Understanding Phonons on a Lattice: Seeking Insight from Niles

[Niles]

Hi all

My book says:

"The reason that phonons on a lattice do not carry momentum is that a phonon coordinate (except for wavevector K=0) involves relative coordinates of the atoms".

I can't quite figure this statement out. I understand the words, but I cannot see why it is an explanation.

Can you shed some light on this topic?


Niles.

 

[elduderino]

Hi.

Read into something called the Born approximation for this lattice model where each atom is considered a harmonic oscillator. Basically, the mean position of the vibrating atom (its COM) does not change over time.

Such approximation is needed when making such a model. So you can consider every atom (each with its own fixed co-ordinate system) to be a classical harmonic oscillator, executing small oscillations about its mean position (origin of the relative co-ordinate system for that particular atom).

I guess now it can be understood. Phonons have energy, but no net momentum, as they keep going back and forth about a fixed origin, and in each cycle the momentum cancels itself out.

 

[kanato]

I don't really understand that. Phonons definitely have a well-defined pseudomomentum, and Umklapp processes which reduce the total momentum of phonons are responsible for keeping thermal conductivity finite in a perfect crystal. Is there more context to the statement in the book? Are they talking about only k = 0 phonons?

 

[Niles]

k=0 phonons are mentioned in the quote as the only phonons with actual momentum, which is 0.

But you say pseudo-momentum?

 

[kanato]

Yes.. in a periodic crystal you don't have continuous translational symmetry, so conservation of momentum doesn't hold. But due to the discrete translational symmetry there is a conservation law of the pseudomomentum vector k, where k is a vector in the first Brillouin zone.

 

[Niles]

Phonons definitely have a well-defined pseudomomentum, and Umklapp processes which reduce the total momentum of phonons are responsible for keeping thermal conductivity finite in a...

Ok, so Umklapp processes change the total phonon pseudomomentum. But then how can we talk about conservation of phonon-momentum?

 

[kanato]

Conservation of total momentum you can't talk about, it doesn't exist in a system with an external potential. Pseudomomentum is conserved, even in Umklapp processes, but pseudomomentum is only well defined up to the edge of the first Brillouin zone, or more precisely, any function in the lattice of f(k) = f(k + K) where K is any integer combination of reciprocal lattice vectors. Only the value of k (the pseudomomentum in the first Brillouin zone) is conserved, if some process adds multiple pseudomomenta and gets a value outside the first BZ it will be translated back in by a vector K.