Density of State for a two atom basis

[Mattszo]

Homework Statement

I have a problem on my assignment in which I am required to find the specific heat of a two atom basis (diatomic) using the Debye model. My problem is coming up with the density of states for a diatomic setup in 1D.

Homework Equations

Density of state: g(w)=$\frac{L}{\pi}$$\frac{1}{\frac{dw}{dq}}$ in 1D
g(w) = $\frac{L}{π\upsilon_{s}}$ where $\upsilon_{s}$ is speed of sound, L is a length of lattice constant in 1D
dispersion relationship w=$\upsilon_{s}$q in 1D

The Attempt at a Solution

My book says the Density of States equation shares properties with the monatomic setup.
I don't know if repeating the same equation is right because density of state depends on dimentionality (1D) and dispersion relation, which seem the same for monatomic and diatomic.

 

[ardie]

the density of states tells you how many states there are for each of your particles to occupy. the dispersion relation in for 2 particles, has a similar form, and i believe it is for you to solve the wave equations to get the 2 dimensional dispersion relation.
as a result your group velocity v_s is different than the one for the mono-atomic case. it will be a tensor in general, but you are only interested in its magnitude, and since vectors add, assuming the two particles have the same magnitude but independent wave vectors, the group velocity will be twice that of the mono-atomic case, and so the density of states will be halved.

 

[Mattszo]

I have to find it in one dimension. I tried something else and that was to take the integral g(w)dw with limits 0 to the Debye frequency and equate it to = (# of dimensions) times N where N is the number of particles in the lattice. In this case, the integral is equal to 1N.

Anyways I handed in my assignment already and I guess I will just see what the actual answer is. Thank you for the help!