Acoustic and optical phonon branches

For r atoms per primitive cell in a three-dimensional crystal, the dynamical matrix has 3r eigenvalues at each independent wavevector. In a mechanically stable unpinned bulk crystal with no extra accidental zero modes, three branches approach zero frequency at the zone centre because of rigid translations. These are acoustic branches. The remaining $3r-3$ are optical branches, describing internal cell motion with finite zone-centre restoring frequencies. A monoatomic Bravais crystal therefore has no optical branches in its primitive description.

For a periodic sample with Nc primitive cells, each branch has Nc independent wavevectors. Multiplying gives $3rN_c$, exactly three times the total atom count. The acoustic zero modes at the zone centre still belong in the count of mechanical coordinates, although uniform translations are not finite-frequency oscillators. Enlarging the chosen cell folds branches and can make a primitive acoustic mode appear as a finite-frequency folded branch. That label change does not create a new physical coordinate or a new species of vibration.

At long wavelength an acoustic mode moves atoms within each cell nearly together, while neighboring cells vary slowly in phase. In an isotropic elastic limit one acoustic polarization is longitudinal and two are transverse. In anisotropic crystals the eigenvectors may be only approximately longitudinal or transverse away from special symmetry directions. Acoustic does not mean longitudinal: transverse shear waves are acoustic as well. Nor does an acoustic branch have to remain linear throughout the Brillouin zone. Its low-frequency character is a statement about the zone-centre limit.

An optical zone-centre mode has relative motion of basis atoms. For a two-atom one-dimensional example, its physical displacement amplitudes obey zero cell centre-of-mass motion, so the heavier atom has a smaller amplitude of opposite sign. Electromagnetic optical activity requires an oscillating dipole or another allowed coupling, determined by charges, symmetry, and polarization. Some optical modes are infrared inactive, while Raman scattering has different selection rules. The historical branch name is therefore not a guarantee of visibility in a particular optical measurement.

Quantization assigns a mode energy $\hbar\omega(n+1/2)$, where n is a nonnegative integer. The available mode count is fixed by the mechanical coordinates; temperature changes their mean occupation rather than creating additional branches. Low-frequency acoustic modes are readily thermally excited, while a high-frequency optical mode can remain weakly populated when its energy quantum exceeds thermal energy. The zero-point contribution remains even when the mean excitation number tends to zero, but its temperature-independent part does not contribute to harmonic constant-volume heat capacity.

Polar crystals add another distinction: a long-wavelength longitudinal optical displacement can create a macroscopic electric field and acquire a different frequency from a transverse optical displacement. Correct treatment requires the effective charges and dielectric response, not simply the short-range spring matrix. Degeneracies, branch crossings, and mixing can complicate labels at finite wavevector. Follow the eigenvectors and symmetry as well as the plotted frequencies, and specify whether a claimed optical splitting follows from a physical long-range interaction or merely from a chosen toy force model.