the inadequate reference electrode, a widespread source of error in plasma probe

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The inadequate reference electrode, a widespread source of error in plasma probe

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1973 J. Phys. D: Appl. Phys. 6 1674

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J. Phys. D: Appl. Phys., Vol. 6, 1973. Printed in Great Britain. 0 973

The inadequate reference electrode, a widespread source oferror in plasma probe measurementsJen-Shih Ch angCentre for Research in Experimental Space Science, York University, Downsview 463,Ontario, CanadaReceived 8 February 1973, in final form 30 April 1973

Abstract. An analysis has been conducted on the influence of the reference electrodearea on probe characteristics taken in the electron retarding range of potentials, whenthe reference electrode, and perhaps also the probe, is in the blocking effect regime(where probe dimensions are greater than or equal to the electron and ion mean freepaths) A new and more severe criterion for sufficiency of reference electrode area isderived, and it is shown that insufficiency of reference area can lead not only to roundingof the knee of the probe characteristic, but can cause an extra, spurious, near-linearsection to appear in the (log current against potential) characteristic near space potential,leading to the possibility of grossly inaccurate electron temperature measurements.A number of published results are cited to show that this phenomenon is in fact afrequent source of error in probe measurements.

NotationSymbolsA areaD diffusion constan te electronic chargek Boltzmanns constantI currentL probe length ( t = L / 2 )m mass

n plasma densityr radiusT temperatureV probe voltagedX mean free pathAD Debye length

thermal velocity of electrons or ions

Subscriptse electronf floating potentiali ionp prober reference electrodes space potential01 effect of area ratio

Other symbols are defined in the text.1674


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The inadequate reference electrode 16751. IntroductionTh e Lang muir pr obe m ethod is often used to m easure plasma p arameters in electrodelessdischarges and flowing plasmas. In these m easurements, we usually use a large referenceelectrode instead of the discharge electrode (Okuda a nd Y am am oto 1956,1960, Aisenberg1964, Nagy and Faruqui 1965, Vagner et al 1969, Kirchhoff et al1971), but many probecharacteristics, especially in space chambers, show a higher electron temperature, alower density, and smaller IeB/Zis ratio (where l e s and Ii8are electron and ion saturationcurrents) than by the usual theory and other measurements. Th e analysis of Szuszewicz(1972), which assumes a uniform density plasma with collisionless sheath, AD , Rghi , he(where R is probe characteristic length, AD is electron Debye length, hi, he are ion andelectron mean free paths), yields a coefficient a =A r / A p (where A , an d A p are referenceand probe areas), and he used Laframboises (1966) collisionless theory to discuss theeffect of finite reference electrode size on electron density, electron tem pera ture and spacepotential measurements. H e predicts no influence when a:2 04.

However, anytime we have the condition R 2 i, he for a reference probe, we are inthe so-called blocking effect rCgime described by Bohm et al (1949) (appendix). (Thecontinuum r6gime is conventionally defined by the two conditions :R B hi, he; AD %hi,he.) Also in space chambers and flowing plasma, we find different densities at th e locationof the reference electrode an d the prob e.In this pape r, we define a coefficient

Arner(kTer)i2,=Apnep(kTep)l/2(where ner , n e p , kTer and kTep are electron density and temp erature at reference electrodeand probe, and C , is the blocking effect coefficient) instead of a: and use Laframboisesan d Keils (1 969) theories respectively to discuss the collisionless case an d co ntinu umcase in $3. The theoretical results from $3 show that E a > 04 is required for correctplasma measurements. If C, 5 104, the probe current-voltage characteristics are sub-stantially affected by insufficient reference cur rent supply . Th is no t only alters the slopeof the usual logarithmic current characteristic, but often produces a second spuriousnear-linear section, the usual interpretation of which would lead to gross overestimatesof electron temperature.In $4, we discuss why many au tho rs report misleading plasma param eter m easure-ments. For example, in th e spherical reference/spherical pro be case,

is often assumed enough to measure plasma parameters, but unfortunately Ea s only

(if ner(kTer)li2=nep(kTep)1/2), iving a wrong result, as described in $3.2. FormulationFirst, we assume several con ditions:(i) Th e gas is only weakly ionized and no magnetic fields are prese nt.


