Function doppler_broaden 

pub fn doppler_broaden(
    energies: &[f64],
    cross_sections: &[f64],
    params: &DopplerParams,
) -> Result<Vec<f64>, DopplerError>

Apply FGM Doppler broadening to cross-section data.

The cross-sections are broadened in velocity space using the exact Free Gas Model integral from SAMMY manual Eq. III B1.7.

Arguments

• energies — Energy grid in eV (must be positive and sorted ascending).
• cross_sections — Unbroadened cross-sections in barns at each energy point.
• params — Doppler broadening parameters (temperature and AWR).

Returns

Doppler-broadened cross-sections in barns on the same energy grid.

Algorithm

1. Convert energy grid to velocity space (v = √E).
2. Build extended grid including negative velocities for the FGM integral.
3. Compute the integrand Y(w) = w · s(w) on the extended grid.
4. For each output velocity, evaluate the Gaussian convolution integral.
5. Transform back: σ_D(E) = result / √E.