The use of Langmuir Probe Methods for Plasma Diagnostic in Middle Pressure Discharges

Contrib. Plasma Phys. SO (1990) 2, 167-184

The Use of Langmuir Probe Methods for Plasma Diagnostic in Middle Pressure Discharges

P. DAVID') (a), M. &CHA (a), M. TICH+ (a), T. KOPICZYNSKI (b), 2. ZAEF~ZEWSKI (b)

(a) Department of Electronin, and Vacunm Physics, Faculty of Hathematics and Physics,

(b) Instltute of Fluid-Flow Xachines, Polish Academy of Sciences, gen. J. Fiszera 14, 80-952 Gdansk, Charles Universlty, V €IoldoviEk&ch 2, 18000 Praha 8, Czechoslovakia;

Poland

Abstract

The article attempts to summarize the most important knowledge about the analysis of a Langmuir probe a t middle pressures. First it briefly reviews the recent theories which deal with interpretation of Langmuir probe data at such conditions and then it applies the theory to the measurements in a Neon discharge a t pressures 1.86 and 2.46 kPa. The charged particle density and the electron temperature obtained from the same probe data using different theories is compared against each other and a satisfactory agreement is obtained. The recommended procedure of probe data interpretation is given a t the end of the paper.

Introduction

The Langmuir probe is a useful toc for the experiments investigations of plasma properties. The main advantage of the probe method is that it gives local values of plasma parameters in the vicinity of the probe that cannot be obtained by other plasma diagnostic techniques. However, in order to interpret the probe measurements in the proper way i t is important to have a realistic theoretical model which converts the measured probe characteristic to the plasma parameters. One of the phenomena which be- come important for the correct interpretation of the probe characteristic in a middle pressure discharge is the influence of collisions of charged particles with neutrals in the space charge sheath surrounding the probe. The LANCIM~R and MOTT-SMITH basic theory [l] and the subsequent theories by ALLEN et al. [2], BERNSTEIN and RABIXO- WITZ [3] and LAFRAMBOISE [4] have been developed under assumption that collisions of charged particles with neutrals within the probe sheath can be neglected. ZAKRZEWSKI and KOPICZYNSKI [5] showed that these collisions may play an important role in the positive ion collection even in the case in which the proportion of the charged particles which collide with neutrals is small. Thus if the probe technique is used in the plasma where the gas pressure is more than several hundred Pa the effect of collisions must be taken into account in the interpretation of both electron and ion probe current compo- nents. Several attempts a t constructing theories for the probe response in a plasma a t middle gas pressure have already been made.

I) Permanent address: Department of Physics and Mathematics, Faculty of Pedagogy, Jeronf- mova 10, 371 15 Cesk6 BudBjovice, Czechoslovakia

1 Contrlb. Plasma Phys. 30 (1990) 2

168 DAVID, P., sf-, M., TIC&, M., et al., Langmuir Probe Methods

The aim of this work is for experimentalists who want to use the probe method to briefly review the recent theories which deal with interpretation of Langmuir probe characteristic in a middle pressure discharge and to demonstrate their applicability. For the latter purpose we performed also probe measurements in the middle pressure neon discharge and compared the results obtained using various methods of the probe characteristic interpretation.

Theories of Lsngmuir Probe in Middle Pressure Plasma

The most complete theory appears to be that of &on et al. [6]. This theory makes use of a moment approximation of the Boltzmann equation with a Krook-type collision integral. In this theory no restriction are placed on the electron or ion Knudsen number Kd,i = ,l,,i/Rp is the electron or ion mean free path respectively) and this theory holds for a wide range of values of the ratio of the probe radius Rp to the Debye length I D (the so-called Debye number DJ. For the positive ion collection by a Langmuir probe the theory of CHOU et al. is supplemented by that of ZAERZEWSKI and KOPICZYNS~I [5] extended to the middle gas pressure range by DAVID [7].

a ) E l e c t r o n c u r r e n t componen t The principal result of the CHOU et al. analysis [6, 81 of the normalized dimensionless

electron current I-* to a moderately negatively biased spherical probe for the case of the Maxwellian distribution of electron energies is the following expression :

where 1

J , = J exp (7) dZ, 0

Z = Rp/r , r is the distance from the centre of the spherical probe, 7 is the normalized

dimensionless probe potential in the electron retarding region 7 = l:J:l, - V, is the

voltage between the probe and the space potential, T, is the electron temperature, qo is the charge of an electron, k is the Boltzmann constant. The electron probe current I- is normalized with respect to the thermal random current, so that I-* = I-IIJe. For a spherical probe the normalization factor I , , is [8]:

