Find the
invisible angle.
Point an X-ray beam at a crystal. Will a bright diffraction spot appear at every angle? No. Most orientations cancel the scattered waves. Let's find the one that makes them agree.
The equation can wait. First, watch the crystal and the light.
Why isn't every angle bright?
Start with a crystal and an X-ray beam. The atoms are ordered, but their scattered waves don't always point the same way. Watch what happens when we turn the crystal.
The (001) point lies on the Ewald sphere: Bragg condition met.
The curve is |kin + G001| − |kin|. Zero means the point lies on the sphere.
Why a sphere?
The momentum transfer is q = kout − kin. A periodic crystal reinforces scattering when q equals a reciprocal-lattice vector G. Elastic scattering also requires |kout| = |kin|. The Ewald sphere collects the outgoing wavevectors allowed by that last condition.
The diagram is scaled by 2π/d. The reciprocal vectors have integer coordinates, and the incident wavevector length is d/λ = 1/(λ/d). Moving the wavelength slider therefore changes the sphere's radius.
This is an ideal geometric construction for a simple-cubic lattice and monochromatic elastic scattering. Points are exact reciprocal-lattice points; the visual proximity band is not a physical peak width. We do not calculate structure factors, finite-crystal peak shapes, dynamical scattering, measured intensities or detector resolution.