Phonons in a lattice

Have you already checked this http://en.wikipedia.org/wiki/Phonon" [Broken]?

 

Discretization could mean many things here.

For a finite lattice, phonons can only have discrete k values.

There are several w(k) branches for each value of k.

Phonons can have quantized energy levels like a particle in a harmonic (or anharmonic) well.

Which one are you talking about?

 

The quantization of phonon energies arises not only out of the quantization of matter but more importantly, out of the boundary conditions that are applied to the differential equation describing the displacements.

 

Do you think elementary particles could arise in this way?

 

Quantization of energy levels for phonons occurs for just about the same reason that quantization of energy levels for a particle in a potential well --except that a phonon mode corresponds to atomic vibrations across the whole crystal.

For a harmonic system (and crystals are typically near-harmonic) Schrodinger's eqn. becomes separable into 3N non-interacting harmonic oscillator wells, where each well corresponds to a particular normal-mode of vibration. The energy in a each well is just (n+1/2) hbar omega, where omega is the frequency in the that well.

Oh- and the distinction between phonon modes and normal modes (say in a molecule) is that phonons have wave-vectors corresponding to the periodicity of the lattice.

 

The first. The vibrational modes are quantized. In a perfectly harmonic crystal, the vibrational mode corresponding to a well of frequency w0 would have energy levels (n+1/2)hbar w0. The 0-1 transition corresponds to a phonon of energy hbar w0, the 0-2 transition corresponds to a phonon of frequency 2hbar w0, which classically corresponds to an overtone.