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Quanta of collective motion. An elementary excitation is a low-energy departure from the many-body ground state that can often be treated as a quasiparticle. Near an ordered equilibrium, small collective motions form normal modes. Quantization turns a harmonic mode of frequency ${\omega }_{\mathbf{k}}$ into bosonic excitations with energy $\hslash {\omega }_{\mathbf{k}}$. Their number is not an independently conserved thermodynamic variable in equilibrium, so their chemical potential is zero andHere $\nu$ labels branches, and $U_0$ contains the ground-state contribution, including phonon zero-point energy where appropriate. The approximation is useful while interactions and damping of the excitations are weak enough that the modes remain identifiable.

Phonons. A phonon is a quantum of an elastic lattice-vibration mode. Long-wavelength acoustic phonon branches have $\omega \simeq c_{\nu }k$, with one longitudinal and two transverse polarizations in a three-dimensional monatomic solid. A multi-atom primitive cell can also support optical phonon branches, generally with nonzero frequency at $k=0$. Linear acoustic dispersion gives $D(ϵ)\propto {ϵ}^2$, so the energy integral has $U-U_0\propto T^4$ and the heat capacity obeys the Debye $T^3$ law. In the Debye model,At temperatures high compared with all phonon energies but below structural changes, the classical harmonic limit gives approximately $C_V=3N{k}_B$. Optical modes are frozen out at temperatures well below their gaps.

Spin waves. A spin wave is a spatially varying collective precession of ordered spins; its quantum is a magnon. In a simple isotropic ferromagnet, a long-wavelength gapless branch has energy ${ϵ}_{\mathbf{k}}\simeq D_s{k}^2$, where the spin stiffness $D_s$ has units $\mathrm{J}{\mathrm{m}}^2$. State counting gives $D(ϵ)\propto {ϵ}^{1/2}$. For one isotropic branch in three dimensions,This exponent is specific to a gapless quadratic ferromagnetic branch. Long-wavelength antiferromagnetic spin waves can be linear and instead give a $T^3$ heat-capacity law in three dimensions. Magnetic anisotropy or an applied field can open a gap, causing activated behaviour at still lower temperatures. Near a magnetic ordering transition, a dilute noninteracting spin-wave description is no longer adequate.

These examples illustrate the low-temperature energy of gapless bosonic excitations: for $ϵ\propto k^s$ in $d$ dimensions, $D(ϵ)\propto {ϵ}^{d/s-1}$, hence $U-U_0\propto T^{1+d/s}$ and $C_V\propto T^{d/s}$. The exponent records both the spatial dimension and the dispersion, making calorimetry a probe of elementary excitations.