Editing Langmuir probe (section)

==Cylindrical Langmuir probe in electron flow==

Most often, the Langmuir probe is a small sized electrode inserted into a plasma which is connected to an external circuit that measures the properties of the plasma with respect to ground. The ground is typically an electrode with a large surface area and is usually in contact with the same plasma (very often the metallic wall of the chamber). This allows the probe to measure the [[I-V characteristic]] of the plasma. The probe measures the characteristic current <math>i(V)</math> of the plasma when the probe is biased with a potential <math>V</math>.

[[File:ShunkoLPFig01W.tif|thumb|Fig. 1. Illustration to Langmuir Probe I-V Characteristic Derivation]]

Relations between the probe [[I-V characteristic]] and parameters of isotropic plasma were found by [[Irving Langmuir]]<ref>{{cite journal |author1=Mott-Smith, H. M.  |author2=Langmuir, Irving |date=1926 |title=The Theory of Collectors in Gaseous Discharges |journal=Phys. Rev. |volume=28 |issue=4 |pages=727–763 |doi=10.1103/PhysRev.28.727 |bibcode = 1926PhRv...28..727M }}</ref> and they can be derived most elementary for the planar probe of a large surface area <math>S_z</math>  (ignoring the edge effects problem). Let us choose the point <math>O</math>  in plasma at the distance <math>h</math>  from the probe surface where electric field of the probe is negligible while each electron of plasma passing this point could reach the probe surface without collisions with plasma components: <math>\lambda_D \ll\lambda_{Te}</math>, <math>\lambda_D</math> is the [[Debye length]] and <math>\lambda_{Te}</math>  is the electron free path calculated for its total [[cross section (physics)|cross section]] with plasma components. In the vicinity of the point <math>O</math> we can imagine a small element of the surface area <math>\Delta S</math> parallel to the probe surface. The elementary current <math>di</math> of plasma electrons passing throughout <math>\Delta S</math> in a direction of the probe surface can be written in the form

{{NumBlk|:|<math>di = q_e\Delta Sdn(v, \vartheta)v\cos \vartheta</math>,|{{EquationRef|1}}}}

where <math>v</math>   is a scalar of the electron thermal velocity vector <math>\vec{v}</math>,

{{NumBlk|:|<math>dn(v,\vartheta)=nf(v)\frac{2\pi\sin \vartheta}{4\pi} dv d\vartheta</math>,|{{EquationRef|2}}}}

<math>2\pi\sin \vartheta d\vartheta</math> is the element of the solid angle with its relative value <math>2\pi\sin \vartheta d\vartheta / 4\pi</math>, <math>\vartheta</math> is the angle between perpendicular to the probe surface recalled from the point <math>O</math> and the radius-vector of the electron thermal velocity <math>\vec{v}</math> forming a spherical layer of thickness <math>dv</math> in velocity space, and <math>f(v)</math> is the electron distribution function normalized to unity

{{NumBlk|:|<math>\int\limits_0^\infty f(v)dv = 1</math>.|{{EquationRef|3}}}}

Taking into account uniform conditions along the probe surface (boundaries are excluded), <math>\Delta S \rightarrow S_z</math>, we can take double integral with respect to the angle <math> \vartheta </math>, and with respect to the velocity <math> v </math>, from the expression ({{EquationNote|1}}), after substitution Eq. ({{EquationNote|2}}) in it, to calculate a total electron current on the probe

{{NumBlk|:|<math> i(v) = q_enS_z \frac{1}{4\pi} \int\limits_{\sqrt{2q_eV/m}}^\infty f(v)dv \int\limits_0^\zeta v\cos \vartheta 2\pi \sin \vartheta d\vartheta</math>.|{{EquationRef|4}}}}

where<math>V</math>   is the probe potential with respect to the potential of plasma <math>V = 0</math>,  <math>\sqrt{2q_eV/m}</math>  is the lowest electron velocity value at which the electron still could reach the probe surface charged to the potential <math>V</math>, <math>\zeta</math>   is the upper limit of the angle <math>\vartheta</math>  at which the electron having initial velocity <math>v</math>  can still reach the probe surface with a zero-value of its velocity at this surface. That means the value <math>\zeta</math>  is defined by the condition

