3D Helmholtz Far-Field Scattering

This module visualises the far-field scattering pattern of monochromatic X-rays from a finite 3D crystal. The crystal is built from N×N×N unit cells centered at the origin, illuminated by a Gaussian beam, and the far-field intensity is displayed on a sphere.

Physics

Under the first Born approximation the far-field scattered amplitude in direction k̂out factorises as

where Scell is the unit-cell structure factor and L is the Gaussian-weighted lattice sum. For orthogonal unit cells, L factors into three independent 1D sums (O(3N) instead of O(N³)).

Visualisation

The scattering pattern is computed on a cube-map grid (6 faces) to avoid latitude/longitude distortions, then mapped as textures onto a sphere in the Three.js 3D scene. Atom positions from the crystal structure are shown as small spheres at the origin. The incoming beam is shown as a cylinder with Gaussian transparency matching the beam profile. Colours use the viridis colourmap on either a log-intensity or linear-intensity scale.