Relevance of mono- and diatomic chain in Solid State Physics

I started studying Solid State Physics in Kittel's book but I experienced that it did not provide physical inside and an intuitive idea on the subject, and also the order of the different subjects was confusing sometimes. Hence, I started to read a lot of other books on Solid State Physics (Ashcroft & Mermin, Hook & Hall, etc.), a few of them started with the "heat capacity dilemma" (which can somehow be considered as the beginning of solid state physics).

They started with Boltzmann who treated the atoms in a solid as harmonic oscillators and in this way he was able to reproduce the las of Dulong-Petit. However, it turned out that his model could not account for the decrease in heat capacity at low temperatures. Hence, Einstein used Boltzmann's model but he also introduced quantum ideas to describe the behavior of the heat capacity at low temperatures. He considered the atoms in a solid as independent identical harmonic oscillators. In this way he could account for the heat capacity $C \to 0$ as $T \to 0$. Unfortunately this decay of the heat capacity $C$ at low $T$ was an exponential decay while experiments suggested a $T^3$ dependence of $C$ at low $T$. It was Debye who came with the solution: he considered the collective motion of the atoms, and this collective motion could be considered as a long wavelength sound wave through the solid, resulting in the idea of a solid composed of atoms each desribed by a harmonic oscillator but they should no longer be identical and thus they could have different frequencies. This led Debye to confirm the $T^3$ dependence of $C$. However, this model gave an infinite value of $C$ at high $T$ so he introduced a cutoff to solve this problem.

I hope this historical explanation is correct so far. But at this point I get confused by the different books. This is how I think they pursue: Debye's model could also be described in terms of a phonon gas (in the same way as Planck did) where the density of states is simply constant since we have a soundwave with frequency $\omega = vk$ (dispersion relation) where $v$ is the speed of sound and $k$ is the wavevector. As next step they start to introduce the monoatomic and diatomic chain. The introduction of these concepts is not very clear to me. My guess is the following:

To study the distribution of modes in function of the forces between the atoms. From this we obtain insight on how the structure and the forces between the atoms in the solid become important, we discover that the density of states is no longer constant.

Can someone confirm this or tell me what the physical insight of these concepts is and why they are interesting to study? Can we conclude that the most correct way to describe a solid (excluding the metals) is by a phonon gas whose density is determined by the dispersion relation which depends on the structure and the forces between the atoms.

• Mechanical waves are pedagogical as a preparation for the more abstract electronic wave functions. The gap at the zone boundary of diatomic chains is good to understand band gaps in the electronic band structure. – Pieter Mar 31 '18 at 20:40