Statistical Mechanics/Debye Theory

Say we have a material with vibrating modes visually akin to that of jello. In other words, we are looking at the phononic modes. The energy equation for a specific mode of a phonon is identical to that of a photon. The only difference between the phonon and photon case is that the phonon case has a possible upper limit (because the highest frequency occurs at the shortest wavelength: when one molecule is at its amplitude and the neighbours are at their troughs), and that we have another possible mode of polarization (two transverse, one longitudinal), and that we are dealing with v (the speed of sound in the material) instead of c. Thus, when we transform the integral as in the photon derivation (another similar density of states problem) we get:

$U={\frac {3\pi }{2}}\int _{0}^{n_{D}}dn{\frac {n^{2}\hbar \omega _{n}}{exp(\hbar \omega _{n}/T)-1}}$

Where n_D is this maximal case.

We define something now called the Debye Temperature (a physical property of a material):

$\theta =(\hbar v/k_{B})(6\pi ^{2}N/V)^{1/3}$

Note: the k_B is apparent because we've been working in energy-represented T, instead of the typical Kelvin-scaled T, k_B effectively means the T's here-on are in Kelvin-units)

If we use this, and take the low-temperature limit of T << θ, we can approximated the upper limit as we continue the integration as ∞ (which physically means that there's a bunch of higher-energy levels states that the solid is not reaching) we can evaluate almost exactly as before:

$U(T)=3\pi ^{4}Nk_{B}T^{4}/5\theta ^{3}$

and:

$C_{V}:=\left({\frac {\partial U}{\partial T}}\right)_{V}={\frac {12\pi ^{4}Nk_{B}}{5}}\left({\frac {T}{\theta }}\right)^{3}$

This is known as Debye's T³ law, and follows well with experiment.