Phonon density of states ($g(\omega )=dN/d\omega$)

The number of lattice-vibration modes per unit angular frequency. For a three-dimensional isotropic monatomic crystal with three linear acoustic branches of a common speed $v_s$, wavevector-shell counting gives $g(\omega )=3V{\omega }^2/(2{\pi }^2{v}_s^3)$ below a suitable cutoff. Different branch speeds replace $3v_s^{-3}$ by $v_L^{-3}+2v_T^{-3}$.

For the nearest-neighbour monatomic-chain dispersion relation, ${\omega }_m=2\sqrt{\alpha /M}$ and $d\omega /dq=(a/2)\sqrt{{\omega }_m^2-{\omega }^2}$ on the positive half-zone. The folded wave-number count gives the displayed density of states, including both propagation directions. Its upper-edge Van Hove singularity is integrable: ${\int }_0^{{\omega }_m}g(\omega )d\omega =N$. A finite chain has discrete modes; the continuous density describes its large-size limit.

A single acoustic branch $\omega =v|k|$ has thermal energy per length $U_T/\ell =\int dk\hslash v|k|/[2\pi (e^{\beta \hslash v|k|}-1)]=\pi (k_{B}T{)}^2/(6\hslash v)$. Differentiating gives the displayed linear heat capacity. The regime requires temperatures below the microscopic phonon cutoff but above the finite-size level spacing, with the translational zero mode treated separately.