What is the significance of crystal momentum?

[EternusVia]

Hello,

Could someone please explain crystal momentum in understandable terms?

Thanks,
EV

 

[Hardik Batra]

crystal momentum is in every regard the same as usual momentum except the fact that it takes values only in Brillouin zone (as a consequence of discrete symmetry of lattice; or more precisely its continuum limit). So the answer to your question is: for most purposes crystal momentum is a real momentum.

Now, the term crystal momentum is being used here in two different meanings and that is probably where confusion arises. Your citation uses it as a total momentum of the crystal. This is obviously zero for phonons (which are just harmonic modes of the material) because on average the atoms of the crystal don't move (they just oscillate around stable positions). And that is precisely why nobody uses the term in this way (and I don't understand why your book does).

But locally energy and momentum are still being transferred (jumping from one atom to the next as they interact). So in fact, phonon is a wave that propagates in the material in some direction and carries some energy. Obviously this is a very physical wave with physical energy and physical momentum. It is this latter momentum which is usually referred to as crystal momentum

 

[EternusVia]

That makes perfect sense! Thank you.

Perhaps you could elaborate on why a phonon is described as a quasiparticle? Why even attribute any particle-like characteristics to it at all, being that it is a wave?

Also, I have no idea what the Brillouin Zone is...

 

[ZapperZ]

That makes perfect sense! Thank you.

Perhaps you could elaborate on why a phonon is described as a quasiparticle? Why even attribute any particle-like characteristics to it at all, being that it is a wave?

Also, I have no idea what the Brillouin Zone is...

This is why we strongly recommend that when people, especially new members, ask a question, he/she must first of all describe what he/she knows, what he/she has attempted to understand, and then elaborate at what level he/she can comprehend. Otherwise, we will get a situation like this where you are not able to comprehend the responses you receive, and our members would have wasted time and effort in putting out a response that went over your head! (It also explains why this was posted in the General Physics forum instead of the more appropriate Solid State physics forum, where it has been moved to).

Phonon is, by definition, quantized lattice vibrations. That is why we can attribute a "quasiparticle" to it, just like we assign such quantization to the electromagnetic field and call it "photons". The background knowledge to understand this is Quantum Field Theory (QFT).

Brillouin zone is the "reciprocal lattice space" of a material, as opposed to real space where the crystal lattice reside. (I fully expect a follow up question "what is reciprocal space?"). It is the equivalent of the Wigner-Seitz cell in reciprocal space.

There is a lot of "prerequisite knowledge" that is needed here to answer your question, and I see this happening repeatedly where every time one makes a response and try to make a step forward, we have to take two steps back just to explain our explanation. Again, this can easily be dealt with had you indicate what you know and don't know.

Zz.

Zz.

 

[DrDu]

crystal momentum is in every regard the same as usual momentum

I don't quite agree on this. Usual momentum is a consequence of translational homogeneity of space via Noethers theorem, while crystal momentum is a consequence of the permutation group of the nuclei forming the lattice. So the two momenta have a completely different origin.

In general, they can have completely different values:
E.g. honons don't carry true momentum, but they carry crystal momentum

 

[stevendaryl]

I don't quite agree on this. Usual momentum is a consequence of translational homogeneity of space via Noethers theorem, while crystal momentum is a consequence of the permutation group of the nuclei forming the lattice. So the two momenta have a completely different origin.

In general, they can have completely different values:
E.g. honons don't carry true momentum, but they carry crystal momentum

I refreshed my memory about crystals by looking them up on Wikipedia, and it seems that crystal momentum is neither the same as regular momentum, nor completely different.

An electron traveling in a periodic lattice has a wave function (or can be written as a superposition of such wavefunctions) of the form $e^{i\ k \cdot r} u(r)$. The expectation value of the momentum of such a wavefunction is $\langle p\rangle = \hbar k + \langle u|-i \hbar \nabla |u\rangle$

So the crystal momentum contributes to the momentum of the electron, but it's not the total momentum. On the other hand, the total momentum is not conserved (because there are periodic forces acting on the electron), while the crystal momentum is conserved (actually, it is only conserved up to a multiple of the inverse lattice vectors).

 

[DrDu]

On the other hand, the total momentum is not conserved (because there are periodic forces acting on the electron)

Ashcroft and Mermin contains a discussion of the conservation of total momentum in solids in one of the appendices.

I don't recommend the article on crystal momentum in wikipedia. E.g. it is simply not true that crystal momentum refers only to the electrons in a lattice. It is also important for the description of phonons and spin waves or photons, to mention just a few.