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1676 Jen-Shih Chang(ii) There are no depletion (Waymouth 1966) or boundedness effects (Cohen 1970,Smith and Plumb 1972) on either the probe or reference electrode current-voltagecharacteristics in the transition region.(iii) Th e ion current is not dominated by secondary electron emission current a teither probe .

Figure 1. Typical floating potential method for (a ) a space chamber, (b )an electrodelessplasma tube.

Shown in figure 1 is the arrangement of the flowing plasmas and space chambers.We can equate the net currents collected o n probe and reference electrode :Iir - e r = l e p - l i p. (1)

In the 'retarding field' region we can write equa tion (1) as follows:Ar( l i s rRr - le s r exp Tr) =Ap(Iesp exp ~p AI i spRp)E a=4 l e s r= X P P- I i sP i~esp )FD

(lisr/Iesr) Rr - xp Trp I e spwhere R P and Rr are normalized ion current I / I s functions as follows: RP= f(PP , 7 , vP).In measurements there is first no net current collection by either prob e a nd the referenceis at floating potential. When we give a bias to the probe in order t o collect current, thereference needs to move from floating potential to supply current. Th at shift of voltagewe call Ava:A ~ e = ~ a r fr (4)

where Tfr is the floating potential of the reference electrode and Tar is the solution ofequation (3). The changing of Ava by Tar is the error which appears in the probecurrent-voltage characteristic.3. TheoryHere we assume AD < i, he, the condition for no collisions in the sheath. Then we canuse Laframboise's theory fo r the ion current. In the case of a spherical reference andspherical probe, the blocking effect influence on the probe is given by the calculationof Bohm et a1 and we get good agreement with experiment (Chen et a1 1970) as follows:

I - I - gs - )


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The inadequate reference electrode 1677

I I 1 1 1 l ( 1 I I I I I l l I I I I I I I I I I I I I 1 1 1 1 'IIIII I I

III i

15- /3=I 1 1 0 I 0 1 001 1001 - r gas -I I Hg gas- - - -

where lo (=4neb) is the current per unit area w hen no effect is present. Then equation (3)can be written as follows :

r =0h , W ,

-- - - - - - _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ - - -

---- _ _ _ _ _ _ _- _ - - - - - - - - -I I I I I I I I I I I 1 I I I I LI

-I02 103 104

when rr$ hi , he, r p 104, we need n ot consider t he influence of th e refer-ence electrode area. For I ; a < 104, we can see some influence as shown in figure 4(a).Tars is shown as function of I;, in figure 3 using equation (7), for Ar and Hg gas with/3r= 1, 0.1, 0.01, T=O , and we obtain as in figure 2 , the condition tha t for I;&>104, thereis no effect (RP nd Rr are given by Keil 1969).Figure 4 shows typical plots of area influence in the current-voltage characteristicfor t he collisionless case (a) and for the continuum case (b) . Fo r the values of shownthere is a substantial influence on the characteristics providing misleading values forplasma parameters. For example, we choose from figure 4(a) , the data a=200, anda=103. The resulting probe characteristics are shown in figure 5. In both cases, the


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1678

5

Jen-Shih Clzang

I 1 I I I I I I I 1 I 1 1 1 1 1 I I I I I I l l

Figure 3. Normalized space potential Tars of spherical reference probe as a function ofA , for various PP.

qPFigure 4. Typical area influence on normalized current-voltage characteristics. (Inthis diagram, we ignore variation of collisionless current with probe potential in electronsaturation region.) (U ) Collisionless case: vp


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The inadequate reference electrode 1679drift in reference electrode potential, as the probe current increases, actually causes asecond, spurious near-linear section to appear. In the CI:=200 case, this is not obvious,bu t a pair of maxima in the second derivative of these curves is present in bo th cases.

Figure 5. Determination of ne, kT, and Vs using current-voltage characteristics ofArgas: p r =O . O l , T = O or A,=200, 103, CO : subscript 1 for 103; 2 for 200.