I,, = 4nRp2 Fnqo 2nm,

where T, is the electron temperature, me, qo is the mass and charge of an electron respectively and n is the charged particles density. From equation (1) the normalized electron current correction factor due to the collisions of electrons within the probe sheath is [9] :

According to the DRUWESTEYN formula [ 101 the electron energy distribution function G(V,) in the collisionless case is proportional to the second derivative of the electron

Contrib. Plasma Phys. 30 (1990) 2

probe current component I- with respect to the probe potential V , :

169

Case [ 111 made use of the CHOU et al. theory to determine the expression for the second derivative of the electron probe current component for collisional case with an arbitrary disribution function G( V,) :

where G(r]), G ( E ) is the electron energy distribution function with respect to the normalized potential 7. Since even a t the assumption of a Maxwellian distribution of electron energies the second derivative yields interpretable data for the electron temperature determination TICH? [ 151 obtained the correction factor ( for the second derivative in the form:

d*I-* G(r]) - (=- dr12 I/;

where 6 = 5 - 25’ + 5” and the prime represents the derivative with respect to r ] . Thus the correction factor can easily be found by taking twice the derivative of equation (1) either analytically as in [15] or numerically.

The integral J, has been calculated numerically for several sets of parameters but the scope of the numerical work was not enough for purpose of an experimentalist. TALBOT and &IOU [8] worked out a method for an approximative evaluation of this integral. This method consists in the determination of the integral J , in two limiting cases: J,, for the collisionless case (&/;io --f 00) and J,, for the continuum case (&/AD +- 0 ) and in application of the ad hoc interpolation formula :

Je = J c o + (1 + Ke1-l (Jeo - Jem) (6) It should be noted that the method of the evaluation of the integral J,, proposed in [8] is valid for q 2 2.

According [8] similar equations as (1) and (6) are valid for cylindrical probe. The differ- ences are in the current normalization, where the factor I,, is replaced by I,, defined as

I , , = 2nRpLp fE 2nm, nqo

where L, is the length of a cylindrical probe and in the definition of the “collision” integral J , which becomes

0

The correction factors 5, ( are functions of the normalized plasma potential r ] , the electron Knudsen number K , = ;Ic/Rp, the ratio of electron to ion temperature t = T, /Ti , the Debye length AD and for a cylindrical probe also of the “aspect ratio” of the probe L,IR,. Via the dependence on the Debye length they also depend slightly on the electron density n. For this reason when calculating the correction factors C, t the electron concentration must be estimated by another independent method. Computer calcula-

1*

170 DAVID, P., &ma, M., TICHP, M., et a]., h n g m u i r Probe Methods

2 5 10

-2 Fig. 1.' Correction factor on collisions of the electron probe current 5 and of the electron current second derivative [ with respect to the normalized probe potential. Calculated according CHOU e t al. theory [6] by using the interpolation formula by TALBOT and CHOU [a]. Cylindrical probe dimensions: Rp = 50 pm, L, = 5.6 mm. Ne gas pressure 1.86 kPa, discharge current 8 mA

tions of the factors 5 , f have been made in [7, 91 and typical results are shown in Fig. 1. The calculations show that the correction factors can reach unity for a sufficiently negative probe as predicted for a cylindrical probe by CASE [ll]. Thus the experimentally obtained electron probe current component of the probe characteristic and its second derivative can be corrected by using the correction factors C and f to take into account the effect of collisions between electrons and neutrals in the vicinity of the probe. HOW- ever from the Fig. 1 it is also seen that for our experimental conditions the correction factor of the probe characteristic electron current component rapidly increases for the probe potentials r ] below 2+3 and that of the second derivative rapidly increases for the probe potentials r] below 3 +4. This fact limits in practice the use of the correction method based on CROU et al. theory because the error in determination of the corrected probe characteristic and its second derivative rapidly increases for lower probe potential q. The correction of the probe characteristic and its second derivative is therefore practicable just above mentioned limits.

The electron current component can be determined from the probe characteristic if the ion current component is known. For the purpose of determination of the electron current component from the experimental probe characteristic it is usually assumed that the ion saturation current can be expressed by semiempirical expression [ 131.

I+ = I+d + 7)" ( 7 ) The unknown parameters I+,, and x it is possible to estimate by means of an experimentally determined dependence of the saturated ion current I+ with respect to ( I + r ] ) for

Contrib. Plasma Phys. 30 (1990) 2 17 1

a sufficiently negative probe where it is possible to assume that I- 0. The absolute value of the electron probe current component (I-! is then obtained by adding the estimated absolute values of the ion current II+l to the total measured probe current I p since I p = II-I - II+I (assuming that the probe characteristic plot I p vs V, is drawn in the usual way, i.e. with negative I , on the positive part of the ordinate axis - see e.g.

