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What is a phonon?

In hindsight, I suppose that I should have addressed phonons earlier. A phonon is a quantized sound wave - a collective vibrational mode of a solid (or liquid). In a crystalline solid, the idea is that the atoms in the solid are displaced, at any given instant, from their equilibrium positions. For a single phonon, the instantaneous displacements are periodic in space (that is, there is some wavelength, where atoms separated by an integer number of wavelengths are displaced the same amount). The displaced atoms feel restoring forces due to their interactions with neighbors, and will tend to oscillate in time around their equilibrium positions. When the wavelength is much longer than the interparticle separation, the frequency of those oscillations times the wavelength gives the speed of sound for the material - phonons propagate along at the speed of sound. In general, the speed of sound can depend on the direction of propagation as well as the direction of the direction of the displacement. If the displacement is along the direction of propagation, the sound is longitudinal; if the displacement is normal to the direction of propagation, the sound is transverse.

The quantum nature of phonons comes in when one discusses their energy content. In a classical mechanical oscillator, you can dump in as much energy as you want; the energy content is proportional to the square of the amplitude of the oscillation, and that can be varied continuously. In a quantum mechanical oscillator of frequency f, the energy content of that oscillator can only take on discrete values, (n + 1/2)hf, where n is a nonnegative integer. This is a subtle yet hugely important distinction. Mathematically it explains a major contribution to the heat capacity of crystalline solids at low temperatures (and it's very strongly related to the form of blackbody radiation when one is worrying about photons rather than phonons).

Because they have a wavelength and therefore a wavevector (and an effective momentum) as well as an energy, one can think about processes that involve the emission, propagation, and scattering of phonons - they have particle-like attributes in that sense.

(For a layperson discussion, I'm avoiding subtle distinctions like acoustic vs. optical phonons. If you really care, in acoustic phonons all the atoms within a unit cell move together, while for optical phonons different atoms within a single unit cell move by different amounts.)

## 4 comments:

These explanations continue to be awesome.

I've seen it a few times in the literature, but would you mind explaining how a phonon dispersion diagram is generated? Is it similar to a density of electronic states?

I'm wondering, is there some way you can tag or categorize these "What is ...?" posts, the way the "Basics" posts over at dot physics are. It seems like that these conversational descriptions might be quite useful for people stuck on the ideas more than the math (e.g., students tackling ideas from PHYS 533/534 and similar).

And here's a reader request, in case you're amenable to taking them. Were you thinking of doing one of these for Bloch waves? I never quite seem to have gotten a mental grip on those, not at the level of things like orbital hybridization and Coulomb blockading and such.

Hi Joel - Thanks for the kind words. The phonon dispersion diagram (that is, \omega vs. k for allowed states) is generated much the same way as the electronic dispersion diagram. In one dimension (for a single-atom basis), you can consider N point masses separated by a distance a and coupled by springs, and write down the equations of motion (basically Newton's laws in this case). Assume a harmonic time dependence (exp(-i \omega t)) and a spatial part of the solution that looks like exp(i k r) where r = n a, where n is an integer. Then you need to assume some kind of boundary conditions, usually periodic. After a little algebra with the trial solution, the equations of motion end up giving you an eigenvalue relationship between \omega and k. See here, for example.

Anon. - Yeah, I really should start tagging my posts, but I've been lazy. At some point I'll read the tag documentation for blogger and add them. I hadn't been thinking about doing Bloch waves exactly, since I'd been focusing on excitations rather than more general concepts, but I'll think about whether I can come up with some nice Bloch wave discussion.

Hello,

Thanks for the info. I have a question concerning phonons.

As I am relatively new to this, would you mind explaining from a phonon point of view how electrical resistivity increases due to increase in temperature for crystalline metals, and that electrical resistivity of non-crystalline metals (i.e. metallic glass) have a linear decrease due to heating?

Thank you for your time.

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