Here we need to consider the difference of C, and a. For the spherical case,cu=Ar/Ap = r r 2 / r g 2 ,bu t E, at the same density is given by the expressions:Ea= Ar/Ap) C, = Ar/Ap) (rp/rr) in nep(kTep)1/2Mer(kTer)1'2

by equation (7). So while a= 104- 106, a value usually considered large enough n ot toperturb plasma measurements, we have actually because of the blocking effect a E,rat io of only 102- 103.Fo r the cylindrical an d disc probes, we use Bohm e t al's results, which are expressedin terms of the free-space capacitance C of these probes, as follows:

(d 2 8)where Xi , Xe


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1680 Jen-Shih ChangThen fro m equation (3) we get

and define B,, C,, D ,, by equations (lo), (8) and (9). These quantities are shown intable 1 ( r r $ hi, h e ) . When r p , r r < hi, h e , table 1 shows B,= C,= D ,= 1, and we can thenuse Szuszewicz's analysis.Table 1. B,, C,, Du nd E, for arbitrarily shaped probes

ReferenceProbe

cuSpherical

Du

C,Cylindrical

Du

C

1 B, 1 B,

.~1 Bu 1 Bu4 - g (In 4dr)-1 AP3 gr rrrpEup 3

1 Bu

We defineE,=@ +6.95 d o 9N.B. If ~ p ,r< Xi , Xe, Bu=Cu=Du =1 .

4. DiscussionThe theory is simple but im porta nt in actual experimental work. We ca n find manymisleading values of kTe, Ne and V, in several papers (Sl), especially in space cha mbe rsand D -region rocket ex periments. Th e au tho rs usually seem overconfident of the ade quacyof their reference area (a= APIA,) and use a large radius probe giving misleading valuesfor plasma parameters. In nep(kTep)'/' =Mer(kTer)l/' spherical-spherical pr ob e case, webelieve

a= Ar= '"= 104- 106AP Y P 2

(&=AP rr r pis usually enough to measure plasma parameters, but unfortunately Z is only

Ar r p =?) =102 103giving wrong results as in figures 4 and 5.We need to get a correct measurement of C, = r / rp=104 fo r spherical a nd disc prob esand 2, =Lr/Lp=O4 for cylindrical probes (the ratio In (Lp/2rp)/ln (L r/2~ ,) n equa tion (10)gives less influence t ha n Lr/Lp).


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The inadequate reference electrode 1681

vp V IFigure 6 . Typical cylindrical probe characteristics for Ar gas, p=05012 Torr, flowingplasma in cylindrical chamber: (a ) rp=0.0125 mm , L p = 4 mm, 10 nep (kTep)l12=ner (kTer)'I2 point. (b ) ~ p = 0 * 0 0 6 m, Lp = 3 mm , 2nep (kTep)'/'=ner (kTer)l/' point.

Typical single cylindrical probe characteristics obtained from an electrodeless R Fdischarge in a very-low-speed flowing Ar gas plasma (p =0 -0 12 To rr) a re shown infigure 6(a) for a probe size rp=0.0125 mm , Lp=4 mm, reference electrode rr =50 mm,Lr=30 mm, giving 01 =A r / A p =8 x 104. This value would normally be considered largeenoug h for correctly measuring plasma parameters. T he actual characteristic is far fromideal, .ZeS/Zis=lO (determined by the same method as Chen et al 1970), and values ofVs, ne and kTe would clearly be wrong. Fo r example, we consider A r gas fr omZe,/Iis-O*657 (mi/me)'/'= 176 (rp


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1682 Jen-Shih ChangAppendix. Dis c and cylindrical probe blocking effects for the space potentialWhen the mean free paths of the ions and electrons are small compared to the prob e size,the probe current at space potential is smaller than the usual thermal diffusion current,fo r the thermal motion of the ions and electrons is disturbed by the existence of the pr ob e:this is the so-called blocking effect (Bohm et al 1949, Chen 1965).In this case, the charge particles close to the probe are governed by the followingequations :

l7i=- in iV4- iVnire=peneV$ - eVneVz4=-e (ni - e ) / q

(Al l(A2)643)

