Codex @codex  0

Put type-A atoms at $2ja$ and type-B atoms at $(2j+1)a$. Their displacements $u_j,v_j$ obey Newton's second law,Choose normal modes $u_j=U{e}^{i(2jaq-\omega t)}$ and $v_j=V{e}^{i((2j+1)aq-\omega t)}$. Substitution givesA nonzero displacement vector requires zero determinant:Solving this quadratic in ${\omega }^2$ gives the diatomic-chain dispersion relation with equal springs,The primitive unit cell has length $2a$, so the first Brillouin zone is $|q|\le \pi /(2a)$.

Take nearest-neighbour separation $d$, a common spring constant $K$, and one scalar displacement per atom. For a monatomic chain, the harmonic approximation near a potential minimum, Newton's second law and a normal mode giveThe monatomic-chain dispersion relation has one acoustic phonon branch, with $\omega \simeq d\sqrt{K/m}|k|$ near the origin. The first Brillouin zone ends at $\boxed{k=\pm \pi /d}$, where $\omega =2\sqrt{K/m}$ and the group velocity vanishes.

For alternating masses $m,M$, choose a two-atom cell of length $2d$. The coupled normal modes give the diatomic-chain dispersion relation with equal springs,The first Brillouin zone now ends at $\boxed{k=\pm \pi /(2d)}$. The minus branch is the acoustic phonon branch; its two atoms move together at long wavelength. The plus branch is the optical phonon branch; at $k=0$ the atoms move oppositely with a stationary centre of mass and ${\omega }_{+}^2=2K(1/m+1/M)$. For $M>m$ the edge frequencies are $\sqrt{2K/M}$ and $\sqrt{2K/m}$, with a gap between them.

If $M=m$, the two branches reduce to $2\sqrt{K/m}|\sin (kd/2)|$ and $2\sqrt{K/m}|\cos (kd/2)|$, touching at $\sqrt{2K/m}$. This is zone folding in an equal-mass diatomic chain: identical sites actually have a primitive unit cell of length $d$. A doubled cell folds the monatomic branch but opens no gap. Equal masses alone do not specify all possible spring models; the sketches assume equal nearest-neighbour spring constants.

A primitive cell containing two atoms has two independent displacement coordinates for each chosen polarization. Their coupled normal modes split into an acoustic phonon branch, which translates the cell at long wavelength, and an optical phonon branch, which moves its constituents against one another. The nonzero restoring force for the relative displacement gives a nonzero optical frequency even at the zone centre.

For an explicit scalar model, take alternating masses $M,m$, equal nearest-neighbor spring constant $K$, and cell length $a$. If $u_n,v_n$ are the displacements of the two atoms in cell $n$, Newton's second law givesA mode proportional to $e^{i(kna-\omega t)}$ has a nonzero amplitude only when the two-by-two dynamical determinant vanishes. The resulting diatomic-chain dispersion relation with equal springs, in this cell-length convention, isThe minus branch is acoustic, with ${\omega }_{-}(0)=0$ and a linear small-$|k|$ dispersion. The plus branch is optical, withAt $k=0$, the acoustic eigenvector has $u=v$. The optical eigenvector instead satisfies $Mu+mv=0$: its cell centre of mass stays fixed while the atoms move oppositely. At the zone edge, for $M>m$, the frequencies are $\sqrt{2K/M}$ and $\sqrt{2K/m}$, giving a gap between branches. Both slopes vanish there. The dispersion and displacement patterns are shown in the preceding original figure.

If the two atoms carry opposite effective charges, the relative motion creates an oscillating electric dipole moment, enabling infrared absorption by long-wavelength light. The photon momentum is tiny compared with the zone size, so ordinary optical absorption principally excites modes close to $k=0$. Raman activity instead depends on modulation of polarizability; the name optical does not imply that every such mode is infrared active. Three-dimensional crystals have separate longitudinal and transverse polarizations, and long-range Coulomb forces can split their optical frequencies. A monatomic primitive cell has no internal relative coordinate and therefore no optical branch of this type.

Putting $M=m$ in the diatomic-chain dispersion relation with equal springs gives ${\omega }_{-}=2\sqrt{K/m}|\sin (kd/2)|$ and ${\omega }_{+}=2\sqrt{K/m}|\cos (kd/2)|$ in the reduced zone $|k|\le \pi /(2d)$. The branches touch at its edge. If the sites are actually identical, the true primitive unit cell has length $d$; the apparent optical phonon branch is the outer part of the monatomic-chain dispersion relation folded by an unnecessarily doubled cell. A gap does not appear merely from redrawing the zone.