Editing Langmuir probe (section)

=== Electron saturation current ===

When the electrode potential is equal to or greater than the plasma potential, then there is no longer a sheath to reflect electrons, and the electron current saturates. Using the Boltzmann expression for the mean electron velocity given above with <math>v_{e0} = 0</math> and setting the ion current to zero, the '''electron saturation current density''' would be

<math>
j_e^{max} 
= j_i^{max}\sqrt{m_i/\pi m_e} 
= j_i^{max} \left( 24.2 \, \sqrt{\mu_i} \right)
</math>

Although this is the expression usually given in theoretical discussions of Langmuir probes, the derivation is not rigorous and the experimental basis is weak. The theory of [[Double layer (plasma)|double layers]]<ref>{{cite journal |author=Block, L. P. |date=May 1978 |title=A Double Layer Review |journal=Astrophysics and Space Science |volume=55 |issue=1 |pages=59–83 |bibcode=  1978Ap&SS..55...59B|url=http://articles.adsabs.harvard.edu//full/seri/Ap+SS/0055//0000065.000.html |access-date=April 16, 2013 |doi=10.1007/bf00642580|s2cid=122977170 }} (Harvard.edu)</ref> typically employs an expression analogous to the [[Debye sheath#The Bohm sheath criterion|Bohm criterion]], but with the roles of electrons and ions reversed, namely

<math>
j_e^{max} 
= q_en_e \sqrt{k_B(\gamma_eT_e+T_i)/m_e}
= j_i^{max}\sqrt{m_i/m_e} 
= j_i^{max} \left( 42.8 \,  \sqrt{\mu_i} \right)
</math>

where the numerical value was found by taking ''T''<sub>''i''</sub>=''T''<sub>''e''</sub> and γ<sub>''i''</sub>=γ<sub>''e''</sub>.

In practice, it is often difficult and usually considered uninformative to measure the electron saturation current experimentally. When it is measured, it is found to be highly variable and generally much lower (a factor of three or more) than the value given above. Often a clear saturation is not seen at all. Understanding electron saturation is one of the most important outstanding problems of Langmuir probe theory.