# How can phonons "travel" if they are excitations of normal modes?

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• confused_man
In summary, phonons are excitations of normal modes in a crystal that can behave as particles due to the wave-particle duality. They can be defined as wavepackets made from standing waves or traveling waves, depending on the desired basis functions for describing crystal vibrations. Visualizing phonons can help with understanding their behavior and properties in different materials. However, their fundamental description requires the use of quantum field theory and second quantization techniques.
confused_man
My understanding is that you can describe the complicated motion of atoms in a crystal as a sum of standing waves (normal modes). A phonon is an excitation of a normal mode in the sense that it increases the vibration amplitude of that normal mode and the energy of that mode by a quantized amount ##\hbar\omega## (here ##\omega## is the vibration frequency of the normal mode).

That makes perfect sense to me until people start describing phonons as being "particles" that can propagate through the crystal. Aren't phonons existing in all parts of the crystal at once given that they are excitation of these normal modes? At this point people usually wave their hands and invoke the wave-particle duality to say that phonons can behave as particles too. This isn't very satisfying to me.

Does anyone have a good conceptual explanation of how you can define phonons as particles that propagate? What does that actually look like in practice? Is it simply that we can also describe crystal vibrations using a different set of basis functions like traveling waves instead of standing waves, and then define a phonon as a wavepacket made from the standing waves?

DrClaude
Before we get into phonons and quasiparticles, do you have the same problem (How can waves "travel" if they are excitations of normal modes? ) with traveling waves, as in a rope?

hutchphd and DrClaude
confused_man said:
My understanding is that you can describe the complicated motion of atoms in a crystal as a sum of standing waves (normal modes). A phonon is an excitation of a normal mode in the sense that it increases the vibration amplitude of that normal mode and the energy of that mode by a quantized amount ##\hbar\omega## (here ##\omega## is the vibration frequency of the normal mode).

That makes perfect sense to me until people start describing phonons as being "particles" that can propagate through the crystal. Aren't phonons existing in all parts of the crystal at once given that they are excitation of these normal modes? At this point people usually wave their hands and invoke the wave-particle duality to say that phonons can behave as particles too. This isn't very satisfying to me.

Does anyone have a good conceptual explanation of how you can define phonons as particles that propagate? What does that actually look like in practice? Is it simply that we can also describe crystal vibrations using a different set of basis functions like traveling waves instead of standing waves, and then define a phonon as a wavepacket made from the standing waves?

https://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1343&context=phy_fac

Zz.

DrClaude and dRic2
Out of curiosity: what is the advantage of "visualizing" phonons (or photons for that matter...)?

I mean, I am an engineer with an interest in physics so I really like to get practical as soon as I can, but when it comes to phonos (or stuff like that) I really don't see a reason to waste energies in trying to visualize something so abstract. I saw some of the "math" behind the concept and I confirmed with my own eyes that the Hamiltonian is just the same Hamiltonian of lots HOs and that was good enough for me.

Probably it was good enough only because it is as far as I can go...

dRic2 said:
Out of curiosity: what is the advantage of "visualizing" phonons (or photons for that matter...)?

I mean, I am an engineer with an interest in physics so I really like to get practical as soon as I can, but when it comes to phonos (or stuff like that) I really don't see a reason to waste energies in trying to visualize something so abstract.

Probably it was good enough only because it is as far as I can go...
Phonons are described by quantum field theory. There is a non-relativistic field theory, believe it or not ;-) It absolutely makes sense to describe excitations in a very fundamental way in field theories.
I am not sure in which field of engineering your will work but the likelihood to have to do with this subject directly is tiny. Nevertheless, you may keep in mind that fundamental properties of condensed matter are only described well using quantum (field) theory where the phonon as a pseudo particle is a very good picture as you can have phonon scattering etc.

dRic2 said:
Out of curiosity: what is the advantage of "visualizing" phonons (or photons for that matter...)?

I mean, I am an engineer with an interest in physics so I really like to get practical as soon as I can, but when it comes to phonos (or stuff like that) I really don't see a reason to waste energies in trying to visualize something so abstract. I saw some of the "math" behind the concept and I confirmed with my own eyes that the Hamiltonian is just the same Hamiltonian of lots HOs and that was good enough for me.

Probably it was good enough only because it is as far as I can go...
You should play a bit: https://henriquemiranda.github.io/phononwebsite/phonon.html to build an intuition. There you can visualize in 3D any kind of phonon in a plethora of materials by clicking on the phonon dispersion curves to see to which kind of oscillations they correspond. So maybe visualizing phonons can actually help you understand deeper phonon dispersion relations.

Omega0
The conduction band gaps near the Brillouin zone edges in a regular crystal involves exactly the lack of traveling wave eigensolutions because of the strong coherent backscatter from the periodic potential.
.
Where they exist the energy eigensolutions for a given k are typically degenerate and so any combination of forward and backward solutions also works as an eigensolution. The choice of a traveling wave basis or a standing wave basis is dictated by convenience, and the word "phonon" need never be spoken. But the dispersion relations invite the definition of the phonon and use of second quantization techniques (which typically do not correspond to localized states).

dRic2
fluidistic said:
You should play a bit: https://henriquemiranda.github.io/phononwebsite/phonon.html to build an intuition. There you can visualize in 3D any kind of phonon in a plethora of materials by clicking on the phonon dispersion curves to see to which kind of oscillations they correspond. So maybe visualizing phonons can actually help you understand deeper phonon dispersion relations.

I thought the OP was asking for a particle-like interpretation of phonons. Of course you can visualize the various modes of the lattice but I don't think that's what the OP meant.

What I meant by "wasting of energies" is that it is useless (to me) to try thinking about a particle moving in the material.

## 1. How do phonons "travel" if they are excitations of normal modes?

Phonons are not physical particles that travel through a material, but rather they are a way to describe the collective motion of atoms in a crystal lattice. When energy is added to a material, the atoms vibrate and transfer this energy to neighboring atoms, creating a wave-like motion that we refer to as phonons.

## 2. Can phonons travel in all directions?

Yes, phonons can travel in all directions within a material. They can travel in a straight line, bounce off of boundaries, or scatter off of defects. The direction and speed of a phonon depends on the material's properties and the temperature.

## 3. How fast do phonons travel?

The speed of phonons varies depending on the material and the temperature. In general, phonons travel faster in materials with higher stiffness and lower density. At room temperature, phonons can travel at speeds ranging from a few hundred meters per second to several kilometers per second.

## 4. Do phonons have a mass?

No, phonons do not have a mass in the traditional sense. They are not physical particles, but rather they are a way to describe the collective motion of atoms in a material. However, they do have an effective mass that is related to the material's properties and the frequency of the phonon.

## 5. How do phonons affect the thermal conductivity of a material?

Phonons play a crucial role in the thermal conductivity of a material. As they travel through a material, they transfer heat energy from one atom to another, contributing to the overall thermal conductivity of the material. The speed and frequency of phonons also affect the thermal conductivity, with higher frequencies and faster speeds leading to higher thermal conductivity.

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