The Ewald construction turns the Bragg condition into pure geometry. Every crystal has a reciprocal lattice of points Ghkl = h·a* + k·b* + l·c* (here a simple cubic lattice with a* = 2π/a). Draw a sphere of radius k = 2π/λ so that the incident wavevector's tail sits on the sphere and its tip touches the origin (000):
Diffraction condition: |G_hkl - (-k·ẑ)| = k
Equivalent Bragg law: λ = 2·d_hkl·sinθ, d_hkl = 2π/|G_hkl|
A reflection "fires" only at the instant a reciprocal lattice point crosses the sphere's surface — this is exactly the rotating-crystal method used at synchrotrons and lab diffractometers.
- Wavelength λ — shrinks or grows the Ewald sphere (k = 2π/λ); a shorter wavelength lets more distant reciprocal points reach the surface, so more reflections are accessible.
- Lattice constant a — sets the reciprocal spacing a* = 2π/a; a larger real crystal cell packs reciprocal points closer together.
- Crystal rotation γ — spins the reciprocal lattice about the vertical axis, sweeping points through the sphere shell one by one.
- Auto-rotate — continuously sweeps γ, reproducing the flashing sequence of spots seen on a rotating-crystal diffraction image.
Real-world relevance: this exact construction is how crystallographers plan data collection — choosing λ (via the X-ray source or monochromator) and rotation range so enough reciprocal lattice points pass through the sphere to solve the structure, from table salt to protein crystals.