# How many types of phonons are there, and what are their characteristics?

I've skimmed through Wikipedia's article, as well as as PSE question What is a phonon?, but I am still entirely left in confusion as to the types of phonons and their corresponding characteristics and differences.

From what I could gather, there are:

1. Acoustic phonons. These are vibration modes of the crystal lattice and are coherent (they move in phase). They can be in the direction of propagation or not. I think these phonons are created when we either displace the whole crystal, or deform it mechanically. In a way, they are the "sound waves" in the crystal/material.

2. Optical phonons. These are out-of-phase vibration modes of the crystal lattice, meaning neighbor atoms or ions are moving in opposite direction. These phonons are usually activated (or created) when light (usually infrared but not strictly) is shone on the crystal. These phonons are responsible for the Raman scattering.

3. Thermal phonons. Unfortunately I am unable to retrieve much information on them, but I suspect they are the two kinds of phonons described above, so maybe this section is not valid.

4. Virtual phonons. From what I could dig on the Internet, they are responsible for superconductivity. In this case, Cooper pairs are deforming (I am not sure whether the deformation is real or not) the crystal lattice, creating these phonons. Unlike the optical and acoustic phonons, these phonons would still exist at absolute zero temperature. In Wikipedia it is written that "At absolute zero temperature, a crystal lattice lies in its ground state, and contains no phonons." This is why I assume that the virtual phonons are not real deformations of the lattice. But then, what are they?

5. Possibly others?

I would greatly appreciate if someone could correct any mistake I wrote above, and shed some light on all the currently known different type of phonons and their corresponding characteristics.

• infinite, as every phonon is slightly different Aug 28 '18 at 8:33
• One should be careful between (1) and (2), since the phase of atomic motion you are relying on to distinguish them varies across $k$ - an optical phonon at the edge of the first zone looks a lot like an acoustic phonon at the zone center. Aug 28 '18 at 14:29

In addition to acoustic and optical phonons, basically any kind of vibration that exists in an elastic continuum material can exist as a phonon (the converse is not true). For example, at the surface of a material there can be surface acoustic waves (e.g. Rayleigh waves).

That said, look at the phonon dispersion relation for a real material. There will be curves that don't fit neatly into any category. Fundamentally, phonons are quantized vibrations, and there are many ways that a system can vibrate. (Basically what Kenshin said.)

I'd say that thermal phonons are the population of phonons you'd expect to exist in a system in thermal equilibrium. So, thermal phonons are not a type of phonon in the same way as acoustic/optical phonons. (Basically what you said.)

Virtual phonons are really no different than any other type of virtual particle that emerge from a quantum field. Again, they're not a type of phonon in the same way as acoustic/optical phonons; you could have a virtual acoustic phonon.

EDIT IN RESPONSE TO COMMENT:

Virtual particles are really just a way of interpreting how fields interact in Quantum Field Theory (QFT). For example, say you have two electrons moving towards each other in a vacuum, and they repel each other because they have the same charge. In classical physics, you’d say that they repelled each other due to an electromagnetic interaction (e.g. the Lorentz force). In QFT, you’d say the two electrons scattered by exchanging virtual photons. So a classical force corresponds to a QFT virtual particle.

Likewise, if an electron is moving through a solid, it attracts the atoms around it, and the displaced atoms can likewise affect other electrons. In this way, electrons can interact with each other using the crystal lattice*; you have essentially created a new “force” by which the electrons can interact thru the lattice. That’s the classical interpretation. Phonons are the quanta of energy in crystal lattice motion (just like photons are the quanta of energy for electromagnetic fields), so phonons are the particle that correspond to this new “force”. Thus the QFT interoperation of this new “force” is that the electrons interact by exchanging virtual phonons. It’s the same thing, just a different language.

So, you’ll often see superconductivity explained in both ways. Strictly speaking, the classical explanation is not correct since it’s, well, classical (altho I believe that the deformation of the crystal lattice is real). That said, the QFT talk about virtual particles is really just a story that we tell ourselves to make sense of the math. It’s arguably intuitive (once you understand the math) and gives the right answers, so why not?

I should add that virtual particles show up in other contexts too, but you seem to mostly be interested in them WRT superconductors.

WRT zero temperature (maybe it's more appropriate to call it the ground state), I suppose these virtual phonons do exist since electrons can still interact with each other thru the lattice. Depending on how you interpret things, regular phonons kind of exist at zero temperature too. The vibrational modes of the lattice have a zero point energy (just like a simple harmonic oscillator), and you could interpret this as a phonon that you can't get rid of.

Reference: Elements of Advanced Quantum Theory by J. M. Ziman --- specifically sections 1.9-1.11, 3.5, and 3.7.

*Strictly speaking, you don’t need a crystal lattice for superconductivity. Some amorphous materials can do it too.

• Could you provide more information about virtual phonons? For instance, at absolute zero degrees for a superconducting material, there are only virtual phonons. Does this mean the lattice contains temporary "vibrations", which would be virtual phonons if I understand well? Once I get this point cleared, I will accept the answer. Sep 2 '18 at 12:37
• I've edited my answer with more information on virtual phonons. I hope that it's helpful. If you're interested in an introduction to QFT that focuses on condensed matter topics (such as phonons), rather than high energy physics, you might want to look at Elements of Advanced Quantum Theory by J. M. Ziman. It's a pretty old-school book (and the style is a bit of an acquired taste), but sections 1.9-1.11, 3.5, and 3.7 are pretty relevant to this discussion. Sep 6 '18 at 16:03