I n a similar manner the second derivative of the electron probe current can be determined from the second derivative of the probe characteristic if the second derivative of the ion current component is known. In order to investigate the contribution of the ion current second derivative component to the total probe current second derivative it is assumed that the ion probe current can be expressed by equation (7). Then the second derivative of the electron current component L"(7) can be obtained from the measured second derivative of the total probe current Ip"(7) by means of the following expression [ 141 :

~ 3 1 ) .

For a Maxwellian electron energy distribution function the correction factor p(7) is according to [ 141 expressed as :

where I-, and I+, are the extrapolated values of the electron current component and the ion current component at plasma potential respectively.

b) T h e pos i t ive ion c u r r e n t The general expression for the ion current to a negatively biased spherical or cylindri-

cal probe according to the theory of CHOU et al. [6] is:

1 1 dZ

where Ji = exp ( -TV) dZ for a spherical and J i = exp (-q) - for a cylindrical s s z 0 0

probe, I:, and I+* are the normalized ion currents in the collisionless and collision case respectively, other symbols have already defined meaning. The measured ion current is normalized in similar manner as was the electron current namely using the factor ISiaci so that I+* = I+/Is i ,c i for a spherical or a cylindrical probe respectively, where

with mi being the mass of an ion, A the surface area of a probe, A = 4xRp2 for a spherical probe and A = 2hRpL, for a cylindrical one. TALBOT and CHOU [8] proposed to determine the integral J i using an interpolation formula similar to (6) :

Ji = Jim + (Ji, - Jim) (1 + Ki)-l (11)

where J i m and Jio are the integrals for collisionless and continuum limit. Similar expressions as (10) and (11) are valid for a cylindrical probe [8]. Basedon the

theory [8] KLAGQE and Tnxr+ (15) carried out a set of numerical calculations of the PO-

172 DAVID, P., Sfcm, M., TICHP, M., et al., Langmuir Probe Methods

sitive ion current to a cylindrical Langmuir probe as a function of the Knudsen number K i with the Debye ratio D1 as a parameter. The ion currents are calculated for the normalized potential of the probe q = (vF + 10). Here the symbol qF denotes the normalized floating potential (for the more precise definition of qF see below). The results are shown in Fig. 2. From this figure it is possible to determine the correction factor (for a particular value of K i and the parameter D1) by means of which the experimental value of the ion probe current can be corrected to include the effect of collisions.

rnF ’ ‘ ‘ “ 1 ‘ ‘ “ ‘ ’ ’ ; . “ ’ ’ ’ {

0,Ol 0,1 1

D RP

Fig. 2. The dependence of the normalized ion current I+*(qF + 10) (qp is normalized floating potential) with respect to the ion Knudsen number Ri = t , /R, . Calculated according the CHOU et al. theory [6] by KLAQQE and TICHP [15]. ---- D1 = 2.8, DA = 100 calculated with Ifoo taken from [4]; - D1 = 0.25, D1 = 0.5, D1 = 1, D1 = 2.5, D1 = 5, D1 = 10, D1 = 20, D1 = 70 calculated with taken from [29]; -.-.- D1 = 0 calculated according [30]; TJT, = 0.1, Re = 10Ki, L,/R, = 100, (rni /rne)”~ = 100

Besides the theory [8] there is another description of the positive ion collection by a Langmuir probe which was developed by ZAKRZEWSLU and KOPICZYNSKI [5]. They assumed two separate collision mechanisms of positive ions within the space charge sheath around the cylindrical probe: destruction of the orbital motion of ions in the sheath and elastic scattering of ions due to collisions with neutrals. The first effect leads to an increase of the ion probe current and dominates for lower pressures. The second one causes a decrease of the ion current and influences the probe current mostly a t higher pressures. The resulting normalised dimensionsless ion current I+* to a cylindrical Langmuir probe is then :

I+* = Y I Y 2 Z L * (12) where y1 is the rate of increase of ion current due to destruction of orbital motion and yz corresponds to the rate of reduction of ion current due to scatt,ering. I,* is normalized

Contrib. Plasma Phys. 80 (1990) 2 173

collisionless ion probe current predicted by L~FRAMBOISE [4] which was expressed in (51 on condition that (T i /TI ) + 0 as:

The normalizing factor for ion current to a cylindrical probe is given by equation (2d) which is rewritten in [5] as:

(14) where all the symbols have been explained already.

