Module lindhard_scharff

Expand description

Lindhard-Scharff electronic stopping.

J. Lindhard and M. Scharff, Phys. Rev. 124, 128 (1961): at ion velocities below v0 Z1^(2/3) the electronic stopping is proportional to velocity.

§Reduced units (Lindhard, Scharff, Schiott, Mat. Fys. Medd. 33 (14), 1963)

screening length a = 0.8853 a0 (Z1^(2/3) + Z2^(2/3))^(-1/2) (Thomas-Fermi, universal-length form; a0 is the Bohr radius),
reduced energy ε = E a M2 / (Z1 Z2 e² (M1 + M2)), with e² meaning e²/(4π ε0),
reduced path length ρ = N x 4π a² M1 M2 / (M1 + M2)²,
reduced stopping (dε/dρ)_e = k_L ε^(1/2), with k_L = ξ_e 0.0793 Z1^(1/2) Z2^(1/2) (A1+A2)^(3/2) / ((Z1^(2/3)+Z2^(2/3))^(3/4) A1^(3/2) A2^(1/2)).

ξ_e = Z1^(1/6) is the Lindhard-Scharff (1961) velocity-proportionality factor (ξ_e ≈ Z1^(1/6), order 1 to 2). Deviation from the issue text: the formula quoted there omits ξ_e; without it the Z1 dependence does not agree with the dimensional LS form below (off by Z1^(1/6)), so it is included here. The 0.0793 is the rounded coefficient; a test checks the whole expression against the dimensional form S_e = Z1^(1/6) 8π e² a0 Z1 Z2 / (Z1^(2/3)+Z2^(2/3))^(3/2) · v/v0 to 1 %.

A per-element multiplicative correction f(Z2) may be attached (LindhardScharff::with_correction); it defaults to 1. It must be finite and non-negative (zero switches the element’s stopping off); anything else is rejected at evaluation time with StoppingError::InvalidParameter.

Structs§

LindhardScharff: Lindhard-Scharff model with optional per-element correction factors.

Functions§

k_l: The Lindhard-Scharff coefficient k_L including ξ_e = Z1^(1/6) (see module docs).
reduced_energy: Reduced energy ε for ion energy energy_ev on a target of mass m2_amu.
reduced_to_si_factor: Factor converting reduced stopping dε/dρ to a cross section in J m².
screening_length: Lindhard-Scharff screening length a, m.