Faraday cup in plasma diagnostics: Difference between revisions

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Revision as of 22:45, 29 July 2013

I merged it to Faraday cup according your remarks Evgeny E. V. Shun'ko (talk) 22:45, 29 July 2013 (UTC)

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 {{Underlinked|date=May 2013}}
+I merged it to [[Faraday cup]] according your remarks
-[[Faraday cup]] utilizes a physical principle according which the electrical charges delivered to the inner surface of a hollow conductor are redistributed around its outer surface due to mutual self-repelling of charges of the same sign – phenomenon discovered by [[Faraday]] (angl. Michael Faraday; Sept. 22 1791, Aug. 25, 1867)<ref>{{cite journal |author=Frank A. J. L. James |booktitle=“Faraday, Michael (1791-1867)”, Oxford Dictionary of National Biography, |publisher=Oxford University Press, Oxford, 2004|volume= |issue= |pages= |doi=}}</ref>.
+Evgeny [[User:E. V. Shun&#39;ko|E. V. Shun&#39;ko]] ([[User talk:E. V. Shun&#39;ko|talk]]) 22:45, 29 July 2013 (UTC)
-The conventional [[Faraday cup]] is applied for measurements of ion (or electron) flows from plasma boundaries and comprises a metallic cylindrical receiver-cap – 1 (Fig. 1) closed with, and insulated from, a washer-type metallic electron-suppressor lid - 2 provided with the round axial through enter-hollow of an aperture with a surface area <math>S_F=\pi D^2_F/4</math>.  Both the receiver cup and the electron-suppressor lid are enveloped in, and insulated from, a grounded cylindrical shield - 3 having an axial round hole coinciding with the hole in the electron-suppressor lid - 2. The electron-suppressor lid is connected by 50 Ω RF cable with the source <math>B_{es}</math>  of variable DC voltage <math>U_{es}</math>. The receiver-cup is connected by 50 Ω RF cable through the load resistor <math>R_F</math> with a sweep generator producing saw-type pulses <math>U_g(t)</math>. Electric capacity  <math>C_F</math>   is formed of the capacity of the receiver-cup - 1 to the grounded shield - 3 and the capacity of the RF cable. The signal from <math>R_F</math> enables an observer to acquire an [[I-V characteristic]] of the [[Faraday cup]] by oscilloscope. Proper operating conditions: <math>h\geq D_F</math>  (due to possible potential sag) and <math>h\ll \lambda_i</math>, where <math>\lambda_i</math> is the ion free path. Signal from <math>R_F</math> is the [[Faraday cup]] [[I-V characteristic]]  which can be observed and memorized by oscilloscope
-[[File:Faraday Cup.tif|безрамки||thumb|Faraday Cup]]
-<math>i_\Sigma(U_g)=i_i(U_g)-C_F\frac{dU_g}{dt}</math> .     (1)
-In Fig. 1: 1 – cup-receiver, metal (stainless steel). 2 – electron-supressor lid, metal (ctainless steel). 3 – grounded shield, metal (stainless steel). 4 - insulator (teflon, ceramic). <math>C_F</math> - capacity of Faraday cup. <math>R_F</math> - load resistor.
-	Thus we measure the sum <math>i_\Sigma</math>  of the electric currents through the load resistor <math>R_F</math>:  <math>i_i</math> ([[Faraday cup]] current) plus the current <math>i_c(U_g)=-C_F(dU_g/ dt)</math>  induced through the capacitor <math>C_F</math> by the saw-type voltage <math>U_g</math>of the sweep-generator:
-The current component <math>i_c(U_g)</math> can be measured at the absence of the ion flow and can be subtracted further from the total current <math>i_\Sigma(U_g)</math> measured with plasma to obtain the actual [[Faraday cup]] [[I-V characteristic]] <math>i_i(U_g)</math> for processing.
-All of the [[Faraday cup]] elements and their assembly that interact with plasma are fabricated usually of temperature-resistant materials (often these are stainless steel and teflon or ceramic for insulators).