The ion makes on average x = S/ l i collisions within the sheath [5] if one denotes the thickness of the sheath as S. KOPICZYNSEI [16] determined the thickness S based on numerical calculations carried out by BASU and SEN [17] and obtained:

ICi = R,L,n 19,) (2nkT,/mi)1/2

fi x =.- DlKi

where = a(q + 3.5) - 4 and E ri 0.59 + 1.86(Dl)0*47.

ZYXSKI in [5, 161: The rate coefficients y l and yz can be estimated as shown by ZAERZEWSXI and KOPIC-

y1 = 1 + ( IA*/IL* - 1) x for x < 1

y1 =IA*]IL* for x 2 1

where I A * is the normalized ion current a t the sheath edge which is approximately given by ALLEN et al. theory [2] as:

IA* = 0.5 (Lr 0.17 . .

where

The rate coefficient y2 has been estimated in [5 ] following the results of SHULTZ and BROWN [ 181 and JAKUBOWSKI [ 191 :

a = 0.65(D1)-0.18

y2 = [3 - 2 exp (-x)]/[l + 251

yz = [3 - exp (-2)]/[2(1 + x)]

for 2 < 1

for z >= 1

For normalized probe potential q = + 15 KOPICZYNSIU [ 161 calculated the dependence of the probe ion current on the ion Knudsen number K,, the Debye number Dl, and the dimensions of the cylindrical probe. DAVID [7] extended these calculations towards lower Knudsen numbers K , occuring in a middle pressure discharge. The numerical calculations carried out by DAVID for normalized probe potential r ] = +15 are shown in Fig. 3a. Using the graph presented in Fig. 3a it is possible to determine the concentration of charged particles from experimental values of the ion probe current. For several values of ion Knudsen number the dependence of charged particles concentration with respect to the measured positive ion probe current is depicted in Fig. 3 b.

c) T h e p la sma po ten t i a l I n the collisionles regime the plasma potential has been mostIy determined from the

“knee” of the probe characteristic or from the zero-crossing point of the probe characteristic second derivative. These methods, however, are subject to EL serious source of

174 DAVID, P.. &aa, M., TI&, M., et el., Langmuir Probe Methixh

loo}

t

1 I 0,001 0,or OJ 1 a

Fig. 3a. The dependence of the ion probe current I+*(? = 15) with respect to the ion Knudsen number Ri. Calculated according ZAERZEWSKI and KOPICZYNSKI theory [5, 161 by DAVID [7] for several values of the parameter (R,,*n/V,)

Fig. 3b. The dependence of the charged particles concentration n with respect to the ion probe current l i ( q = 15) for several values of Ki. Obtained by DAV-KD [7] from the data depicted in Fig. 3a


errors which is related to the phenomena at the probe surface: change of the work function over the probe surface, the presence of layers on the probe surface due to contami- mtions, the reflection of electrons at the probe surface, secondary emission of electrons from the probe surface [20,21]. These phenomena at the probe surface round the “lmee” of the probe characteristic and the zero-crossing point of its second derivative does not exactly correspond to the plasma potential.

In the case in which the collisions of charged particles with neutrals in the probe space charge sheath play an important role in collection of charged particles by the probe the collisions also “contribute” to the rounding of the probe characteristic “knee”. KLAQQE and TICHP [ 151 found that the influence of collisions on the estimation of the plasma potential can be neglected provided that the following relation is valid :

In the case that the relation (19) is not fulfilled the determination of the exact value of the space potential from the zero-crossing point of the second derivative is rather diffi-

To demonstrate the effect of collisions on the probe characteristic second derivative KLAQOE and TICEP [15] calculated also probe characteristic accounting for the effect of collisions. In the region q 2 2 and for q = 0 they determined the electron current component of the probe characteristic based on TALBOT and Chon theory [8]. As mentioned already the method for calculation of the collision effect proposed in [8] is not valid in the region 0 c q 5 2. Thus in the region 0 < r ] 5 2 the polynomial Newton interpolation between q = 0 and q = 2 is used in [15]. Such probe characteristic is twice numerically differentiated to obtain a rough idea of the position of the second derivative zero-crossing point relative to the space potential. The results for K , = 0.36 and L,/R,

. cult.

Fig. 4. The calculated second derivative I / ( q ) for K , = 0.36 and L,/R, = 100 made by KLAQQE and !C’IOH$ [16]. Curve for DA = 0 calculated according [30]; for DA = 1,100 calculated with 1:- taken from [a]; for D1 -+ ca calculated under assumption J , = 1


= 100 and various values of D1 presented in [15] are shown in Fig. 4. Although this method can be looked at only as a rough approximation it is evident from Fig. 4 that a significant difference between the space potential and the second derivative zero-crossing point arises if the effect of collisins starts to be important. Hence in such conditions the estimation of the space potential based on the probe characteristic second derivative is very difficult.

