The purpose of this post is to describe a theoretical model of solids based on the quantization of sound waves in solids into quasiparticles known as phonons due to Debye which culminates in a successful explanation for the experimentally observed heat capacity behavior \(C_V\propto T^3\) as \(T\to 0\) in a wide range of materials.
Just as photons are the discrete quanta of the electromagnetic field, it turns out that the acoustic fields that propagate within solids are also associated to discrete quanta called phonons (I guess because “phono” is the Latin stem for sound, but it’s nice that it also sounds similar to “photon”). For both photons and phonons, \(E=\hbar\omega\), but for photons the dispersion relation is \(\omega=ck\) with \(c\) the speed of light whereas for phonons the dispersion relation is \(\omega=c_sk\), with \(c_s\) the speed of sound (this is a slight lie; the dispersion relation for phonons is only linear at low momenta \(k\to 0\)).
In light of the wavenumber quantization \(\textbf k=\frac{2\pi}{L}\textbf n\) for \(\textbf n\in\textbf Z^3\) arising from periodic boundary conditions in a cubic solid of side length \(L\) and volume \(V=L^3\) containing \(N\) atoms, a simple change of variables reveals that the volume element \(d^3\textbf n\) in lattice space can be expressed by virtue of the dispersion relation in angular frequency space as \(d^3\textbf n=g(\omega)d\omega\) where the density of states \(g(\omega)=\frac{3V}{2\pi^2c_s^3}\omega^2\) grows quadratically in the angular frequency \(\omega\) (e.g. there are four times as many degenerate accessible microstates for each phonon at twice the energy). Note the factor of \(3\) is due to the \(3\) linearly independent polarization states for a phonon (two transverse and one longitudinal, which is unintuitive to me since I had expected just a single longitudinal polarization state; after all, isn’t that what sound is?).
Besides the fact that \(c_s\neq c\) and the \(3\) instead of just \(2\) polarization states, there is one more crucial difference between phonons and photons, namely that there is no upper bound on the frequency \(\omega\) of a photon, but there is an upper bound \(\omega_D<\infty\) on the frequency of phonons in solids simply because of the lattice nature of the atomic spacing. This upper bound frequency \(\omega_D\) is called the Debye frequency and should be thought of as constraining the frequency domain \([0,\omega_D]\subseteq\textbf R\) over which the density of states function \(g(\omega)\) is well-defined. Consequently, the total number of accessible single-phonon microstates is not infinite, but rather:
\[\int_0^{\omega_D}g(\omega)d\omega=\frac{V\omega_D^3}{2\pi^2 c_s^3}\]
Now, the key insight here, due to Debye, is that there exists a bijection between each of these accessible single-phonon microstates and the \(3N\) vibrational degrees of freedom of the lattice of \(N\) atoms (in the cube of volume \(V=L^3\)), since after all phonons are vibrational quanta. So we arrive at an explicit formula for the Debye frequency \(\omega_D\):
\[\omega_D=\left(\frac{6\pi^2 N}{V}\right)^{1/3}c_s\]
Heuristically, we could have anticipated this result given \(\omega_D=c_sk_D=2\pi c_s/\lambda_D\) with the Debye wavelength \(\lambda_D\approx (V/N)^{1/3}\) on the order of the lattice spacing as that was the whole point of the cutoff frequency \(\omega_D\). To recap, the highest-energy phonons in a solid are those with the Debye energy \(E_D:=\hbar\omega_D\) and these are excited at the Debye temperature \(T_D:=E_D/k\) which for most materials is within \(100\text{ K}\) of room temperature \(T_D\in[150, 350]\text{ K}\).
Another point of similarity this time between phonons and photons is that neither is conserved in quantity. If we imagine a fictitious thermodynamic ensemble in which the frequency of all phonons in a solid is constrained to some fixed \(\omega\in[0,\omega_D]\), then the canonical partition function \(Z_{\omega}\) tailored to this specific angular frequency \(\omega\) would be given by the geometric series over all possible phonon populations \(n\in\textbf N\):
\[Z_{\omega}=\sum_{n=0}^{\infty}e^{-n\beta\hbar\omega}=\frac{1}{1-e^{-\beta\hbar\omega}}\]
The total partition function \(Z\) is therefore (because phonons like photons are non-interacting):
\[\ln Z=\int_0^{\omega_D}g(\omega)\ln Z_{\omega}d\omega=-\frac{3V}{2\pi^2c_s^3}\int_0^{\omega_D}\omega^2\ln\left(1-e^{-\beta\hbar\omega}\right)d\omega\]
which is integrable in terms of polylogarithms. The energy \(E\) of the phonon system is found by differentiating under the integral sign:
\[E=-\frac{\partial\ln Z}{\partial\beta}=\frac{3V\hbar}{2\pi^2c_s^3}\int_0^{\omega_D}\frac{\omega^3}{e^{\beta\hbar\omega}-1}d\omega\]
Changing to the dimensionless variable \(x:=\beta\hbar\omega\) means that:
\[E=\frac{9NkT^4}{T_D^3}\int_0^{T_D/T}\frac{x^3}{e^{x}-1}dx\]
The heat capacity at constant volume is:
\[C_V=\frac{\partial E}{\partial T}=9Nk\left(4\left(\frac{T}{T_D}\right)^3\int_0^{T_D/T}\frac{x^3}{e^{x}-1}dx-\frac{T_D}{T(e^{T_D/T}-1)}\right)\]
In the high-temperature limit \(T\gg T_D\), one can check that \(\lim_{T/T_D\to\infty}C_V=3Nk\) asymptotes to a constant heat capacity of \(3Nk\) called the Dulong-Petit heat capacity. In hindsight, the Debye argument for the total number of accessible microstates equalling \(3N\) can be rationalized as giving this experimentally known high-\(T\) Dulong-Petit heat capacity. The genuinely new and experimentally verified prediction of the Debye model of solids however is the low-temperature \(T\ll T_D\) asymptotic behavior of the heat capacity which one can check goes cubically to \(C_V(T=0)=0\) like \(C_V=\frac{12\pi^4}{5}\left(\frac{T}{T_D}\right)^3Nk\) in agreement with the \(3\)rd law of thermodynamics.