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.
Answer
The reason to introduce the monoatomic and diatomic chains is that they are the simplest models which capture the essential physics of acoustic and optical phonons. That is, crystals have multiple phonon mode solutions that generally include acoustic modes, which have states for which energy goes to zero in the long wavelength limit. The acoustic modes are primarily responsible for the heat capacity of materials. These models are a pedagogical tool to introduce a microscopic model of phonons so that you have a reason to believe that they are real, and to give you a flavor of how you might go about solving for the real modes in a real material (i.e. you need the crystal structure, the masses of the atoms, and the coupling between them).
We can conclude that the best way to describe mechanical vibrations in a solid (including metals) is as a gas of phonons whose density is determined by the dispersion relation and the occupation probability (i.e. the Bose-Einstein distribution).
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