The plasma potential can also be determined from the floating potential, see e.g. [21]. At the floating potential VF ( VF is measured with respect to the plasma potential) the total probe current vanishes and thus is can be simply measured. Furthermore from the Fig. 1 it is seen that for sufficiently large normalized probe potential 7 the correction factors for both the electron component of the probe current and its second derivative reach nearly unity. It means that a t nF = IqoVF/kT,j which is a moderately large probe potential the influence of collisions of electrons with neutrals on the probe characteristic is small and the uncorrected electron current component of the probe characteristic is approximately equal to the corrected one. However this does not hold for the ion current since ions have a much shorter mean free path lli than electrons and hence are subject to many more collisions withm the same distance. Thus the floating potential is possible to calculate if the uncorrected electron current at qF is put equal to the ion current calculated according the ZAKRZEWSKI and KOPICZYXSKI collisional model [5, 161 of the ion probe current [22] :

where I+*(vF) is normalized ion current a t floating potential calculated according [16] (see equations (12) to (18)). The normalized floating potential vlF calculated numerically from the transcendental equation (20) for Ne discharge as a function of the Debye number D1 = R,/ID with the ion Knudsen number K i = LJRp as a parameter is depicted in Fig. 5. The described method of determination of t,heplasmapotentialfrom the measured floating potential UF and calculated normalized potential qF using the equation (20) has got, however, one inherent difficulty : The electron temperature which is determined either from the slope of the corrected probe characteristic or from the slope of its second derivative depends via the corrections on the plasma potential which is not known

t K,= 0,12 I -

6,d 0,7 ow 03 l,o - Oh

Fig. 5. The dependence of the normalized floating potential q p with respect to the D1 for several values of Ki as a parameter (Ri = 0.12, 0.14, 0.16, 0.18, 0.20) in a Ne discharge


For this reason at the first approximation the plasma potential is taken at the zero- crossing point of the probe second derivative and then the described procedure is used in an iterative manner to get the final value of the plasma potential.

d) T h e doub le p robe t echn ique The double probe response in the collisional case, i.e. when the collisions of charged

particles with neutrals in the vicinity of the probe cannot be neglected has been studied by KIRCKHOFF et al. [23], KLAGGE and TICHP [15] and BRADLEY and ~ T T H E W S [24]. The first two papers are based on the CHou et al. [6] analysis while the last one used Su and L m continuum theory [25]. In the continuum theory it is assumed that the gas pressure is sufficiently high so that both ions and electrons suffer numerous collisions with neutrals before being collected by the probe. The motion of charged particles is in this case described by the continuity equations.

.' h= 0 _/ _/-- ----- I-

t

Fig. 6. T b e dependence of the correction factor for correction of the electron temperature determined from a double probe characteristic with respect to K, for several values of DA as a parameter calculated by KLAGOE and TIOE+ [15]. ----- D1 = 0 calculated according [30]. D1 = 2.8, D1 = 100 calculated with I*+m taken from [4];

calculated with I$, taken from [17], r = 10, K, = log, , L,/R, = 100, (m,/m,)'P = loo

In order to det,ermine the electron temperature by means of a double probe technique, KLAGGE and 'Red [ 151 calculated a correction factor ,uT for determination of the electron temperature from the double probe data in dependence on the Knudsen number K , and the ratio RJID as depicted in Fig. 6. The value of the electron temperature T, obtained from the measured double probe characteristic using the collisionless double probe theory must be multiplied by this correction factor to obtain the correct value of T, in a collisional case. It can be seen from the Fig. 6 that the correction factor is small. For RJAD 2 2 it is generally less than 10% for a broad range of K , (from 0.01 up to 10). Si- milar results have been obtained by other authors [13, 141. These results mean that the


/ Ii0) / i

Fig. 7a. "he typical double probe characteristic. Iiol and Iiop are the ion probe currents a t

Vd = 0, Ie l and I,, the electron probe currents a t probe potential difference v,; - and - the derivatives of the ion currents

Fig. 7 b. Transformation of the single probe characteristic to the double probe characteristic. I, is the selected probe current, vd, is the potential difference between two points on the single probe characteristic which correspond to the selected values of the probe current I, and -Il , Vd potential difference between two probes, V F floating potential

dIi, dI,, dvd dV,j


electron temperature derived from a double .probe characteristic using the collisionless theory are resonably accurate under the collisional condition too.

