# Mathematical reference
These pages define the mathematical models currently implemented by the
published Rust crates. Equations are rendered as plain, copyable text so the
reference works on docs.rs and in offline `cargo doc` output without external
JavaScript or a math-rendering service.
## Coverage
| [`peak_profiles`] | normalized pseudo-Voigt, TCH mixing, CW broadening, wavelength components, FCJ asymmetry, neutron TOF |
| [`crystallography`] | cell/reciprocal metrics, exact symmetry, reflection conditions, scattering factors, displacement factors, structure factors, LP corrections, Bragg positions |
| [`pattern_composition`] | multi-phase and multi-wavelength sums, size, microstrain, preferred orientation, derivative products |
| [`backgrounds`] | Smooth Bruckner preprocessing, polynomial, Chebyshev, point, amorphous, and composite backgrounds |
| [`refinement`] | residuals, R factors, constraints, Le Bail redistribution, damped Gauss–Newton, covariance, and quantitative phase analysis |
Every section names the Rust module or type that owns the equation. Models not
implemented by the current API—such as magnetic scattering, absorption,
extinction, and symmetry-refined anisotropic displacement—are not described as
available mathematics.
## Shared notation
- `x` is a sampled pattern coordinate.
- `φ = 2θ` is a diffraction angle; angular Rust APIs use degrees unless a
formula explicitly says radians.
- `λ` and direct-cell lengths are in ångströms.
- `h = (h,k,l)^T` is a Miller-index column vector.
- `G` and `G* = G^-1` are direct and reciprocal metric tensors.
- `q² = h^T G* h = 1/d²`; `s = sqrt(q²)/2 = sin(θ)/λ`.
- `J v` and `J^T u` denote Jacobian-vector and transpose-Jacobian-vector
products. They are analytical products, not finite differences.
[`peak_profiles`]: crate::guide::mathematics::peak_profiles
[`crystallography`]: crate::guide::mathematics::crystallography
[`pattern_composition`]: crate::guide::mathematics::pattern_composition
[`backgrounds`]: crate::guide::mathematics::backgrounds
[`refinement`]: crate::guide::mathematics::refinement