Function density_effect_single_oscillator 

pub fn density_effect_single_oscillator(
    beta_gamma_sq: f64,
    plasma_energy_ev: f64,
    mean_excitation_ev: f64,
) -> f64

Density-effect correction δ of a single-oscillator dielectric model.

Takes the squared product (βγ)², the target plasma energy ħω_p and the mean excitation energy I (both eV). Returns 0 below the threshold (βγ)² = (I/ħω_p)² and otherwise

δ = ln((βγ)² (ħω_p/I)²) − 1 + (I/ħω_p)² / (βγ)².

Source and status

This is the Fermi (Phys. Rev. 57, 485 (1940)) / Sternheimer (Phys. Rev. 88, 851 (1952)) dielectric formulation, which in the form given by Fano (Ann. Rev. Nucl. Sci. 13, 1 (1963)) reads δ = Σ f_i ln(1 + L²/ω_i²) − L² (1−β²)/(β² ω_p²), with L fixed by Σ f_i ω_p² / (ω_i² + L²) = 1/β² − 1. We specialise it to ONE oscillator (f = 1, ω_0 = I/ħ), where the constraint solves in closed form (L² = ω_p² (βγ)² − ω_0²), giving the expression above. The reduction was done by us and is not quoted from a paper; equation numbers of the sources are not cited because they were not checked. It is verified in the tests by its limits: δ → 0 continuously (value and slope) at threshold, and δ → 2 ln(βγ ħω_p/I) − 1 at large βγ, the standard high-energy form.

Validity

Exact in the high-energy limit when I is the true mean excitation energy. Near and above threshold a single oscillator is a crude stand-in for the real oscillator spectrum, and it contains no conductor (free-electron) terms, so the transition region is only approximate. It is zero below βγ = I/ħω_p, which is about 5 to 6 for Si (a proton of about 5 GeV). Real materials switch the effect on much earlier (around βγ ≈ 1.5 for Si in the published parameterisations), because their oscillator spectrum is spread; this model therefore UNDER-estimates δ through the transition and, in particular, is identically 0 over the whole range BetheBloch::validity advertises (up to 1 GeV/u). Its value is the correct asymptote for ultra-relativistic ions only.