{{NumBlk|:|<math>v\cos\zeta = \sqrt{2q_eV/m}</math>. |{{EquationRef|5}}}}

Deriving the value <math>\zeta</math>  from Eq. ({{EquationNote|5}}) and substituting it in Eq. ({{EquationNote|4}}), we can obtain the probe [[I-V characteristic]] (neglecting the ion current) in the range of the probe potential <math>-\infty <V\leq 0 </math> in the form

{{NumBlk|:|<math>i(V)=\frac{q_enS_z}{4}\int\limits_\sqrt{2q_eV/m}^\infty f(v)\left ( 1 - \frac{2q_eV}{mv^2}\right ) vdv</math>.|{{EquationRef|6}}}}

Differentiating Eq. ({{EquationNote|6}}) twice with respect to the potential <math>V</math>, one can find the expression describing the second derivative of the probe [[I-V characteristic]] (obtained firstly by M. J.  Druyvestein <ref name="Druyvesteyn1930">{{cite journal|vauthors = Druyvesteyn MJ|title=Der Niedervoltbogen|journal=Zeitschrift für Physik|volume=64|issue=11–12|year=1930|pages=781–798|issn=1434-6001|doi=10.1007/BF01773007|bibcode=1930ZPhy...64..781D|s2cid=186229362 }}</ref>

{{NumBlk|:|<math>i^{\prime \prime} (V) = \frac{q_e^2 nS_z}{4m}\frac {1}{V}f\left ( \sqrt{2q_eV/m}\right ) </math>|{{EquationRef|7}}}}

defining the electron distribution function over velocity  <math>f\left ( \sqrt{2q_eV/m}\right ) </math> in the evident form. M. J. Druyvestein has shown in particular that Eqs. ({{EquationNote|6}}) and ({{EquationNote|7}}) are valid for description of operation of the probe of any arbitrary convex geometrical shape. 
	Substituting the [[Maxwellian distribution]] function:

{{NumBlk|:|<math>f^{(0)} (v) = \frac{4}{\sqrt{\pi}}\frac{v^2}{v_p^3}\exp \left (-v^2/v_p^2\right )</math>,|{{EquationRef|8}}}}

where <math>v_p = \langle v\rangle \sqrt{\pi}/2</math>   is the most probable velocity, in Eq. ({{EquationNote|6}}) we obtain the expression

{{NumBlk|:|<math>i^{(0)} (V) = \frac{q_en\langle v \rangle}{4}S_z\exp \left (-q_eV/\mathcal{E}_p \right )</math>.|{{EquationRef|9}}}}

[[File:IV Ar 0.058 370 z12.tif|thumb|Fig. 2. I-V Characteristic of Langmuir Probe in Isotropic Plasma]]

From which the very useful in practice relation follows

{{NumBlk|:|<math>\ln \left ( i^{(0)} (V)/i^{(0)} (0)\right ) = -q_eV/\mathcal{E}_p </math>.|{{EquationRef|10}}}}

allowing one to derive the electron energy <math>\mathcal{E}_p  = k_B T</math>   (for [[Maxwellian distribution]] function only!) by a slope of the probe [[I-V characteristic]] in a semilogarithmic scale. 
Thus in plasmas with isotropic electron distributions, the electron current <math>i_{th} (0)</math>   on a surface <math>S_z = 2\pi r_z l_z </math>   of the cylindrical Langmuir probe at plasma potential <math>V = 0</math>   is defined by the average electron thermal velocity <math>\langle v \rangle </math>   and can be written down as equation (see Eqs. ({{EquationNote|6}}), ({{EquationNote|9}}) at  <math>V = 0</math>)

{{NumBlk|:|<math>i_{th}(0) = q_en\langle v\rangle\frac {1}{4}\times 2\pi r_z l_z</math>,|{{EquationRef|11}}}}

where <math>n</math> is the electron concentration, <math>r_z</math> is the probe radius, and <math>l_z</math> is its length. 
It is obvious that if plasma electrons form an electron '''''wind''''' ('''''flow''''') '''''across''''' the '''''cylindrical''''' probe axis with a velocity <math>v_d\gg \langle v\rangle</math>,  the expression

