= Solution
At long <wavelength>, the <acoustic phonon branch> moves the two atoms in a cell nearly in phase: $u_M\simeq u_m$. In the $q\to0$ limit this is rigid translation and $\omega\to0$. The <optical phonon branch> moves them oppositely, with stationary cell <centre of mass>: $Mu_M+mu_m=0$. Its $q=0$ <frequency> is finite because adjacent unlike atoms change their separation. For unequal masses the lighter atom has the larger optical displacement. Three-dimensional crystals have three acoustic branches and, with two atoms per primitive cell, three optical branches; longitudinal and transverse directions describe displacement relative to the <wavevector>. The name optical does not mean that every <optical phonon branch> is infrared-active: coupling also depends on effective charges and symmetry.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2012/paper-2-diatomic-displacements.png]
{title=Original long-wavelength acoustic and optical displacement sketches for unequal masses, with optical centre-of-mass cancellation}
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