Why is the change in momentum of a crystal include reciprocal lattice vectors?

In summary, the change in momentum of a crystal involves a reciprocal lattice vector because we are dealing with crystal momentum, not true momentum. Phonons do not carry true momentum, only crystal momentum. The book by Ashcroft & Mermin is a good resource for understanding this concept. Previous discussions on this topic can also be found on the Physics Forums website.
  • #1
cooev769
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So i don't really understand why the change in momentum of a crystal involves a reciprocal lattice vector. Surely it is just the change in momentum due to the change in the number and frequency of the phonons before and after whatever event/scattering/collision takes place. Can somebody please explain this to me in as analogous to classical mechanics way as possible, not just a purely mathematical explanation please. I'm not proficient with Fourier transformations momentum spaces etc. Thanks ;)
 
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  • #2
This is a tricky question. The first important step is to realize that you are talking about crystal momentum, not true momentum. E.g. phonons do not carry true momentum but only crystal momentum.
The book by Ashcroft & Mermin is a good source on this topic.
We also had an interesting discussion in this forum before:
https://www.physicsforums.com/threads/do-phonons-conserve-momentum-or-not.611109/
 

FAQ: Why is the change in momentum of a crystal include reciprocal lattice vectors?

Why is momentum important in crystal structures?

Momentum is important in understanding the behavior and properties of crystal structures because it describes the movement of particles within the crystal lattice. The momentum of a particle can affect its position, energy, and interactions with other particles in the lattice, which ultimately determines the physical and chemical properties of the crystal.

What is the relationship between crystal lattice vectors and momentum?

The crystal lattice vectors represent the periodic arrangement of atoms in a crystal structure. Momentum, on the other hand, is a measure of the motion of particles within the crystal. The change in momentum of a crystal includes reciprocal lattice vectors because they describe the periodicity of the crystal, and any change in momentum must also take into account the periodicity of the crystal lattice.

How do reciprocal lattice vectors affect the change in momentum of a crystal?

Reciprocal lattice vectors are important in describing the periodicity of the crystal structure. When there is a change in momentum within the crystal, it affects the periodicity of the lattice, and thus, must also be described by the reciprocal lattice vectors. This allows for a more accurate understanding of the dynamics and properties of the crystal.

Can the change in momentum of a crystal be measured or observed?

Yes, the change in momentum of a crystal can be measured or observed through various experimental techniques, such as X-ray diffraction, neutron diffraction, and electron diffraction. These techniques allow for the analysis of the crystal lattice and its reciprocal lattice vectors, providing information about the change in momentum within the crystal.

Why is it important to consider reciprocal lattice vectors when studying crystal structures?

Reciprocal lattice vectors provide essential information about the periodicity and symmetry of a crystal structure. By considering these vectors, scientists can better understand the behavior and properties of crystals, as well as make predictions about their physical and chemical properties. In addition, reciprocal lattice vectors are crucial in the design and development of new materials with specific properties, such as semiconductors and superconductors.

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