= Solution
\b[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> $\hbar\omega_{\mathbf k}$. Their number is not an independently conserved thermodynamic variable in equilibrium, so their <chemical potential> is zero and
$$
\bar n_{\mathbf k}=\frac1{e^{\hbar\omega_{\mathbf k}/(k_BT)}-1},\qquad
\boxed{U-U_0=\sum_{\mathbf k,\nu}\hbar\omega_{\mathbf k,\nu}\bar n_{\mathbf k,\nu}.}
$$
Here $\nu$ 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.
\b[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 $D(\epsilon)\propto\epsilon^2$, 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>,
$$
\boxed{C_V\simeq\frac{12\pi^4}{5}Nk_B
\left(\frac T{\Theta_D}\right)^3\quad(T\ll\Theta_D).}
$$
At temperatures high compared with all <phonon> energies but below structural changes, the classical harmonic limit gives approximately $C_V=3Nk_B$. Optical modes are frozen out at temperatures well below their gaps.
\b[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> $\epsilon_{\mathbf k}\simeq D_s k^2$, where the spin stiffness $D_s$ has units $\mathrm{J\,m^2}$. State counting gives $D(\epsilon)\propto\epsilon^{1/2}$. For one isotropic branch in three dimensions,
$$
\frac{U-U_0}{V}=\frac{\Gamma(5/2)\zeta(5/2)}{4\pi^2D_s^{3/2}}(k_BT)^{5/2},
\qquad\boxed{C_V^{\rm magnon}\propto T^{3/2}.}
$$
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.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2017-q2-excitation-dispersions.png]
{title=Original schematic linear acoustic-phonon and quadratic ferromagnetic-magnon dispersions with their three-dimensional low-temperature heat-capacity powers}
These examples illustrate the <low-temperature energy of gapless bosonic excitations>: for $\epsilon\propto k^s$ in $d$ dimensions, $D(\epsilon)\propto\epsilon^{d/s-1}$, 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.
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