Heat capacity from phonon dispersion curves

Question

I have phonon dispersion curves for a crystal with two atoms per unit cell.
The following figure is phonon dispersion curves for the same crystal as above. $q_{BZ}$ denotes the wavevector at the BZ boundary. The direction of $q$ is antisymmetric, so the LA, LO branches are doubly degenerate. LA/TA, LO/TO denote longitudinal or transverse and acoustic and optical branches.
How would the temperature dependence of the lattice contribution to the heat capacity of this crystal be, in the low temperature limit and high temperature limit? The final page of this document shows the longitudinal and tranverse, acoustic and optical branches.
If we know that $$C_v = 3frac{V}{(2pi)^3} k_B 4pi int_{0}^{k_D} frac{(hbar omega (k)/k_B T)^2e^{hbaromega (k)/k_B T}}{(e^{hbar omega (k)/k_B T}-1)^2}$$
What is the simplification for $omega(k)$ that I use to solve for $C_v$?

Artem Alexandrov · Accepted Answer

Finally, as I understand the question is: how can one calculate the dispersion relation for diatomic 1D chain and then derive the heat capacity?
The quite detailed derivation of the dispersion relation can be find here (p. 8-9).
Having obtained the analytical expressions for the dispersion relation, we can compute the internal energy of our crystal,
$$Uproptointfrac{domega,omega}{e^{betaomega}-1},$$
where, for simplicity, we can consider the limit of small momenta and expand $omega(q)$ in terms of $q$. Then, perform change of variables, it seems possible to evaluate appearing integral. Finally, we use
$$Cproptofrac{partial^2U}{partial T^2}$$
to obtain the desired answer.

user137289 · Answer

The simplest approximations might be all that is required:

the Einstein model for the optical branches
the Debye model for the acoustical branches

So estimate $omega_E$ and $omega_D$ from the dispersion curves and add their contributions to $c_v$.

Heat capacity from phonon dispersion curves

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