Editing Langmuir probe (section)

===Triple probe===

An elegant electrode configuration is the triple probe,<ref name="Chen">{{cite journal |author1=Sin-Li Chen |author2=T. Sekiguchi |date=1965 |title= Instantaneous Direct-Display System of Plasma Parameters by Means of Triple Probe|journal=  Journal of Applied Physics|volume=36 |issue= 8|pages=2363–2375 |doi=10.1063/1.1714492 |bibcode = 1965JAP....36.2363C |doi-access=free }}</ref> consisting of two electrodes biased with a fixed voltage and a third which is floating. The bias voltage is chosen to be a few times the electron temperature so that the negative electrode draws the ion saturation current, which, like the floating potential, is directly measured. A common rule of thumb for this voltage bias is 3/e times the expected electron temperature. Because the biased tip configuration is floating, the positive probe can draw at most an electron current only equal in magnitude and opposite in polarity to the ion saturation current drawn by the negative probe, given by :

<math>
-I_{+}=I_{-}=I_i^{max}
</math>

and as before the floating tip draws effectively no current:

<math>
I_{fl}=0
</math>.

Assuming that: 
1.) The electron energy distribution in the plasma is Maxwellian,
2.) The mean free path of the electrons is greater than the ion sheath about the tips and larger than the probe radius, and
3.) the probe sheath sizes are much smaller than the probe separation,
then the current to any probe can be considered composed of two parts{{spaced ndash}}the high energy tail of the Maxwellian electron distribution, and the ion saturation current:

<math>
I_{probe} = -I_{e} e^{-q_e V_{probe}/(k T_{e} )} + I_i^{max}
</math>

where the current ''I<sub>e</sub>'' is thermal current. Specifically,

<math>
I_{e} = S J_{e} = S n_{e} q_e \sqrt{kT_{e}/2 \pi m_{e}}
</math>,

where ''S'' is surface area, ''J<sub>e</sub>'' is electron current density, and ''n<sub>e</sub>'' is electron density.<ref>{{cite journal |last1= Stanojević|date=1999 |first1= M. |last2= Čerček |first2= M. |last3= Gyergyek |first3= T. |title= Experimental Study of Planar Langmuir Probe Characteristics in Electron Current-Carrying Magnetized Plasma|journal=Contributions to Plasma Physics |volume=39 |issue=3 |pages=197–222 |doi=10.1002/ctpp.2150390303 |bibcode = 1999CoPP...39..197S |s2cid=122406275 |doi-access= free }}</ref>

Assuming that the ion and electron saturation current is the same for each probe, then the formulas for current to each of the probe tips take the form

<math>
I_{+} = -I_{e} e^{-q_e V_{+}/(k T_{e} )} + I_i^{max}
</math>

<math>
I_{-} = -I_{e} e^{-q_e V_{-}/(k T_{e} )} + I_i^{max}
</math>

<math>
I_{fl} = -I_{e} e^{-q_e V_{fl}/(k T_{e} )} + I_i^{max}
</math>.

It is then simple to show

<math>

\left(I_{+} - I_{fl})/(I_{+} - I_{-}\right) = \left(1-e^{-q_e(V_{fl}-V_{+})/(k T_{e})}\right)/ \left(1-e^{-q_e(V_{-}-V_{+})/(k T_{e})}\right)

</math>

but the relations from above specifying that ''I<sub>+</sub>=-I<sub>−</sub>'' and ''I<sub>fl</sub>''=0 give

<math>

1/2 = \left(1-e^{-q_e(V_{fl}-V_{+})/(k T_{e})}\right)/ \left(1-e^{-q_e(V_{-}-V_{+})/(k T_{e})}\right)
</math>,

a transcendental equation in terms of applied and measured voltages and the unknown ''T<sub>e</sub>'' that in the limit ''q<sub>e</sub>V<sub>Bias</sub> = q<sub>e</sub>(V<sub>+</sub>-V<sub>−</sub>) >> k T<sub>e</sub>'', becomes

<math> 
(V_{+}-V_{fl}) = (k_BT_e/q_e)\ln 2 
</math>.

That is, the voltage difference between the positive and floating electrodes is proportional to the electron temperature. (This was especially important in the sixties and seventies before sophisticated data processing became widely available.)

More sophisticated analysis of triple probe data can take into account such factors as incomplete saturation, non-saturation, unequal areas.

Triple probes have the advantage of simple biasing electronics (no sweeping required), simple data analysis, excellent time resolution, and insensitivity to potential fluctuations (whether imposed by an rf source or inherent fluctuations). Like double probes, they are sensitive to gradients in plasma parameters.