For the collisionless case there exist two methods for the determination of the electron temperature from the double probe characteristic. By means of the first one i t is possible to calculate the electron temperature V , = IkT,/pol from the slope of t,he double probe characteristic a t v d = 0 (Vd is the potential difference between the probes in a double probe technique) :

where (dId/dVd)vd=o is the slope of the double probe characteristic a t v d = 0, Iil and Ii, are the ion currents to the first and second probe respectively, Iiol and Iioz are the same currents extrapolated to v d = 0 (see Fig. 7a). If the surface areas of both probes are the same then I;, = I;,.

By means of the second method is the electron temperature V, determined from the following dependence of so called r function with respect to the potential difference vd :

where r = (I,JIe2), I,, and I,, are electron currents to the first and second probe respectively (see Fig. 7a), A , and A, are the surface areas of the first and second probe respectively.

It is important to note that the double probe characteristic can be simply obtained from the single probe Characteristic. The procedure consists in the estimation of the voltage difference vd, between two points on the single probe characteristic which correspond to the same absolute value of the chosen probe current I, but with opposite signs; as depicted in Fig. 7b. The plot of 1, vs Vd, ( v d , plays the role of the voltage v d between two probes in a double probe technique) then creates one half of the double probe characteristic; the other is obtained by turning the first half around the origin by 180".

Exporimental Arrangement

The probe measurements have been performed in a positive column of glow discharge plasma in Ne a t pressures 1.86 kPa, 2.19 kPa and 2.46 kPa. In order to perform measurements in a stable plasma a small amount of mercury has been mixed with Ne. A small glass vessel connected to the discharge tube contained the amount of about 2 cm3 of Hg and has been kept a t room temperature thus the partial pressure of Hg vapours was N 1.6.10-l Pa. The ratio of the partial pressure of Hg to that of Ne was less than The spherical and cylindrical probes have been used.

In the present paper a method of a direct mertsurement of the probe characteristic and its second derivative has been used. The method of the second derivative measurements is based on the measurements of the second harmonic of the AC component of the probe current if a small-amplitude pure harmonic AC voltage is superimposed on the probe bias. The second harmonic component of the probe current has been measured by means of an analog comelator [26]. A sawtooth oscillator has been used to slowly vary the probe bias voltage. The probe characteristic and its second derivative have been recorded by means of the X- Y recorder.

A standard experimental set up has been used for measurements of the double probe characteristic. To reduce the leakage currents the experimental set up has been precisely isolated from earth.

180 DAVID, P., Sfcaa, M., TICHP, M., et al.J Langmuir Probe Methods

Fig. 8. The typical semilogarithmic plot of the single probe characteristic. Cylindri- cal probe dimensions: Rp = 50 pm, L, = 5.6 mm; Ne gas pressure 1.86 kPa, discharge current 10 mA. - measured probe characteristic; ----- electron probe current; - . -a - electron probe current corrected according CHOU et al. theory [6] by using TALBOT and CHOW interpolation formula [8] . V,, is the zero-crossing point of the probe characteristic second derivative

Results and Discussion

The example of the semilogarithmic plot of the probe characteristic is shown in Fig. 8. The electron current component has been obtained by subtraction of the ion current component from the measured probe characteristic under the assmuption that the ion saturation current can be approximated by equation (7). The electron current component was then corrected according &on et al. theory for collisions in the space charge sheath. Since for both the ion current and the collisional corrections the knowledge of the space potential is required the following method of successive iterations has been used. At the first step the plasma potential has been chosen a t the zero-crossing point of the measured probe second derivative. Then the electron current component has been extract- ed from the measured probe characteristic and corrected for collisions in the space charge sheath. From the corrected electron current semilogarithmic plot the electron density and temperature has been estimated as the first iterative step. In order to make the second iterative step the plasma potential has been determined from equation (20) from the measured floating potential using the values of plasma parameters obtained in the first iterative step and the described procedure has been repeated. The next succesive iterative step has also been made and an example of results after three iterative steps is shown in Fig. 8.

From Fig. 8 it is seen that the probe potential which corresponds to the zero-crossing point of the probe characteristic second derivative is negative with respect to the plasma


- # M3-

g -2 m

t m -

u

5 0 -

potential obtained by means of the described method. This is in agreement with investigations made by KKAGGE and T I C H ~ [ 151 (see Fig. 4) and with conclusions made by SEIFERT et al. in [Zl].

The example of the semilogarithmic plot of the second derivative is presented in Fig. 9. The second derivative of the electron current component has been determined from the measured second derivative of the probe characteristic using the equation (8). After that the second derivative of the electron current has been corrected to include the effect of collisions using CHou et al. theory. For purpose of this correction the value of the plasma potential has been used which has been obtained by means of the above described probe characteristic processing. The corrected second derivative is also shown in Fig. 9.

