Photons and phonons

Question

A few months ago I asked about phonons. I got some very good answers but I still have difficulty getting an intuition for phonons, while somehow photons, which in many ways are similar and which I realize I hardly understand anything about, seem more accessible to intuition.

In Ron Maimon's answer to the question "What exactly is a quantum of light?" he asserts that

A quantum of light of wavelength λ is the minimum amount of energy which can be stored in an electromagnetic wave at that wavelength

and

the classical wave is a superposition of a large number of photons

Translating this to vibrations in a crystal lattice, could we say that a phonon is the minimal amount of energy which can be stored in an lattice vibration in a given mode and that a classical vibration is a superposition of a large number of phonons?

I hope I am correct when I say that the electromagnetic field can interact with matter through the absorption of a photon, and it is this interaction that makes the photon into something particle-like. Do we have the same for phonon-interactions? I.e. that when a crystal vibration interacts with matter it does so by the creation/destruction of whole phonons at a time, which may also get absorbed at more or less precise locations, e.g. the energy of a single phonon is absorbed by a localized electron.

Finally I would like to understand how phonon exchange can effectively establish an attractive force between electrons, but I cannot say I have any intuition for how photons mediate the electromagnetic force either. I am afraid that for the moment this is beyond the scope of my background.

Answer 1

Here are some wordy, math-free answers.

a phonon is the minimal amount of energy which can be stored in an lattice vibration in a given mode

Sounds good.

I.e. that when a crystal vibration interacts with matter it does so by the creation/destruction of whole phonons at a time, which may also get absorbed at more or less precise locations, e.g. the energy of a single phonon is absorbed by a localized electron.

A phonon is a periodic motion of the atoms in a solid, so I'd argue that it's always interacting with matter since it is matter in motion.

Localization of phonons is a tricky business. The textbook derivations for phonons result in vibrations (waves) that extend through the whole material. However, they're usually treated as localized.

You can make any function by adding up waves of different wavelengths (the waves form a basis), so you can build up localized phonon "packets" from lattice vibrations of different frequencies. Unlike photons, phonons have non-linear dispersion relations -- meaning that waves of different frequencies travel at different speeds (unlike light where all frequencies travel at the same speed, at least in a vacuum), so the packets will eventually fall apart if left alone. However, they can stick together long enough that they can be thought of as particles. If the frequencies in the packet are of a narrow range, you can think of the packet as having a frequency equal to the average frequency of its constituent waves.

This localization makes sense if electrons are likewise localized. If an electrons scatters with a phonon, and that electron is localized, that means the electron is only really interacting with nearby atoms. So, any lattice vibration the electron creates should be initially localized to that region too.

I should add that a major form of phonon scattering is with other phonons. It turns out that you can't have two-phonon processes (two phonons colliding and create two other phonons); you can only have three-phonon processes and higher (e.g. two phonons merge to create a third). You don't have to think of these processes as being localized in space.

Finally I would like to understand how phonon exchange can effectively establish an attractive force between electrons

The atoms in a lattice are charged, so they can pull on nearby electrons. If several atoms are pulling on one electron, then the atoms are effectively pulling on each other and are brought closer together. If the electron is moving, it can leave a wake of atoms that are closer together (a phonon). Atoms being closer together means more positive change in an area, and that in turn can draw in another electron -- effectively attracting the electrons together.

See

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/coop.html