# Critical Hall parameter (fully ionized gas)

## Description

The electrothermal instability (also known as the ionization instability) is a magnetohydrodynamic (MHD) instability appearing in magnetized non-thermal plasmas used in MHD converters. This instability is a turbulence of the electron gas in a non-equilibrium plasma (i.e. where the electron temperature Te is greatly higher than the overall gas temperature Tg). It arises when a magnetic field powerful enough is applied in such a plasma, reaching a critical Hall parameter βcr.

The critical Hall parameter βcr greatly varies according to the degree of ionization α :

α = n_i/n_nwhere ni is the ion density and nn the neutral density (in particles per cubic metre).

The electron-ion collision frequency νei is much greater than the electron-neutral collision frequency νen.

Therefore with a weak energy degree of ionization α, the electron-ion collision frequency νei can equal the electron-neutral collision frequency νen.

This formula calculates the Critical Hall parameter for a fully ionized gas (Coulombian plasma, when νei > νen) when combined with the formula :

“'s’ parameter for the Critical Hall parameter”

NB: The term “fully ionized gas”, introduced by Lyman Spitzer, does not mean the degree of ionization is unity, but only that the plasma is Coulomb-collision dominated, which can correspond to a degree of ionization as low as 0.01%

Related formulas## Variables

β_{cr} | Critical Hall parameter (fully ionized gas) (dimensionless) |

s | 's' parameter (see formula : "'s' parameter for the Critical Hall parameter") (dimensionless) |