1D Phonon density of state derivation

[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

 

[BigJDubs]

, p. 239:"In one dimension, the wavevector k is related to the angular frequency ω by k=ω/v, where v is the velocity of sound. Because there are two values of k corresponding to each angular frequency, the number of modes per unit range of frequencies is 2L/(2π). Thus the density of states is D(ω)=2L/(2π)".This approach is consistent with higher dimensions, as the wavevectors in 1D (k) and 3D (K) are related to the angular frequency by k=ω/v and K=ω/V, respectively. In both cases, there are two values of the wavevector corresponding to each angular frequency, so the number of modes per unit range of frequencies is 2L/(2π) in both cases, and D(ω) can be calculated as dN/dK * dK/dω.