Module doppler 

Doppler broadening via the Free Gas Model (FGM).

The FGM treats target atoms as a free ideal gas at temperature T. The Doppler-broadened cross-section is obtained by averaging the unbroadened cross-section over the Maxwell-Boltzmann velocity distribution of the target atoms.

SAMMY Reference

• Manual Section III.B.1 (Free-Gas Model of Doppler Broadening)
• fgm/mfgm1.f90 subroutine Dopfgm (quadrature in mfgm2.f90 Modsmp/Modfpl)

Method

We implement the exact FGM integral in velocity space (SAMMY Eq. III B1.7), including its w/v integrand weight:

v²·σ_D(v²) = (1/(u√π)) ∫ exp(-(v-w)²/u²) · w² · s(w) dw

where v = √E, u = √(k_B·T / AWR), and: s(w) = σ(w²) for w > 0 s(w) = -σ(w²) for w < 0

This is the same kernel weighting as SAMMY’s Dopfgm, which multiplies the normalized Gaussian quadrature weights by w² and divides the integral by E = v² (fgm/mfgm2.f90 Modsmp/Modfpl Wts·Velcty**2, mfgm4.f90 val/Em). The quadrature itself differs: NEREIDS integrates the Gaussian exactly over piecewise-linear segments of Y, while SAMMY uses the Modsmp/Modfpl point rules — both discretize the same Eq. III B1.7 integral. Two analytic consequences (both pinned by kernel_error_scales_pinned_vs_full_fgm_reference): a constant σ is broadened to σ·(1 + u²/2v²) — the physical low-energy upturn — and a 1/v cross-section is preserved exactly. (An earlier revision omitted the w/v weight, which skewed Doppler-broadened resonance flanks by a first-order ~u/v; the pinning test fails loudly on any regression to that kernel.)

The key advantage of the velocity-space formulation is that u is independent of energy, making it a true convolution.

Doppler Width

The SAMMY Doppler width at energy E is: Δ_D(E) = √(4·k_B·T·E / AWR)

Structs

DopplerParams

Doppler broadening parameters.

Enums

DopplerError

Errors from Doppler broadening computation (not parameter construction).

DopplerParamsError

Errors from DopplerParams construction.

Functions

doppler_broaden

Apply FGM Doppler broadening to cross-section data.

doppler_broaden_with_derivative

Doppler-broaden cross-sections AND compute the analytical temperature derivative ∂σ_D/∂T in a single pass.