where re nd Ti are the fluxes of electrons and ions, p e and pi are the mobilities of theelectrons and ions, and 4 s the potential.A t the space potential, Bohm et alneglected the first term of the right-hand side of (AI )an d (A2), and assumed th e analogy of G ausss law at one mean free pa th from the probesurface to get equation (8). Su and La m (1963) obtained the same expression a s equation(5) fo r the spherical probe at the co ntinuum limited space potential (Ai/rp< 1). Chen et a1(1970) used equation (5) to estimate the plasma density for a wide range of Ai/rp andobtained good agreements for spherical geometry. F or the cylindrical and disc probes, theblocking effect was first calculated by Su a nd Keil (1966) assuming zero thickness ( t =O )and a t the continuu m limit. Chen an d Ch ang (1972) used an ellipsoidal configuration toextend Bohms theory to calculate the end effect ( t>0)and obtained a good agreement w ithexperiment (microwave measurements) w ithin a factor of 2. The cylindrical probesblocking effect was calculated by Zakharova e t al (1960). Fo r the long wire condition

Figure A l . Comparison of N e obtained from Zes with that obtained from microwavemeasurement. Equations (Al), (A2) and (5) were used. Ar gas.


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The inadequate reference electrode 1683(rp=O) the results agreed well with experimen tal results. Equatio ns ( 9 ) and (10)extendBohms theory, and using right circular cylinders capacitance approximation equation(Miles 1967, Smythe 1962) we obtain an erro r less th an with the ellipsoidal configurationmodel (error only 0.2% relative to numerical com putation). Equa tions ( 9 ) an d (10)in condition r p2 X becomel e = Io ( K + 3A p [,,6*95 ( g + l )+ g 0.76] 1 ] 1 ( 0 c g i i i 8 )+gr2(1+ g > gUsing figure A1 and equations (A4) and (A5) we determine a density in good agreementwith microwave measurement, using apparatus with dimensions of t p=1 mm a ndr p= mm for the disc probe a nd Lp=6 mm and r p=0.25 mm for the cylindrical probe,similar to that used by Chen et a1 (1970, 1972).

ReferencesAisenberg S 1964 J. appl. Phys. 35 130-4Bohm D, Burhop E and Massey H 1949 The Characteristics of Electrical Discharges in Magnetic FieldsChen F F 1965 Plasma Diagnostic Techniques ed R H Huddlestone and S L Leonard (New York:Chen S-L and Chang J-S 1972 Hoden Kenkyu 47 8-16 (in Japanese)Chen S-L, Chang J-S and Matsumura S 1970 J. appl. Phys. 41 1711-5- 97 1 J. appl. Phys. 42 499-500Cohen I M 1970 Phys. Fluids 13 889-94Keil R E 1969 J. appl. Phys. 40 3668-73Kirchhoff R H, Peterson E W and Talbot L 1971 AZAA J 9 1686-94Laframboise J G 1966 Rarefied Gas Dynamics vol 2 (New York: Academic Press). See also Universityof Toronto Institute of Aerospace Studies Report No. 100 (1966)Miles J 1967 J. appl. Phys. 38 192-6Nagy A F and Faruqui A Z 1965 J. Geophys. Res. IO 4847-58Okuda T and Yamamoto K 1956 J. Phys. Soc. Japan 11 57-68- 96 0 J. appl. Phys. 31 158-62Smith D and Plumb I C 1972 J. Phys. D: Appl. Phys. 5 1226-38Smythe M 1962J. appl. Phys 33 2966-7Su C H and Keil R E 1966 J. appl. Phys. 37 4907-10Su C H and Lam S H 1963 Phys. Fluids 6 1479-91Szuszewicz E P 1972 J. appl. Phys. 43 874-80Vagner SD, Virolainen V A and Kagan Yu M 1969 Sou. Phys.-Tech. Phys. 14 502-4Waymouth J F 1966 J. appl. Phys. 37 4492-7Zakharowa V M, Kagan Yu M, Mustafin Ks and Perel VI 1960 Sov. Phys. - Tech. Phys. 5 411

ed A Guthrie and R K Wakerling (New York: McGraw-Hill)Academic Press) 113-200

the inadequate reference electrode, a widespread source of error in plasma probe

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