Editing Langmuir probe (section)

===Ion saturation current density===

Consider first a surface biased to a large negative voltage. If the voltage is large enough, essentially all electrons (and any negative ions) will be repelled. The ion velocity will satisfy the [[Bohm sheath criterion]], which is, strictly speaking, an inequality, but which is usually marginally fulfilled. The Bohm criterion in its marginal form says that the ion velocity at the sheath edge is simply the sound speed given by

<math> c_s = \sqrt{k_B(ZT_e+\gamma_iT_i)/m_i}</math>.

The ion temperature term is often neglected, which is justified if the ions are cold. ''Z'' is the (average) charge state of the ions, and <math>\gamma_i</math> is the adiabatic coefficient for the ions. The proper choice of <math>\gamma_i</math> is a matter of some contention. Most analyses use <math>\gamma_i=1</math>, corresponding to isothermal ions, but some kinetic theory suggests that <math>\gamma_i=3</math>. For <math>Z=1</math> and <math>T_i=T_e</math>, using the larger value results in the conclusion that the density is <math>\sqrt{2}</math> times smaller. Uncertainties of this magnitude arise several places in the analysis of Langmuir probe data and are very difficult to resolve.

The charge density of the ions depends on the charge state ''Z'', but [[Plasma (physics)#Plasma potential|quasineutrality]] allows one to write it simply in terms of the electron density as <math>q_e n_e</math>, where <math>q_e</math> is the charge of an electron and <math>n_e</math> is the number density of electrons.

Using these results we have the current density to the surface due to the ions. The current density at large negative voltages is due solely to the ions and, except for possible sheath expansion effects, does not depend on the bias voltage, so it is
referred to as the '''ion saturation current density''' and is given by

<math>j_i^{max} = q_{e}n_ec_s</math> where <math>c_s</math> is as defined above.

The plasma parameters, in particular, the density, are those at the sheath edge.