lot similar to that of 1-dim electrons, except that two electrons are allowed to be in one state by Pauli exclusion principle. To elaborate,

$dN=\frac{dk}{\frac{2π}{a}}=\frac{a}{2π}dk$ for phonons,

$dN=2⋅\frac{dk}{\frac{2π}{a}}=\frac{a}{π}dk$ for electrons.

But in Kittel's solid state physics, the latter is described as a phonon case. What's wrong with my procedure?

Are you sure ? My understanding is that two electrons are not allowed to be in one state by the Pauli exclusion principle

 

[cozycoz]

Are you sure ? My understanding is that two electrons are not allowed to be in one state by the Pauli exclusion principle

?
I meant two electrons with two different spin types..up and down. That's why we should multiply 2 in electron case. What I calculated by $\frac{dk}{\frac{2π}{a}}$ is spatial part, not including spin part.

 

[DrClaude]

What's wrong with my procedure?

Hard to say if you don't post how you derived the density of states.

 

[cozycoz]

Hard to say if you don't post how you derived the density of states.

okay

I want to get how many phonon states(dN) are in [K, K+dK] in 1d K-space. For a state occupies small length $\frac{2π}{a}$ by periodic boundary condition,
$dN=\frac{dK}{\frac{2π}{a}}$.