Ewald sphere vs Brillouin Zone

[Darkmisc]

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

 

[DrDu]

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.

 

[Darkmisc]

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?

 

[johng23]

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.