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I am trying to understand diffraction a little better and eventually Kikuchi lines. I am confused about something -- namely the difference between the Ewald sphere and the so-called sphere of reflection -- and the role of Brillouin zone boundaries.

In order to satisfy constructive interference conditions due to (elastic) diffraction we must conserve energy -- so the incident beam and diffracted beam must have the same amplitude: $|k_i| = |k_f|$.

We also know that if the path difference between a $k_i$ and $k_f$ is equal to a reciprocal lattice vector G we get constructive interference.

If we combine these conditions we get: $2 k_i \cdot G= G^2$ which describes Brillouin zone boundaries -- since in essence we are bisecting $G$. Does this mean that every Ewald Sphere is centred at a Broullin Zone Boundary?

My final goal is to understand the difference between the Ewald Sphere and the Sphere of Reflection which explains kikuchi lines.

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I don't think it necessary has to be.

The Laue's rule describes Brillouin zone boundary in the sense that it describes the Bragg planes and their intersections define the BZ boundaries. But the Bragg planes can extend outside the BZ. And you can use any points on any region of these planes as centres for the Ewald spheres (i.e. the diffraction condition is met by a vector from BZ origin to any point on the Bragg planes).

I think the Ewald sphere can be centred at the BZ boundaries but it doesn't have to.

BZvsEwaldSphere

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