Use the Ewald sphere to calculate h,k,l?

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Discussion Overview

The discussion revolves around the use of the Ewald sphere in calculating the Miller indices (h, k, l) and the lattice constant (a) in crystallography. Participants explore the relationship between the Ewald sphere, the lattice structure, and diffraction patterns, particularly in the context of cubic lattices.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether the Ewald sphere can be used solely to calculate the lattice constant "a" without knowing the Miller indices (h, k, l) or the interplanar spacing (d_hkl).
  • Another participant asserts that it is necessary to know either the lattice constant or the interplanar spacing to use the Ewald sphere effectively, drawing an analogy to needing grating spacing to determine wavelength.
  • A later reply emphasizes that the Ewald sphere is fundamentally linked to the wavelength of the incident radiation and does not provide information about the crystal structure itself.
  • One participant explains that the Ewald sphere represents energy conservation in scattering, indicating that momentum conservation can be altered by reciprocal lattice vectors due to the periodicity of the crystal lattice.

Areas of Agreement / Disagreement

Participants express differing views on the utility of the Ewald sphere for determining lattice constants and Miller indices. There is no consensus on whether the Ewald sphere can provide information about the crystal structure without additional data.

Contextual Notes

Participants highlight the dependence on specific definitions and assumptions regarding the relationship between the Ewald sphere, lattice constants, and Miller indices. The discussion remains open regarding the implications of these relationships.

Lars Ph
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TL;DR
Is there a way to use the ewald sphere to calculate h,k,l?
I am unsure wether or not all I can use the Ewald sphere for is to calculate d_hkl for the diffracted wave vector. For cubic lattices for example d= a/sqrt(h^2+k^2+l^2). To determine the lattice constant "a" you would then need to know exactly what your h,k and l are or you use lattice-dependent requirements like "h+k+l are an even or odd number" in conjunction with the determined d_hkl to find your a (I think). My question is, is there a different way to find the lattice constant solely by using the Ewald sphere?
 
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You mean without knowing d ? No. You need to know either a to find d or vice versa. This is the same as needing to know the grating spacing to determine the wavelength for light You also need to know the defracted order. In 3D, it is a bit more complicated but the physics is the same.
 
Lars Ph said:
TL;DR Summary: Is there a way to use the ewald sphere to calculate h,k,l?

I am unsure wether or not all I can use the Ewald sphere for is to calculate d_hkl for the diffracted wave vector. For cubic lattices for example d= a/sqrt(h^2+k^2+l^2). To determine the lattice constant "a" you would then need to know exactly what your h,k and l are or you use lattice-dependent requirements like "h+k+l are an even or odd number" in conjunction with the determined d_hkl to find your a (I think). My question is, is there a different way to find the lattice constant solely by using the Ewald sphere?
The Ewald sphere depends only on the wavelength of the incident radiation so it has nothing to do with the lattice. If you look at different crystals with the same radiation the Ewald sphere is always the same. That means that it must be impossible to deduce anything about the crystal from the Ewald sphere alone.
 
Yes. The Ewald sphere is simply a statement about energy conservation in the scattering process where the periodicity of the (~infinitely massive) crystal lattice has allowed momentum to not be strictly conserved (it can change by any reciprocal lattice vector) .
 

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