# I 1D Phonon density of state derivation

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1. Jun 8, 2017

### mshr

I'm having some trouble finding consistent results for the derivation of the 1D phonon density of state. I'm applying periodic boundary conditions to a 1D monatomic chain.

I can work through and find that D(K)=L/(2π). This is the same result as given by Myers (1990, p. 127). Myers uses only periodic boundary conditions.

Both Kittel (8th ed, pp. 109-110) and Blakemore (1974, pp. 99-100) calculate D(K) using both fixed and periodic boundaries.

Kittel says that the number of modes per unit K-range is L/π in the fixed case, and L/(2π) in the periodic case, and then goes on to use only L/π.
Blakemore says that the number of modes per unit K-range is L/π in the fixed case, and then asserts that it is L/π in the periodic case, even though he shows that K=+/- 2nπ/L.

Di Bartolo & Powell (p. 239) use periodic boundary conditions and say that D(K) = 2 * L/(2π), as there are "two values of corresponding to each angular frequency".

What's going on?

Using L/(2π) makes sense for higher dimensions, as you can then say N = (L/(2π))3*(4πK3/3) for 3D, and find D(K) as dN/dK, and from there D(ω) as dN/dK * dK/dω. (as per Kittel (pp. 111-112))

How can I maintain consistency between the approaches used in the 1D and 3D (and 2D) cases?

Di Bartolo & Powell (1976) Phonons and Resonances in Solids

2. Jun 13, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.