1. The Debye model assumes that atoms in materials move in a collective fashion, described by normal modes / sound waves with a dispersion relation $ω = v_{\rm s}|\mathbf{k}|$.
2. The normal modes have a constant density of $(L/2π)^3$ in the reciprocal space.
2. The normal modes have a constant density of $(L/2π)^3$ in $k$-space.
3. The total energy and heat capacity are given by integrating the contribution of the individual modes over $k$-space.
4. The density of states $g(ω)$ counts the number of modes per unit frequency. $g(ω)$ is proportional to $ω^2$ for 3D bosons with a dispersion relation $ω = v_{\rm s}|\mathbf{k}|$.
5. At low temperatures the phonon heat capacity is $∼T^3$.
5. At low temperatures the phonon heat capacity is $\propto T^3$.
