How to interpret huge number of phonons?

In my condensed matter class we have seen how to treat the problem of vibration in a lattice in quantum mechanics. After heavy calculation we derive that the number of phonons with a given crystal momentum q and in a band $$j$$ is

$$n_j(\textbf{q})=\frac{1}{\exp{\left( \frac{\hbar w_{j\textbf{q}}}{k_BT}\right)}-1}$$

Where $$w_{nq}$$ is the frequency associated with that phonon. I don't see directly the meaning of it. My interpretation is something like this: we have in our material a huge number of vibrational modes with a specific frequency, therefore the entire lattice will vibrate. We can localize an oscillation on a lattice taking a linear combination of those modes. Suppose that $$u_{in}$$ is the projection along the i-axis of the displacemente with respect to the Bravais Lattice of the $$n$$-atom, then we will have:

$$u_{in} ( \textbf{x},t) = \sum_{j}\int{d^3\textbf{q} \quad c_j(\textbf{q}) \exp\left( i( \textbf{x} \cdot \textbf{q} - w_{j\textbf{q}}t )\right)}.$$

(Question: Is it right to sum also over the band index?)

My idea is that if $$n_j(\textbf{q})$$ is a high number ($$k_b T \propto \hbar w_{j\textbf{q}})$$ the the amplitude $$c_j(\textbf{q})$$ will be dominant. Is it right? Also if the phonon dispersion relation is given is there a way to compute those $$c_j(\textbf{q})$$ if our material is at temperature $$T$$?

• "we have in our material a huge number of vibrational modes with a specific frequency, therefore the entire lattice will vibrate." Can you clarify what you mean by this? Why do you expect a huge number of modes with a given frequency? Do you think that the lattice wouldn't vibrate if there were not a huge number of modes? Commented Jul 31, 2023 at 2:05
• I meant something like this: there are a lot of vibrational modes, if we select one (∆k=0) all atoms in the materials will vibrate (∆x->∞). In order to localize a vibration we must sum other vibrational modes Commented Jul 31, 2023 at 8:40

It is indeed correct to sum over the band index $$j$$ in your formula for the atomic displacements. The $$c_j(q)$$ are essentially the phonon eigenvectors you find by diagonalizing the dynamical matrix - they tell you, in the mode characterized by $$q,j$$, which atom is moving in which direction. I don't think it would be possible to reconstruct all the $$c_j(q)$$ from knowledge of the dispersion relation alone.
The formula for $$u_{in}$$ you've written is derived under the harmonic approximation, which assumes the temperature is low enough such that the atomic displacements from their equilibrium positions are small compared to the interatomic spacings. The $$c_j(q)$$ are not functions of temperature in the harmonic approximation. Various approximation schemes exist (eg. the quasiharmonic approximation) which take into account temperature-dependent effects. Hope this helps!