Module bethe 

Bethe-Bloch electronic stopping (relativistic) with Bloch and optional user-supplied shell/density corrections, and a named effective-charge model.

Formula

Per target atom (Bethe 1930/1932; Fano, Ann. Rev. Nucl. Sci. 13, 1 (1963); PDG review “Passage of particles through matter”, the same equation):

S = 4π (e²)² z² Z2 / (m c² β²) · [ ½ ln(2 m c² β² γ² W_max / I²) − β² − δ/2 − C/Z2 + L1 + L2 ]

with e² meaning e²/(4π ε0), z the projectile effective charge, W_max = 2 m c² β²γ² / (1 + 2γ m/M + (m/M)²), I the mean excitation energy, δ the density-effect correction and C the shell correction.

• L2 (Bloch 1933, Ann. Phys. 408, 285): the exact series L2 = −y² Σ_{n≥1} 1 / (n (n² + y²)), y = z α / β.
• I defaults to the Bloch rule I = 10 eV · Z2 (Bloch 1933). This is a rough estimate, good to about 20 % in I (about 2 % in S); pass a measured I (cited) with BetheBloch::with_mean_excitation_ev.
• δ and C default to 0 and can be set by the caller as constants if they come from a cited source. δ can instead be computed with the opt-in single-oscillator model, see density_effect_single_oscillator and BetheBloch::with_density_effect.

Omitted terms

The Barkas term L1 (Ashley-Ritchie-Brandt 1972) needs a tabulated function, and the usual shell-correction parameter sets and the multi-oscillator density-effect parameter sets are published as tables (ICRU 49/37 and Sternheimer-Berger-Seltzer), which this project may not ingest. They are not implemented. See docs/stopping-models.md for the literature findings; expect errors of a few percent in the range 1 to 10 MeV/u until they are added.

Effective charge

EffectiveCharge::BarkasEmpirical: z_eff = Z1 [1 − exp(−125 β / Z1^(2/3))] (W. H. Barkas, Nuclear Research Emulsions I, Academic Press (1963); quoted in Northcliffe, Ann. Rev. Nucl. Sci. 13, 67 (1963)).

Structs

BetheBloch

Bethe-Bloch model.

Enums

EffectiveCharge

Projectile effective-charge model.

Functions

bloch_term

Bloch correction L2(y), y = z α / β (see module docs).

density_effect_single_oscillator

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

plasma_energy_ev

Plasma energy ħ ω_p = ħ sqrt(n e² / (ε0 m)) in eV for an electron density n in m^-3 (the free-electron plasma frequency).