In this book chapter, particle collisions are examined. Electrons and ions can experience elastic collisions by preserving total momentum and energy. Otherwise, they lose their energy in the form of ionization or excitation and this type is named inelastic collision. Electrons and fully stripped ions have only kinetic energy but excited and ionized atoms possess internal energy, analogous to potential energy. While their internal energy is constant, kinetic energy is redistributed between the colliding particles in elastic collisions. For the super elastic collisions, an excited atom can be de-excited and the total kinetic energy becomes larger.
Particle collisions are explored by defining cross section area (σ), mean free path (λ) and collisional frequency (ν). The start point is the particle flux (Γ) describing the density of the incoming particles times their velocity. For an imaginary case, the incident particles collide with the target particles having infinite mass and density, ρ. For the collisions, particles are assumed to be hard spheres. The interaction cross section describes an area defined by a circle with radius, a1+a2. Particle fluxes can be obtained in terms of σ and the number density of gas. Mean free path represents a distance that particles can pass through without a collision and it can be written as 1/ρσ. Collision frequency is calculated regarding the mean free path and the particle velocity. However, we don’t know yet the directions of the particles after they scattered. Therefore, differential scattering cross section (I) is introduced with impact parameter, b, and scattering angle, θ. Total scattering cross section (λ) and momentum scattering cross section (λ) are equal for hard spheres, πa12^2. For the atom-electron collisions, we can ignore the radius of an electron. If both particles are moving, we should define a center of mass coordinate system to handle the velocities and the scattering angles. The distances and masses are written in new coordinate system. That helps to get differential cross section in terms of the relative velocity (v1-v2) and the scattering angle θ1.
Elastic collisions are investigated to get an expression for the energy transfer. First, momentum equations are written in cartesian coordinates, then the energy equation is written for a moving particle and a stable particle. By eliminating some terms appearing in the equations, the fraction of energy lost is obtained by averaging the differential scattering cross sections for the momentum cross section and the total cross section. If we consider a small angle after scattering, then we can redefine the distance from center of mass by assuming a straight trajectory line. For this calculation, time term is included to calculate distance. The force at the center is stated by scaling the distance term, r. The momentum terms in parallel and perpendicular directions are scaled to get the scattering angle (Θ) for the transformed system. The impact parameter b is written in terms of that angle. The differential scattering cross section is obtained for small angles with Θ and the kinetic energy of the center of mass.
Reference: Lieberman, Michael A., and Alan J. Lichtenberg. Principles of plasma discharges and materials processing. John Wiley & Sons, 2005.
Course: AME 60637