Fundamentals of Plasma Physics 2025, 18 - Plasma Parameters
Not knowing any formal plasma physics while repeatedly needing to touch advanced concepts of plasma physics is starting to get to me, so I'm trying to check some basics off my list. For that purpose, I'm using P. Belian's "Fundamentals of Plasma Physics" from 2004. I'm expecting this to have at least some similarity to thermodynamics in structure,
A plasma is characterized by its fundamental parameters: particle density n, temperature of each particle species, and the steady state magnetic field B. From these, several parameters can be derived. As plasmas are a wide category, consisting mostly of difficult to observe or intuit, a general framework for building intuition can be found in the relation between Lorentz equations and Maxwell equations, one deriving trajectory and velocity from the electromagnetic fields, and one deriving the electric and magnetic fields from trajectory and velocity of each particle. The high particle number is usually the point of failure if this is the direct approach, so each approach can be heavily simplified by approximations.
There are then different approaches for slow and fast phenomena. Slow phenomena experience "Debye shielding". This assumes a finite-temperature, large plasma with equal and spatially uniform particle densities. Collisions are assumed to be rare enough to neglect. Each particle species σ is considered its own fluid with a density, temperature, pressure and mean velocity, so it gets its own collisionless e.o.m. In these, the inertial term and inductive electric fields are negligible, and temperature gradient and thermality over the plasma is flat (at equilibrium). With this, the Boltzmann equations can be written down. Its relation assumes that perturbation is very slow (for all species). The displacement of plasma particles is "shielding" a test particle, as it reduces the effectiveness of the test particle field. It's written through Poisson's equation with the source terms being the test particle and its associated cloud. The Boltzmann-character is retained as the test particle is introduced slowly, and this allows substitution of the particle number density, indirectly defining the Debye length.
If the Debye-length is at microscopic scales, then plasmas are close to neutral (but not exactly neutral).
Injecting a test particle into a plasma is expected to make random collisions with plasma particles, altering momentum and energy of the test particle and deflecting it at some angle relative to the initial velocity vector.
Grazing collision occurs outside of even small angle collisions. The larger the collision, the less likely it is. Grazing collisions are then the most likely collision and are expected to have a large cumulative effect. Follow the flux of field particles Γ defined by relative velocity between test and field particles, until the cumulative effect of small angle collisions is equivalent to a large angle one is 1≈Σθ2 = Γt∫2πb db[θ(b)]2 the integral of which is equivalent to the cross section for the cumulative effect of grazing collisions to equivalent to a single large angle scattering. The field of the scattering center is screened out for distances greater than the Debye-length, so small angle scattering only occurs below this.
The ratio between electrons and ions in a plasma affects the kinetic energy and momentum change of incident particles due to collisions between like and unlike particles. The momentum scattering is characterized by the time required for collisions to deflect incident particles by π/2, energy scattering is characterized by the time required for incident particles to transfer its full kinetic energy to a target particle. It separates into relative scalings of 1, (mi/me)1/2, mi/me.
| ~1 | ~(mi/me)1/2 | ~mi/me |
|---|---|---|
| νee | νii | νie |
| νei | νEii | νEei |
| νEee | νEie |
Weakly-ionized plasma adds collisions with neutrals, which differ due to the short-range interaction forces, so the neutral is effectively a hard-body with cross-section of the order of its actual geometry. The scattering can be either elastic or inelastic, depending on the energy transfer to the neutral. Simple transport phenomena can be divided into electrical resistivity in a uniform electric field E. Electrons and ions are accelerated in opposite directions creating a relative momentum between the species. e-i collisions dissipate relative momentum, so the state steadies out where relative momentum creation is balanced by the dissipation through interspecies collisions. Standard random walk leads to (ambipolar) diffusion, showing that particle diffusion coefficients scale with the characteristic step size squared, over the time delta between steps. For electrons, this is usually larger than the ion diffusion coefficients in an unmagnetized plasma. e-e collisions are exclused from this, since they are averaged over electrons with the same momentum.