Monoatomic and diatomic chain phonons

First consider one mass m per primitive interval a, with identical springs of stiffness K joining neighbors. Expanding the bond potential to quadratic order gives force $K(u_{n+1}+u_{n-1}-2u_n)$ on displacement un. Substituting $u_n=u e^{i(qna-\omega t)}$ diagonalizes translation and yields a frequency determined by the phase difference qa. The resulting branch is periodic in q with period two pi over a. Frequencies are nonnegative; the sine must therefore appear with an absolute value when taking the square root.

Near zero wavevector the frequency is proportional to the magnitude of q, with sound speed $a\sqrt{K/m}$. At the zone boundary it saturates, while the group velocity vanishes in this ideal nearest-neighbor model. The phase velocity and group velocity need not coincide away from the long-wavelength limit. These conclusions follow from the chosen force law. Longer-range forces can reshape the dispersion, and the continuum sound approximation cannot be extended blindly to wavelengths comparable with the lattice spacing.

For the diatomic chain, place mass m at na and mass M at na plus a over two, with the same spring K on every nearest-neighbor bond. A cell therefore has two displacement amplitudes and a two-by-two generalized eigenproblem. Its determinant gives the two squared frequencies below. The minus branch goes to zero at q equal to zero; the plus branch remains finite. The primitive zone boundary is now pi over the two-atom cell length a, not pi over the neighbor spacing.

At zero wavevector the acoustic displacement moves the two atoms together. The optical displacement moves them oppositely with zero cell centre-of-mass displacement. When M exceeds m, the boundary frequencies are the square roots of two K over M and two K over m. Thus a mass contrast opens a gap between the two branches in this simple chain. Equal masses close that gap; the true primitive interval then becomes a over two, and the two branches can be understood as a folded monoatomic dispersion rather than two chemically distinct vibrations.

Classical normal-mode frequencies are obtained before quantization. Quantizing each independent harmonic mode makes its energy levels equally spaced by hbar omega, and one excitation is called a phonon. A phonon is not an individual atom moving along a sine curve; it is a quantum of the collective displacement coordinate. Counting branches begins with the number of independent coordinates per primitive cell. These scalar chains consequently have one or two branches, while an actual three-dimensional crystal also has displacement polarizations.

The harmonic approximation assumes small excursions around a mechanically stable periodic equilibrium and neglects damping, defects, and anharmonic scattering. It predicts perfectly sharp mode frequencies but no intrinsic finite lifetime. A simple wave-superposition lab cannot generate this dispersion by changing its phase slider: that instrument fixes its component frequencies. To test a chain model quantitatively, use the equations with declared masses, stiffnesses, cell length, and wavevector, then check zero-wavevector translation, boundary values, and the equal-mass folding limit independently.