-

In Figs. 8 and 9 the corrected semilogarithmic plot of the probe characteristic electron current component and its second derivative vs the probe potential is linear and thus the bulk of the electron energy distribution function should be Maxwellian. This fact justi- fies the use of CHOU et al. theory which assumes the Maxwellian energy distribution function of electrons for the probe collisional correction in our experimental case.

The charged particles density has been determined by three methods. The first one uses for evaluation of the electron density the electron probe current a t the space potential which has been in our w e obtained from the corrected electron probe current extrapolated to the space potential. The second and the third one determines the positive ion density from the positive ion probe current component. In the second method we used the experimentally determined ion probe current corrected according CHOU et al. theory and finally in the third one we applied to the ion probe current the ZAKRZEWSKI

182

10"

DAVID, P., &HA, x., TIC-, M., e t al., h n g m u i r Probe Methods

0

-

x)" c

A

0 0

! *

I

A

0

b

5 10 --"[mAI

Fig. 11. The values of the charged particles concentration n obtained a t the Ne pressure 2.46 kPa with respect to the discharge current I , ; + electron concentration obtained from the extrapolated electron probe current corrected according CHOW et al. [6] to the space potential; o positive ion concentration obtained from the ion probe current corrected according CHOW e t al. [6]; A positive ion concentration obtained by means of ZAKRZEWSKI and KOPICZYNSKI method [5, 161.

and KOPICZYNSKI method. The values of charged particle concentrations obtained by described methods are shown in Figs. 10 and 11 for two gas pressures. As it is seen from Figs. 10 and 11 the agreement between values obt.ained by different methods is quit.e good. This takes place despite the ad hoc interpolation formula used for calculation of integrals J, and J i in the CHOW et al. theory and despite the comparatively crude methods used for estimation of the number of particle collisions x in the space charge sheath by KOPICZYNSKI [16]. From the spread of the results which are presented in Figs. 10 and 11 it is possible to estimate approximately the accuracy of the electron concentration measurements in a middle pressure discharge in our experimental conditions as one signi-

entrib. Plaema Phys. 80 (1990) 2 183

ficant digit. The electron temperature data obtained from single and double probe measurements are presented in Tab. I. In the single probe experiments the electron temperature was determined from the electron probe current and its second derivative corrected for the effect of collisions according Chon et al. theory. Furthermore the electron temperature was calculated from the uncorrected double probe characteristic by means of equations (21) and (22). From the Tab. 1 it is seen that the electron temperatures obtained by means of corrected single probe characteristic are in a good agreement with those obtained from the uncorrected double probe characteristic. This is assumed to be due to that in double probe measurement the probe bias is sufficiently negative causing the electron current collisional correction factor to be near unity (see Fig. 1). The fact that the uncorrected double probe characteristic is possible to use for electron temperature determination in a high pressure plasma was experimentally proved also in [27] and [28] where the double probe method has been compared with the optical one.

From results which are presented in Tab. 1 follows, that the accuracy of electron temperature measurements by means of a Langmuir probe in a middle pressure discharge was in our experimental conditions better than one significant digit.

Table 1 Measured values of electron temperature [Vl Ne gas pressure 1,86 kPa

determined from: discharge current probe second double probe

~~

character. derivative equation

[mAI (21) (22)

Conclusion

The presented results show that it is possible to use the probe technique as a diagnostic tool in a middle pressure plasma. This is in agreement with experimental results presented in [27], [28] and [14] some of which indicate that the probe technique is applicable even for the flame plasma a t atmospheric pressure.

The advantage of the probe method in comparison with optical and microwave ones consists in that it is not necessary to know the particle density profiles. The disadvantage on the other hand is that for correct interpretation of Langmuir probe data at middle pressures a series of correction processes of the measured probe characteristic is necessary. For minimization of the number of correction processes the following way is recommended : 1) The electron temperature is determined from the double probe characteristic. The double probe characteristic is either directly measured or i t is obtained by means of the transformation from the single probe one (see Fig. 7b). 2) At the first iterative step the plasma potential is assumed to be equal to the probe voltage a t which the measured probe characteristic second derivative zero-crossing point occurs. Then from the ion probe'current by means of Z ~ Z E W S K I and KOPICZYNSKI method it is possible to determine the charged particles density (see Bigs. 3 a and 3b). 3) At the second iterative step the plasma potential is determined from the experimental value of the floating potential. To do this the normalized difference between plasma

2 Contrib. Plasma Phys. 30 (lQQ0) 2

184 DAVID, P., Sfmil, J l . , TICHY, M., e t al., Langmuir Probe Methods

and floating potential qF is calculated according the equation (20) (see Fig. 5 ) , converted to difference in volts using the electron temperature estimated in point 1) and added to the measured floating potential. Then it is necessary to determine once more the charged particles density from the ion probe current by means of ZAKRZEWSKI and KOPICZYNSKI method. Since from Fig. 5 it is seen that the difference between the floating and the space potential depends on the Debye number only slightly the third iterative step is usually not necessary.

