Understanding Phonons on a Lattice: Seeking Insight from Niles

In summary: Ok, so the pseudomomentum is conserved, but it's only well-defined up to the edge of the first Brillouin zone. So how do we talk about momentum conservation in a periodic crystal?In summary, the phonons in a periodic crystal do not have net momentum because they keep going back and forth about a fixed origin. Pseudomomentum is conserved, even in Umklapp processes, but pseudomentum is only well defined up to the edge of the first Brillouin zone.
  • #1
Niles
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0
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.
 
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  • #2
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.
 
  • #3
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?
 
  • #4
k=0 phonons are mentioned in the quote as the only phonons with actual momentum, which is 0.

But you say pseudo-momentum?
 
  • #5
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.
 
  • #6
kanato said:
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?
 
Last edited:
  • #7
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.
 

1. What are phonons and how are they related to lattices?

Phonons are quantized lattice vibrations that carry thermal energy through a solid material. They are closely related to lattices because they are the result of collective vibrations of atoms within a lattice structure.

2. How do phonons affect the properties of a material?

Phonons play a crucial role in determining the thermal, electrical, and optical properties of a material. The frequency and amplitude of phonons can greatly influence the overall behavior and performance of a material.

3. What is the significance of studying phonons on a lattice?

Studying phonons on a lattice provides valuable insight into the behavior of materials at a microscopic level. It helps us understand how thermal energy is transferred through materials and how it affects their physical properties.

4. Why is the work of Niles important in the field of phonon research?

Niles' work has greatly advanced our understanding of phonons on a lattice. His research has provided new insights into the behavior of phonons and their role in determining the properties of materials, leading to potential applications in various fields such as energy conversion and storage.

5. How can the knowledge gained from understanding phonons on a lattice be applied?

The understanding of phonons on a lattice has potential applications in areas such as thermoelectric materials, thermal management, and energy storage. It can also provide insights into the behavior of materials at extreme temperatures and conditions, aiding in the development of new technologies and materials.

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