Improved analytic fits of collisional cross-sections

Computing collisional cross-sections σ(ɛ), where ɛ is the energy of the incident electron, is costly. Furthermore, for collisional-radiative models the thermally averaged cross-sections 〈σ〉Te integrated over the distribution function of all electron energies is needed, so that one generally fits the collision strength Ω(ɛ) for 5–20 energies ɛi before an analytical or numerical integration is performed. However, in multi-charged, multi-electron ions, it is usual to find poor fits, which can lead to negative rates, for about 20% of the excitation transitions when one employs the classical Goett et al. fit (GCS)Ω(u=ɛ/ΔE)=A+Dln(u)+c1/(C+u)+c2/(C+u)2even when using the Bethe limit, also called Born or Coulomb-Born limit, at high energy as a constraint. From examples, we shall derive the requirement for an adequate fit and propose a tractable and efficient algebra to replace the GCS formula.