Mathematical Background: Stuhrmann Decomposition
The scattering intensity is expressed as a sum over spherical harmonic channels (truncated at l = 50, giving 2601 modes):
I(q) = 4π · Σ_{l=0}^{50} Σ_{m=-l}^{l} |B_lm(q)|²
where the coefficients B_lm(q) are computed by projecting atomic scattering onto the spherical harmonic basis:
B_lm(q) = Σ_i f_i(q) · j_l(q·rᵢ) · Y*_lm(θᵢ, φᵢ)
f_i(q): complex atomic form factor (f₀ + f₁ + i·f₂) at momentum transfer q
j_l: spherical Bessel function of order l; encodes radial shell information
Y_lm: complex spherical harmonic; * denotes conjugate (analysis projection)
rᵢ, θᵢ, φᵢ: spherical coordinates of atom i relative to the molecular centroid
The 3D surface visualises the angular scattering envelope:
r(θ,φ) = 1 + Re( Σ_{l,m} B_lm(q) · Y_lm(θ,φ) ) / max|B_lm(q)|
Deformations from a unit sphere indicate anisotropic scattering at that q. Coefficients are normalised to unit amplitude so the shape reflects anisotropy, not absolute intensity.
Complexity: O(N · L²) vs O(N²) for the Debye sum, orders of magnitude faster for large molecules.