phonon dispersion
Pluck a thick guitar string and it hums a low note; pluck a thin tight one and it sings high. In a crystal there is a similar rule connecting a vibration's pitch to its size — and laying that rule out as a curve is what we mean by phonon dispersion: how a vibration's frequency depends on its wavelength.
Every possible vibration of the lattice has a wavelength (how far apart the crests of the swaying wave are) and a frequency (how fast each atom shakes). The dispersion is the master map that pairs them up, usually drawn as frequency rising and falling across the range of allowed wavelengths. Long, gentle waves of the acoustic branch sit at low frequency and travel at the speed of sound; as the waves get shorter the curve bends and flattens, and a separate, high-lying optical branch appears for crystals with more than one atom per cell. The whole picture lives inside one fundamental window of wavelengths called the Brillouin zone.
Phonon dispersion matters because, once you know it, you can read off almost everything thermal about a solid — how it stores heat, how fast sound runs through it, how stiff its bonds are. The subtle point is that the curve is not a straight line: short-wavelength vibrations travel slower than long ones, so a sharp pulse of many wavelengths spreads out and blurs as it goes, which is literally what 'dispersion' names.
Physicists measure phonon dispersion by firing neutrons at a crystal and seeing how much energy and direction each one loses: a neutron that creates a vibration of a certain wavelength always gives up a certain matching amount of energy. Map enough collisions and the full frequency-versus-wavelength curve emerges.
Neutron scattering reads out a crystal's phonon dispersion, collision by collision.
Don't picture frequency simply growing forever as wavelength shrinks. In a crystal it levels off at the shortest meaningful wavelength, because two atoms cannot wiggle on a scale finer than the spacing between them.