A phonon is a bosonic quantum of a crystal's normal modes. Because phonon number is not conserved in thermal equilibrium, its chemical potential is zero.
A phonon dispersion relation gives its angular frequency as a function of its wavevector . For an isotropic power law, .
An optical phonon branch has a nonzero frequency as , corresponding to relative motion of degrees of freedom within one primitive cell.
For equal masses separated by and alternating spring constants and , the primitive-cell length is and the two branches areThe minus and plus signs give the acoustic phonon branch and optical phonon branch, respectively.
The Debye model replaces the crystal's vibrational spectrum by a continuum of phonon modes up to a cutoff chosen to preserve the total number of vibrational degrees of freedom.
The Debye frequency is the upper frequency cutoff fixed by equating the continuum mode count to the crystal's number of vibrational degrees of freedom.
The Debye temperature is .
For two polarizations, atoms in area , and , mode counting givesAt low temperature its energy scales as and its constant-volume heat capacity as .
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A phonon is a quantized mode of vibration that occurs in a rigid crystal lattice, such as those found in solid materials. In other words, it is the quantum mechanical description of lattice vibrations. Phonons play a key role in various physical properties of solids, including thermal conductivity and sound propagation. Phonons can be thought of as discrete packets of vibrational energy, similar to how photons are packets of electromagnetic energy.