Editing Langmuir probe (section)

==Electrode configurations==

Once one has a theory of the ''I-V'' characteristic of an electrode, one can proceed to measure it and then fit the data with the theoretical curve to extract the plasma parameters. The straightforward way to do this is to sweep the voltage on a single electrode, but, for a number of reasons, configurations using multiple electrodes or exploring only a part of the characteristic are used in practice.

===Single probe===

The most straightforward way to measure the ''I-V'' characteristic of a plasma is with a '''single probe''', consisting of one electrode biased with a voltage ramp relative to the vessel. The advantages are simplicity of the electrode and redundancy of information, i.e. one can check whether the ''I-V'' characteristic has the expected form. Potentially additional information can be extracted from details of the characteristic. The disadvantages are more complex biasing and measurement electronics and a poor time resolution. If fluctuations are present (as they always are) and the sweep is slower than the fluctuation frequency (as it usually is), then the ''I-V'' is the ''average'' current as a function of voltage, which may result in systematic errors if it is analyzed as though it were an instantaneous ''I-V''. The ideal situation is to sweep the voltage at a frequency above the fluctuation frequency but still below the ion cyclotron frequency. This, however, requires sophisticated electronics and a great deal of care.

===Double probe===

An electrode can be biased relative to a second electrode, rather than to the ground. The theory is similar to that of a single probe, except that the current is limited to the ion saturation current for both positive and negative voltages. In particular, if <math>V_{bias}</math> is the voltage applied between two identical electrodes, the current is given by;

<math>
I 
= I_i^{max} \left( -1 + \,e^{q_e(V_2-V_{fl})/k_BT_e} \right)
= -I_i^{max} \left( -1 + \,e^{q_e(V_1-V_{fl})/k_BT_e} \right)
</math>,

which can be rewritten using <math>V_{bias}=V_2-V_1</math> as a [[Hyperbolic function|hyperbolic tangent]]:

<math>
I = I_i^{max} \tanh\left( \frac{1}{2}\,\frac{q_eV_{bias}}{k_BT_e} \right)
</math>.

One advantage of the double probe is that neither electrode is ever very far above floating, so the theoretical uncertainties at large electron currents are avoided. If it is desired to sample more of the exponential electron portion of the characteristic, an '''asymmetric double probe''' may be used, with one electrode larger than the other. If the ratio of the collection areas is larger than the square root of the ion to electron mass ratio, then this arrangement is equivalent to the single tip probe. If the ratio of collection areas is not that big, then the characteristic will be in-between the symmetric double tip configuration and the single-tip configuration. If <math>A_1</math> is the area of the larger tip then:

<math>
I = A_1 J_i^{max} \left[ \coth\left(\frac{q_eV_{bias}}{2k_BT_e}\right) + \frac{\left(\frac{A_1}{A_2}-1\right)\,e^{-q_eV_{bias}/2k_BT_e}}{2\sinh\left(\frac{q_eV_{bias}}{2k_BT_e}\right)} \right]^{-1}
</math>

Another advantage is that there is no reference to the vessel, so it is to some extent immune to the disturbances in a [[radio frequency]] plasma. On the other hand, it shares the limitations of a single probe concerning complicated electronics and poor time resolution. In addition, the second electrode not only complicates the system, but it makes it susceptible to disturbance by gradients in the plasma.

===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.

===Special arrangements===

Arrangements with four ('''tetra probe''') or five ('''penta probe''') have sometimes been used, but the advantage over triple probes has never been entirely convincing. The spacing between probes must be larger than the [[Debye length]] of the plasma to prevent an overlapping [[Debye sheath]].

A '''pin-plate probe''' consists of a small electrode directly in front of a large electrode, the idea being that the voltage sweep of the large probe can perturb the plasma potential at the sheath edge and thereby aggravate the difficulty of interpreting the ''I-V'' characteristic. The floating potential of the small electrode can be used to correct for changes in potential at the sheath edge of the large probe. Experimental results from this arrangement look promising, but experimental complexity and residual difficulties in the interpretation have prevented this configuration from becoming standard.

Various geometries have been proposed for use as '''ion temperature probes''', for example, two cylindrical tips that rotate past each other in a magnetized plasma. Since shadowing effects depend on the ion Larmor radius, the results can be interpreted in terms of ion temperature. The ion temperature is an important quantity that is very difficult to measure. Unfortunately, it is also very difficult to analyze such probes in a fully self-consistent way.

'''Emissive probes''' use an electrode heated either electrically or by the exposure to the plasma. When the electrode is biased more positive than the plasma potential, the emitted electrons are pulled back to the surface so the ''I''-''V'' characteristic is hardly changed. As soon as the electrode is biased negative with respect to the plasma potential, the emitted electrons are repelled and contribute a large negative current. The onset of this current or, more sensitively, the onset of a discrepancy between the characteristics of an unheated and a heated electrode, is a sensitive indicator of the plasma potential.

To measure fluctuations in plasma parameters, '''arrays''' of electrodes are used, usually one{{spaced ndash}}but occasionally two-dimensional. A typical array has a spacing of 1&nbsp;mm and a total of 16 or 32 electrodes. A simpler arrangement to measure fluctuations is a negatively biased electrode flanked by two floating electrodes. The ion-saturation current is taken as a surrogate for the density and the floating potential as a surrogate for the plasma potential. This allows a rough measurement of the turbulent particle flux

<math>
\Phi_{turb} 
= \langle \tilde{n}_e \tilde{v}_{E\times B} \rangle
\propto \langle 
\tilde{I}_i^{max} ( \tilde{V}_{fl,2} - \tilde{V}_{fl,1} ) 
\rangle
</math>