# Why are lattice vibrations called optical and acoustic? Dispersion relation for a diatomic basis in 1d. Two solutions are obtained: optical and acoustic branch.

The dispersion relation $$\omega(k)$$ of a one-dimensional crystal lattice with a diatomic basis has two branches. These branches result from the solution of two differential equations for this problem of diatomic chains.

The one branch (one solution) $$\omega_-(k)$$ is called acoustic because in this case the lattice planes oscillate in phase, as is the case of acoustic waves.

The other branch (the other solution) $$\omega_+(k)$$ is called optical because this solution of the respective differential equation gives an opposite-phase oscillation of the lattice planes (with $$m_1$$ and $$m_2$$). For example, if the atoms are ions, then they are electrically charged. An antiphase oscillation creates electric dipole moments in the crystal, which affects the optical properties of the crystal.

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