Module scattering 

Classical elastic scattering in a screened-Coulomb potential.

Everything is in the reduced variables of crate::ion::potential: x = r/a, beta = b/a, eps = a E_cm / (Z1 Z2 e^2/(4 pi eps0)). The radial motion in the centre-of-mass frame obeys G(x) = 1 - phi(x)/(eps x) - beta^2/x^2, with turning point G(x0) = 0. The centre-of-mass scattering angle is (Goldstein; ZBL 1985, Ch. 2)

theta = pi - 2 beta * Int_{x0}^{inf} dx / (x^2 sqrt(G(x)))

Three evaluators are provided: direct Gauss-Mehler quadrature (theta_quadrature, accurate everywhere), the Biersack-Haggmark “magic formula” (theta_magic, fast and approximate), and a precomputed ScatteringTable for the hot path.

Structs

MagicConstants

Constants of the Biersack-Haggmark magic formula for one screening function.

ScatteringTable

Precomputed centre-of-mass scattering angle theta(eps, beta).

TableSpec

Specification of a ScatteringTable grid.

Constants

DEFAULT_NODES

Default number of Gauss-Mehler nodes for theta_quadrature.

Functions

closest_approach

Reduced distance of closest approach x0 for reduced energy eps and reduced impact parameter beta > 0.

nuclear_stopping_reduced

Reduced nuclear stopping cross section s_n(eps) = 2 eps Int_0^inf sin^2(theta/2) beta d beta from the quadrature angle (reduced units of ZBL 1985, Ch. 2; the energy transfer is T = gamma E sin^2(theta/2)). See nuclear_stopping_with.

nuclear_stopping_with

Reduced nuclear stopping cross section for an arbitrary angle function theta(beta) at fixed eps (quadrature, magic formula, table, …). Integrated over ln beta with Simpson’s rule, per_decade intervals per decade from beta_min to beta_max; the contributions below beta_min (~ beta_min^2) and above beta_max (screened, exponentially small) are neglected.

theta_magic

Biersack-Haggmark “magic formula” (Nucl. Instrum. Methods 174 (1980) 257) for the centre-of-mass angle:

theta_quadrature

theta_quadrature_n with DEFAULT_NODES nodes.

theta_quadrature_n

Centre-of-mass scattering angle (radians, in [0, pi]) by Gauss-Mehler quadrature with n nodes (Mendenhall and Weller, Nucl. Instrum. Methods B 58 (1991) 11; Gauss-Chebyshev rule of the first kind).