By the described way it is possible to determine the plasma parameters such as electron temperature and charged particles density with a reasonable accuracy without the necessity to calculate the correction factors. For the increase of accuracy of the plasma parameters determination it would be necessary to develop a more sophisticated model of ion collection by a Langmuir probe than that made by ZAERZEWSKI and KOFIC- ZYNSKI. For example (S/,l,) may not be the characteristic parameter for determination of the number of collisions within the space charge sheath around the probe. Also the collisions in the presheath region may have an influence on the orbit motion destruction.

Refersnces

[l] LANQMUIR, I., MOTT-SNITH, H. M., Gen. Elec. Rev. 26 (1923) 731; 2 7 (1924) 449, 538, 616,

[2] ALLEN, J. E., BOYD, R. L. F., REYNOLDS, P., Proc. Phys. SOC. B 70 (1957) 297. [3] BERASTEIN, I. B., RABINOVITZ, J. M., Phys. Fluids 2 (1959) 112. [4] LAFRAMBOISE, J. G., Univ. of Toronto, UTIAS Rept. KO. 100 (1966). [5] ZA~RZEWSKI, Z., KOPICZYNSKI, T., Plasma Phys. 16 (1974) 1195. [6] CHOW, Y. S., TALBOT, L., WILLYS, D. R., Phys. Fluids 9 (1966) 2150. [7] DAVID, P., CSc Thesis,-Charles University, Prague (1985). [8] TALBOT, L., CHOU, Y. S., Rarefied Gas Dynamics, Ac. Press (1966) 1723. [9] TICHP, M., CSc Thesis, Charles University, Prague (1979).

762, 810.

[lo] DRUYVESTEYN, M. J., Z. Physik 64 (1930) 781. [ l l ] CASE, T. C., Proc. X-th ICPIG, London (1971). [12] TrcrrP, M., Czech. J. Phys. B 28 (1978) 1335. [13] NUHX, B., PETER, G. Proc. XIII-th ICPIG, Berlin 97 (1977) 999. [14] SfcH.4, M., Czech. J. Phys. B 89 (1979) 640. [15] KLAQQE, S., TI^, M., Czech. J. Phys. B 35 (1985) 988. [16] KOPICZYNSEI, T., DSc Thesis, Institute of Fluid Flow Machines, Polish Academy of Sciences,

[17] BASU, J., SEN, C., Japan. J. Appl. Phys. 12 (1973) 1081. [18] SHULTZ. G. J., BROWN, S. C., Phys. Rev. 98 (1955) 1642. [19] JAKUBOWSKI, A. K., AIAA Journal 8 (1972) 988. [20] AISBERO, S., Proc. 3-th Symposium on Engeneering Aspect of Magnetohydrodynamics (1963)

1211 SEIFERT, W., JOEANNIXQ, D., LEEVANX, H. R., BAXKOV, D., Contr. Plasma Phys. 26 (1986)

[22] ~ ~ C H Y . , X., SfCHA, M., to be published. [23] K I R C ~ O F F , R. H., PETERSON, E. W., T ~ B O T , L., AIAA Journal 9 (1971) 1686. [24] BRADLEY, D., MATTHEWS, K. L., Phys. Fluids 10 (1967) 1336. [25] Sn, G. H., LAM, S. H., Phys. Fluids 6 (1963) 1479. [26] S A F R ~ E O V ~ , J., NBJIEEEIC, Z., Czech. J. Phys. B 24 (1974) 117. [27] FERDLYAND, J., Proc. XI-th ICPIG Praha (1973). [28] FERDINAND, J., &A&, J.. LEGO, J., Proc. 6 Int. Conf. MHD Munich I (1971) 57. [29] CHEN, F. F., Plasma Phys. 7 (1965) 47. [30] CHANQ, J. S., LAFRAMBOISE, J. G., Phys. Fluids 19 (1976) 25.

Gdansk, Poland.

89.

237.

Received June 6,1988

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The use of Langmuir Probe Methods for Plasma Diagnostic in Middle Pressure Discharges