{{NumBlk|:|<math>i_d = env_d \times 2r_z l_z</math>|{{EquationRef|12}}}}

holds true. 
In plasmas produced by gas-discharge arc sources as well as inductively coupled sources, the electron wind can develop the Mach number <math>M^{(0)} = v_d /\langle v\rangle = (\sqrt{\pi}/2)\alpha \gtrsim 1 </math>  . Here the parameter <math>\alpha</math>   is introduced along with the Mach number for simplification of mathematical expressions. Note that <math>(\sqrt{\pi}/2)\langle v\rangle = v_p</math>, where<math>v_p</math>    is the most probable velocity for the [[Maxwellian distribution]] function, so that <math>\alpha = v_d/v_p</math>  . Thus the general case where <math>\alpha  \gtrsim 1</math>   is of the theoretical and practical interest. 
Corresponding physical and mathematical considerations presented in Refs. [9,10] has shown that at the [[Maxwellian distribution]] function of the electrons in a reference system moving with the velocity <math>v_d</math>   '''''across axis of the cylindrical''''' probe set at plasma potential <math>V = 0</math>, the electron current on the probe can be written down in the form

[[File:CylinderProbeInElectrWind.tif|thumb|Fig.3. I-V Characteristic of the cylindrical probe in crossing electron wind]]

{{NumBlk|:|<math>\frac{i(0)}{enS_z} = \frac{\langle v\rangle}{4} \exp(-\alpha ^{2} /2)I_0 (\alpha ^{2} /2) \left (1+\alpha ^{2} \left (1+I_1(\alpha ^{2} /2)/I_0(\alpha ^{2}/ 2)\right )\right )</math>,|{{EquationRef|13}}}}

where <math>I_0</math>   and <math>I_1</math>  are Bessel functions of imaginary arguments and Eq. ({{EquationNote|13}}) is reduced to Eq. ({{EquationNote|11}}) at<math>\alpha \rightarrow 0</math>   being reduced to Eq. ({{EquationNote|12}}) at <math>\alpha \rightarrow \infty</math>  . 
The second derivative of the probe I-V characteristic <math>i^{\prime \prime}(V)</math>  with respect to the probe potential <math>V</math>   can be presented in this case in the form (see Fig. 3)

{{NumBlk|:|<math>i^{\prime \prime}(x) = enS_z \frac{v_p}{2\pi ^{3/2} (\mathcal {E}_p/e)^2} \frac {1}{\sqrt{x}}\int \limits_0^\pi (\sqrt{x}- \cos \varphi) \exp\left ( -\alpha ^2 (\sqrt{x} - \cos \varphi)\right ) d\varphi</math>,|{{EquationRef|14}}}}

where

{{NumBlk|:|<math>x = \frac{1}{\alpha^2}\frac{V}{\mathcal {E}_p/e}</math> |{{EquationRef|15}}}}

and the electron energy <math>\mathcal {E}_p/e</math>  is expressed in eV.

All parameters of the electron population: <math>n</math>, <math>\alpha </math>, <math>\langle v\rangle </math>  and <math>v_p</math>  in plasma can be derived from the experimental probe I-V characteristic second derivative <math>i^{\prime \prime}(V)</math>   by its least square best fitting with the theoretical curve expressed by Eq. ({{EquationNote|14}}). For detail and for problem of the general case of none-Maxwellian electron distribution functions see.<ref>{{cite journal |author=E. V. Shun'ko |date=1990 |title=V-A characteristic of a cylindrical probe in plasma with electron flow |journal=Physics Letters A |volume=147 |issue=1 |pages=37–42 |doi=  10.1016/0375-9601(90)90010-L|bibcode =  1990PhLA..147...37S}}</ref><sup>,</sup> <ref>{{cite book |vauthors = Shun'ko EV |title=Langmuir Probe in Theory and Practice |publisher= Universal Publishers, Boca Raton, Fl. 2008|pages=243 |isbn=978-1-59942-935-9|year=2009 }}</ref>