Editing Langmuir probe (section)

==''I-V'' characteristic of the Debye sheath==

The beginning of Langmuir probe theory is the [[current–voltage characteristic|''I–V'' characteristic]] of the [[Debye sheath]], that is, the current density flowing to a surface in a plasma as a function of the voltage drop across the sheath. The analysis presented here indicates how the electron temperature, electron density, and plasma potential can be derived from the ''I–V'' characteristic. In some situations a more detailed analysis can yield information on the ion density (<math>n_i</math>), the ion temperature <math>T_i</math>, or the electron energy [[Distribution function (physics)|distribution function]] (EEDF) or <math>f_e(v)</math>.

===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.

===Exponential electron current===

As the voltage of the Debye sheath is reduced, the more energetic electrons are able to overcome the potential barrier of the electrostatic sheath. We can model the electrons at the sheath edge with a [[Maxwell–Boltzmann distribution]], i.e.,

<math>f(v_x)\,dv_x \propto e^{-\frac{1}{2}m_ev_x^2/k_BT_e}</math>,

except that the high energy tail moving away from the surface is missing, because only the lower energy electrons moving toward the surface are reflected. The higher energy electrons overcome the sheath potential and are absorbed. The mean velocity of the electrons which are able to overcome the voltage of the sheath is

<math>
\langle v_e \rangle = \frac
{\int_{v_{e0}}^\infty f(v_x)\,v_x\,dv_x}
{\int_{-\infty}^\infty f(v_x)\,dv_x}
</math>,

where the cut-off velocity for the upper integral is

<math>v_{e0} = \sqrt{2q_{e}\Delta V/m_e}</math>.

<math>\Delta V</math> is the [[voltage]] across the Debye sheath, that is, the potential at the sheath edge minus the potential of the surface. For a large voltage compared to the electron temperature, the result is

<math>
\langle v_e \rangle = 
\sqrt{\frac{k_BT_e}{2\pi m_e}}\,
e^{-q_{e}\Delta V/k_BT_e}
</math>.

With this expression, we can write the electron contribution to the current to the probe in terms of the ion saturation current as

<math>
j_e = 
j_i^{max}\sqrt{m_i/2\pi m_e}\,
e^{-q_{e}\Delta V/k_BT_e}
</math>,

valid as long as the electron current is not more than two or three times the ion current.

===Floating potential===

The total current, of course, is the sum of the ion and electron currents:

<math>
j = j_i^{max} 
\left( -1 + \sqrt{m_i/2\pi m_e}\,e^{-q_{e}\Delta V/k_BT_e} \right)
</math>.

We are using the convention that current ''from'' the surface into the plasma is positive. An interesting and practical question is the potential of a surface to which no net current flows. It is easily seen from the above equation that

<math>\Delta V = (k_BT_e/q_e)\,(1/2)\ln(m_i/2\pi m_e)</math>.

If we introduce the ion [[reduced mass]] <math>\mu_i=m_i/m_e</math>, we can write

<math>
\Delta V = (k_BT_e/q_e)\, ( 2.8 + 0.5\ln \mu_i )
</math>

Since the floating potential is the experimentally accessible quantity, the current (below electron saturation) is usually written as	

<math> 
j = j_i^{max} 
\left( -1 + \,e^{q_{e}(V_{0}-\Delta V)/k_BT_e} \right) 
</math>.

=== 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.