Module bethe 

Nonrelativistic Bethe stopping, the high-energy reference for the dielectric models.

The Bethe equation as given by the Particle Data Group, S. Navas et al., Phys. Rev. D 110, 030001 (2024), review 34 “Passage of particles through matter” (D. E. Groom and S. R. Klein, rev. August 2023), eq. (34.5):

-dE/dx = K z² (Z/A) (1/β²) [ (1/2) ln(2 m_e c² β² γ² W_max / I²) - β² - δ(βγ)/2 ].

Here it is used nonrelativistically and for a unit charge (z = 1, γ -> 1, the -β² and density-effect terms dropped, m_e c² β² = m_e v² = 2E for an electron of kinetic energy E), and as a linear stopping power for an electron density n (the K (Z/A) ρ of the mass stopping power is 4π n r_e² m_e c²):

S(E) = (4π n (e²/4πε₀)² / (m_e v²)) · (1/2) ln(2 m_e v² W_max / I²)
     = (2π n (e²/4πε₀)² / E) · (1/2) ln(4 E W_max / I²).

W_max is the largest energy loss the model allows. The PDG gives eq. (34.5) for heavy particles, where W_max is the kinematic maximum transfer to a free electron; for an incident electron the recommended stopping formula differs (identical-particle exchange). The single-pole Penn model has no exchange and allows losses up to the full kinetic energy (W_max = E), so its high-energy limit is the expression above with W_max = E, S = (2π n (e²/4πε₀)² / E) ln(2E/I). That is the comparison this module exists for; it is not a recommended electron stopping power.

Functions

stopping_power_ev_per_m

Nonrelativistic Bethe stopping power of a unit-charge particle of electron mass (see the module docs), eV/m.