Dispersion Relation and wavelength

[lylos]

I've been working on problems trying to find the dispersion relation of monatomic and diatomic chains. I'm able to find the dispersion relation, but I don't really understand what it is. I'm pretty sure that it's the relation of the frequency to the wave vector.

I'm working through Kittel's Intro to Solid States, and it states that the wave vector is simply 2Pi/wavelength. I'm wondering where the wavelength comes from though. Is that the wavelength of the wave propagating through the crystal lattice? Could someone please explain?

 

[lzkelley]

The dispersion tells you what happens to different frequencies of light in/incident to a material.
For instance, if they will decay, grow, oscillate, reflect, be transmitted etc.

 

[lylos]

Ok. So the wavelength is from the incident wave onto the surface. You can then determine (based on the dispersion relation) whether it will be transmitted or not. Also, would the frequency given by the dispersion relation be that of the individual atoms?

 

[lzkelley]

First part is correct. In dispersion relations there are usually numerous frequencies (i'm not sure what your looks like), but it would necessarily include the incident photon frequency, and most likely key resonance frequencies of the material.

 

[lylos]

Alright, so for a diatomic chain there would be two key normal mode frequencies. Those two frequencies are determined by the incident wavelength. There's the optical branch and the acoustical branch?

Sorry I'm asking so many questions. I haven't had a solid states course and that's what we're doing in my Physics REU...

 

[lzkelley]

Sorry i don't know any of the details of a diatomic chain, i just happened to have been dealing with lots of dispersion relations recently (plasma astrophysics). If there's a particular dispersion equation you're looking at - copy it in, maybe i or someone can give you a hand.
What REU are you working with?

 

[Cthugha]

Alright, so for a diatomic chain there would be two key normal mode frequencies. Those two frequencies are determined by the incident wavelength. There's the optical branch and the acoustical branch?

Generally speaking a dispersion relation just relates the kinetic energy of some wave-like excitation to the momentum of it. Monatomic and diatomic chains are basic models for phonon dispersion relations, so I suppose these are meant here. This means, that there is not necessarily light involved.

As an easy model, which is used in most basic courses, just imagine your atoms as little balls, model the forces between one atom and its next neighbours as a spring and ignore all other forces. Now you got a long chain off balls connected with springs. You can easily imagine, that an atom behaves like a harmonic oscillator, if it is moved out of its equilibrium position and can therefore oscillate. As you have a long chain, you can also imagine a collective oscillation of all atoms around their equilibrium position. In this basic model, this is a phonon.

If you go to a diatomic chain, you can imagine, that there are several basic modes. All atoms can move in phase or the first kind of atoms can oscillate in antiphase to the second kind, for example. If you have charged particles instead of neutral atoms the second mode produces an oscillating dipole moment. Therefore this is called the optical branch.

However, the whole topic is more complicated - you can also take transversal modes into account and more than just 1D chains and such stuff, but for the moment, it should be sufficient to know, that the dispersion relation relates the kinetic energy of this excitation of all atoms to its momentum.

If you need a more technical approach, a google search for phonon dispersion relations might provide you with some info. The books of Kittel and Ashcroft/Mermin will work as well.

 

[Gokul43201]

The dispersion tells you what happens to different frequencies of light in/incident to a material.
For instance, if they will decay, grow, oscillate, reflect, be transmitted etc.

kelley, the question posed by the OP has nothing to do with light incident on a material. This is about the vibrational (phonon) eigenmodes of the chain, as explained by Cthugha.

lylos, if your chains are restricted to live in a 1-dimensional space, then there will only be one (acoustic) branch for the monoatomic chain and two (acoustic and optical) branches for the diatomic chain. However, if the atoms are allowed to oscillate along directions normal to the chain, then each branch will additionally possesses longitudinal and transverse modes. All these results will pop directly out of solving the eigenvalue problem of the elastic chain.

 

[lylos]

Thank you all for the help. I have since developed a better understanding. :) To answer Izkelly's question, I'm working with Boston College REU.

 

[marcusl]

lylos, see if your college library has a copy of Leon Brillouin's wonderful book "Wave Propagation in Periodic Structures." The first couple of chapters deal in detail with exactly your question: wave propagation and dispersion in monoatomic and diatomic solids. It is related to telegraph and transmission lines, coupled pendula and similar systems, and contrasted to sound propagation in gases. The notion and utility of Brillouin zones is then, naturally, introduced. I highly recommend it!