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A Langmuir probe is a device used to determine the electron temperature, electron density, and electric potential of a plasma. It works by inserting one or more electrodes into a plasma, with a constant or time-varying electric potential between the various electrodes or between them and the surrounding vessel. The measured currents and potentials in this system allow the determination of the physical properties of the plasma.

I-V characteristic of the Debye sheath

The beginning of Langmuir probe theory is the 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 (${\displaystyle n_{i}}$), the ion temperature ${\displaystyle T_{i}}$, or the electron energy distribution function (EEDF) or ${\displaystyle f_{e}(v)}$.

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

${\displaystyle c_{s}={\sqrt {k_{B}(ZT_{e}+\gamma _{i}T_{i})/m_{i}}}}$.

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 ${\displaystyle \gamma _{i}}$ is the adiabatic coefficient for the ions. The proper choice of ${\displaystyle \gamma _{i}}$ is a matter of some contention. Most analyses use ${\displaystyle \gamma _{i}=1}$, corresponding to isothermal ions, but some kinetic theory suggests that ${\displaystyle \gamma _{i}=3}$. For ${\displaystyle Z=1}$ and ${\displaystyle T_{i}=T_{e}}$, using the larger value results in the conclusion that the density is ${\displaystyle {\sqrt {2}}}$ 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 quasineutrality allows one to write it simply in terms of the electron density as ${\displaystyle q_{e}n_{e}}$, where ${\displaystyle q_{e}}$ is the charge of an electron and ${\displaystyle n_{e}}$ 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

${\displaystyle j_{i}^{max}=q_{e}n_{e}c_{s}}$ where ${\displaystyle c_{s}}$ 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.,

${\displaystyle f(v_{x})\,dv_{x}\propto e^{-{\frac {1}{2}}m_{e}v_{x}^{2}/k_{B}T_{e}}}$,

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

${\displaystyle \langle v_{e}\rangle ={\frac {\int _{v_{e0}}^{\infty }f(v_{x})\,v_{x}\,dv_{x}}{\int _{-\infty }^{\infty }f(v_{x})\,dv_{x}}}}$,

where the cut-off velocity for the upper integral is

${\displaystyle v_{e0}={\sqrt {2q_{e}\Delta V/m_{e}}}}$.

${\displaystyle \Delta V}$ 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

${\displaystyle ⟨<!--\; ⟨\; -->v_e⟩<!--\; ⟩\; -->=\sqrt{\frac{k_B{T}_e}{2\mathrm{\pi <!--\; \pi \; -->}{m}_e}}{e}^{-<!--\; -\; -->q_e}}$Zdroj:https://en.wikipedia.org?pojem=Langmuir_probe
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