Ewald sphere vs Brillouin Zone

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Discussion Overview

The discussion centers on the relationship and differences between the Ewald sphere and the Brillouin Zone (BZ) in the context of reciprocal space representations, particularly regarding the condition k'=k+G. Participants explore how each construct conveys information about momentum conservation in periodic potentials and whether they provide distinct insights.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether both the Ewald sphere and BZ represent the condition k'=k+G and seeks clarification on their differences and the information they convey.
  • Another participant explains that the Ewald sphere is a sphere in reciprocal lattice space around point k, while the BZ is a volume that depends on the crystal structure and contains essential reciprocal space points.
  • It is noted that the Ewald sphere is a surface whose size is determined by the wavelength of the diffracting radiation, whereas the BZ represents a volume in reciprocal space.
  • A participant asserts that k'=k+G is a general statement of momentum conservation applicable to both constructs, but neither should be viewed solely as an illustration of this condition.

Areas of Agreement / Disagreement

Participants express differing views on whether the Ewald sphere and BZ convey the same information regarding k'=k+G. There is no consensus on the significance of their differences or the implications for understanding momentum conservation in periodic potentials.

Contextual Notes

Participants highlight that the Ewald sphere and BZ serve different roles in reciprocal space, with the Ewald sphere being a surface and the BZ being a volume, but the implications of these differences remain unresolved.

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Is it correct to say that the Ewald sphere and Brillouin Zone are both representations of k'=k+G?

I'm comfortable with the construction of the Ewald sphere, but don't quite see how a BZ represents k'=k+G.

Can anyone explain how the construction of a BZ represents k'=k+G and whether it provides any information that one can't represent with an Ewald sphere?


Thanks
 
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I am not quite sure if I understand your question. The Ewald sphere E is a sphere in the reciprocal lattice around the point k with radius |k|, i.e. all vectors q=k'-k with |k'|=|k|. Obviously, it is possible to fold back the Ewald sphere into the first Brillouin zone. The vectors on q of E which coincide with a lattice vector G are mapped onto the origin 0 of the Brillouin zone.
 
What I mean to ask is whether the Ewald sphere and BZ both represent the condition that k'=k+G.

If so, why do they look different?

Do they convey different information?
 
The Brillouin zone depends only on the crystal structure, and contains the minimum number of reciprocal space points which are required to fully characterize the system's behavior in momentum space. (That may not be the most clear way to state it). The Brillouin zone represents a volume in reciprocal space.

The Ewald sphere is only a surface in reciprocal space, and the interior volume does not have much significance to the construction. Its size depends only on the wavelength of the diffracting radiation (ie x-rays).

k'=k+G is a general statement of momentum conservation in a periodic potential, and as such it is going to show up in either construct. I don't consider either construct to simply be an illustration of k'=k+G.
 

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