# Are the Laue equations only satisfied when the reciprocal lattice point touches the Ewald sphere?

I've derived the Laue conditions from the requirements for constructive interference in a crystal lattice ( J. Drenth (2007), Principles of X-Ray Crystallography 3rd Edition ) and also then equivalated the Laue conditions to the Bragg conditions for diffraction.

From what I understand, the reciprocal lattice gives every possible point where diffraction is possible in a crystal (i.e. when constructive interference between unit cells occurs). Since this is equivalent to the Bragg conditions, shouldn't every single reciprocal lattice point be visible in the diffraction pattern? Why is it that only points that lie on the Ewald sphere meet the conditions for diffraction?

• The answer is rather nicely explained in its Wikipedia article en.wikipedia.org/wiki/Ewald%27s_sphere Commented Apr 20, 2023 at 15:59
• @naturallyInconsistent So can I say that the diffraction conditions are given by the Laue conditions and fulfilled by all points on the reciprocal lattice, but the Ewald sphere adds on the energetic constraints of elastic scattering? Commented Apr 20, 2023 at 16:55
• Yes, but it is better to be approximately correct in a way that is indistinguishable from experiment than trying to get the complete answer in a way that is infinitely more tedious and most of the new contributions are unimportant, leading to a risk of forgetting to evaluate the important contributions first. Commented Apr 20, 2023 at 17:01

Don't forget that the Bragg conditions also depend on the wavelength of the X-ray probe. The diffraction of a beam with a particular wavelength admits Bragg's famous geometrical relation. Not all reciprocal lattice points (i.e. all crystal planes) can be in the diffraction condition at the same time. Only by rotating the sample you can make other points meet the diffraction criteria, that is, hit Ewald's sphere.

This is due to energy being conserved in the scattering process. Laue equation state that the change in momentum must lie in the reciprocal lattice, that is, $$k_{out}-k_{in} = G$$ where $$G$$ is a vector of the reciprocal lattice. In elastic scattering, $$|k_{out}| = |k_{in}|$$. If you fix the origin and magnitude of $$k_{in}$$ (or $$k_{out}$$), then all the possibilities for $$k_{out}-k_{in}$$ describe a sphere. This is Ewald's sphere, and Laue equation can only be satisfied when a lattice point belongs to it.