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Collision cross sections

Important notes on cross sections

A good introduction about the topic of (physical) cross sections can be found on If extending or using the cross section database of PIC-MC some terminology issues should be taken into account. Here are the most common terms and definitions:

  • Differential scattering cross section
    • Energy and angular dependant elastic or inelastic scattering cross section σ(e,θ).
    • Measuring angular scattering distributions seems very demanding. Hence, accordant data is hard to find on the web.
  • Angular integrated or “total” scattering cross section
  • Integration of σ(e,θ) over all angles θ ([0,π]) yields σ(e).
  • Using an accordant formular for the energy dependant scattering anisotropy on this cross section yields the momentum transfer cross section.
  • Momentum transfer cross section
    • Elastic or inelastic momtentum transfer is not the same as elastic or inelastic scattering.
    • You can have a scattering event but the transferred momentum depends on the scattering anisotropy.
    • Momentum transfer cross sections can be obtained from swarm analysis experiments together with the use of Boltzmann equation solvers.
    • The compilations on SIGLO and JILA consists of elastic and inelastic momentum transfer cross sections (not total scattering as I once thought).
  • Effective momentum transfer cross section
    • This is the sum of all, elastic and inelastic, momentum transfer cross sections.
    • Important: We need the elastic momentum transfer cross sections otherwise we double count the effect of inelastic momentum transfer.
    • Important: Most Boltzmann equation solvers use the effective momentum transfer. Thus, you mainly find effective momentum transfer cross sections, e.g. on JILA and SIGLO.
    • Important: Some compilations, e.g. SIGLO, wrongly labeled effective momentum transfer with elastic momentum transfer. These inconsistencies are sometimes mentioned and sometimes not.
  • Scattering anisotropy
  • If there are no differential cross sections available, scattering anisotropy and angular distribution, respectively can be approximated.
  • Check Okhrimovskyy 1) on how to approximate anisotropic electron scattering based on a screened Coulomb potential.
  • Important: For the approach of Okhrimovskyy02 et. al. the total elastic scattering cross section and not the elastic momentum transfer cross section must be used.

Energy sampling in cross section array

Previously we used a linear sampling for the energy in the cross section array, i.e. E(i) = E0 · i / N, whereas E0 is the maximum energy in eV and N is the array resolution. However, using a linear sampling scheme we had to distinguish two simulation cases: (i) DSMC gas flow simulation and (ii) PIC-MC plasma simulation. This was because a higher energy resolution in the low energy regime is needed to correctly compute gas flow dynamics and neutral collisions, respectively. To get rid of this limitation we could either increase the array resolution or use a higher order sampling scheme. The latter case was more pratical. Thus, we chose a second order sampling, i.e. E(i) = E0(i / N)², which gives a better resolution in the low energy regime. The crucial range hereby is the thermal energy around 25 meV, which is sufficiantly resolved using the second order sampling as seen in the figure below.

Energy sampling in cross section array
First and second order sampling of energy in cross section array

Figure 1: First order sampling for dsmc and pic-mc simulation (black and red curve) and second order sampling (blue curve) to combine dsmc and pic-mc simulation.

Cross section database

Database of the the PIC-MC framework

A. Okhrimovskyy: Electron anisotropic scattering in gases: A formula for Monte Carlo simulations, Phys. Rev. E Vol. 65 Nbr. 3, 2002