-	For processing of the [[Faraday cup]] [[I-V characteristic]], we are going to assume that the [[Faraday cup]] is installed far enough away from an investigated plasma source where the flow of ions could be considered as the flow of particles with parallel velocities directed exactly along the [[Faraday cup]] axis. In this case, the elementary particle current <math>di_i</math> corresponding to the ion density differential <math>dn(v)</math> in the range of velocities between <math>v</math> and <math>v+dv</math> of ions flowing in through operating aperture <math>S_F</math> of the electron-suppressor can be written in the form
-<math>di_i=eZ_i S_F vdn(v)</math>,     (2)
-где
-<math>dn(v)=nf(v)dv</math>,            (3)
-<math>e</math> is elementary electric charge, <math>Z_i</math> is the ion charge state, and <math>f(v)</math> is the one-dimensional distribution function of ions over velocity <math>v</math>. Therefore the ion current at the ion-decelerating voltage <math>U_g</math> of the [[Faraday cup]] can be calculated by integrating Eq. (2) after substituting in it Eq. (3)
-<math>i_i(U_g)=eZ_i n_i S_F\int\limits_{\sqrt{2eZ_i U_g /M_i}}^{\infty} f(v)vdv</math>,       (4)
-where the lower integration limit is defined from the obvious equation <math>M_iv^2 _{i,s}/2=eZ_i U_g</math> where <math>v_{i,s}</math>  is the velocity of the ion stopped by the decelerating potential  <math>U_g</math>, and <math>M_i</math>  is the ion mass. Thus the expression (4) represents the [[I-V characteristic]] of the [[Faraday cup]].
-Differentiating Eq. (4) with respect to <math>U_g</math>, one can obtain the relation
-<math>\frac{di_i(U_g)}{dU_g} = -en_i S_F \frac{eZ_i}{M_i}f\left(\sqrt{2eZ_i U_g /M_i}\right)</math>    ,        (5)
-where the value  <math> -n_i S_F (eZ_i/M_i ) = C_i </math>  is an invariable constant for each measurement. Therefore the average velocity  <math>\langle v_i \rangle</math> of ions arriving into the [[Faraday cup]] and their average energy <math>\langle \mathcal{E}_i \rangle</math>  can be calculated (under the assumption that we operate with a single type of ion) by the expressions
-<math>\langle v_i \rangle = 1.389\times10^6 \sqrt{\frac{Z_i}{M_A}}\int\limits_0^\infty i^\prime _i (U_g)dU_g \left ( \int\limits_0^\infty \frac{i^\prime _i}{\sqrt{U_g}}dU_g \right )^{-1}</math>      [cm/s],      (6)
-<math>\langle \mathcal{E}_i \rangle = \int\limits_0^\infty i^\prime _i (U_g) \sqrt{U_g}dU_g \left ( \int\limits_0^\infty \frac{i^\prime _i}{\sqrt{U_g}}dU_g \right )^{-1}</math>                      [eV],     (7)
-where  <math>M_A</math>  is the ion mass in atomic units. The ion concentration <math>n_i</math>  in the ion flow at the [[Faraday cup]] vicinity can be calculated by the formula
-<math>n_i = \frac{i_i (0)}{eZ_i \langle v_i \rangle S_F}</math>    (8)
-which follows from Eq. (4)  at <math>U_g = 0</math>
-<math>\int\limits_0^\infty f(v)vdv = \langle v \rangle</math>      (9)
-[[File:Faraday Cup Fig. 02.tif|безрамки||thumb|]]
-and from the conventional condition for distribution function normalizing
-<math>\int\limits_0^\infty f(v)dv = 1</math> .    (10)
-Fig. 2 illustrates the [[I-V characteristic]] <math>i_i (V)</math>  and its first derivative <math>i^\prime _i (V)</math>  of the [[Faraday cup]] with <math>S_F = 0.5  cm^2</math>  installed at output of the [[Inductively coupled plasma]] source powered with RF 13.56&nbsp;MHz and operating at 6 mTorr of H2. The value of the electron-suppressor voltage (accelerating the ions) was set experimentally at <math>U_{es} = - 170 V</math>, near the point of suppression of the [[secondary electron emission]] from the inner surface of the [[Faraday cup]] <ref>{{cite journal |author=E. V. Shun'ko. |booktitle=“Langmuir Probe in Theory and Practice”, |publisher= Universal Publishers, Boca Raton, Fl. 2008|volume= |issue= |pages=249 |doi=}}</ref>.
-==References==
-{{reflist}}
-==See also==
-* [[Faraday cage]]
-* [[Faraday cup]]
-* [[List of plasma (physics) articles]]
-* [[Plasma parameters]]
-[[Category:Plasma physics]]
-[[Category:Plasma physics facilities]]
-[[Category:Plasma diagnostics]]
-[[Category:Measuring instruments]]