pitts_ppcf_32_1237_1990

Plasma Physics and Controlled Fusion, Vol. 32. h o . 13, pp 1237 to 1248, 1990 Printed in Great Briiain

0741-3335 90 $3.00t .00 IOP Publishing Ltd. and Pergamon Press plc

EXPERIMENTAL TESTS OF LANGMUIR PROBE THEORY FOR STRONG MAGNETIC FIELDS

R. A. PITTS AEA Fusion. Culham Laboratory, UKAEAjEuratom Fusion Association, Abingdon,

Oxon OX14 3DB. U.K.

and

P. C. STANGEBY Institute for Aerospace Studies, University of Toronto, Canada and JET Joint Undertaking.

Abingdon, Oxon, OX14 3EA. U.K. (Receiced 23 April 1990 : and in revised f o r m 13 JulF 1990)

Abstract-The suppression of electron current to a single Langmuir probe immersed in a strongly magnetized plasma has been the subject of theoretical analysis since the original work of BOHM et ai. [1949, in Characteristics ofBlectricalDischa~ges in Magnetic Fields (Edited by A. GUTHRIE and R. K. WAKERLING). McGraw-Hill. New York]. Experimentally, studies of the phenomenon have been restricted to a comparison of measured data with analytic predictions of the I-V characteristic. Both theory and experiment show that the simple exponential law for electron collection assumed in the derivation of T, from the characteristic no longer holds for probe potentials above the floating potential. It is assumed, however, that the portion of the characteristic below floating potential can be used in deriving T,. By monitoring the potential of a small, electrically floating pin as a function of the voltage applied to a much larger plate located just behind it and interpreting the pin voltage as a direct measure of the plasma potential near the plate, the validity of this important assumption has been experimentally confirmed for the first time. Certain theoretical aspects of net electron collection have also been tested and have been found to be in reasonable agreement with experiment, indicating that more effort may be justified in attempting to use this region of the characteristic as a diagnostic tool.

1 I N T R O D U C T I O N DUE TO the extensive application of the single Langmuir probe as a Tokamak edge diagnostic (MANOS and MCCRACKEN, 1986), theoretical study of the probe characteristic from magnetized plasmas has been the subject of a number of papers (BOHM et al., 1949; COHEX, 1978; STANGEBY, 1982). Whilst the ion saturation region of the characteristic is thought to be reasonably well understood, the same cannot be said for the region corresponding to electron collection where the net current is observed to be much less than that which would be obtained in a non-magnetic plasma. Indeed, the ratio Is;t/I&t is often observed to be only - 10% of that pertaining to the field free case (i.e. 6).

One theoretical study (STANGEBY, 1982) models the complete Langmuir probe characteristic and, for probe bias potentials above the floating potential (at which point there is no net current to the probe), finds a marked deviation from the simple exponential law applicable in the field free case. A reduction factor, r , is defined describing the depression of electron current in electron saturation over the B = 0 case [see equation (2)]. This factor contains a number of parameters which are generally unknown to the experimentalist attempting to apply the theory to Langmuir probe data (ion temperature, T,, electron cross-field diffusion coefficient, D y, impurity

1237

1238 R. A. PITTS and P. c. STANGEBY

content, ZeE, in the plasma edge). Nevertheless, several authors (LABOMBARD, 1986 ; BUDNY and MANOS, 1984; LAUX et al., 1989) have made such attempts.

As a consequence of the departure from ideal behaviour above the floating potential, the recommended procedure for use of Langmuir probes in strong magnetic fields is to ignore this region and extract T, from the characteristic for V, d V, where it is thought to be less likely that distortions are occurring. Experimental work on JET (TAGLE et al., 1987) showed that errors in T, occurred if the reduction effect was ignored and too great a fraction of the characteristic used in the conventional analysis (i.e. assuming electron collection is simply exponential with applied potential). This is not, however. confirmation that the ion saturation region gives T, correctly, since the use of standard theory to derive it assumes that all of the probe bias is taken up by the sheath. In fact, there is no a priori reason to assume in a magnetized plasma, where the probe is usually biased with respect to limiter and cross-field diffusion must often close the circuit over long distances in the plasma. that some of the probe voltage difference does not appear across magnetic field lines to drive the circuit current. The question, therefore, is whether T, can be reliably extracted from any portion of the I-V characteristic for a probe in a strong magnetic field.

In this paper, we present the results of a new approach to the problem of inves- tigating the magnetized Langmuir probe characteristic. By concentrating on the behaviour of the potential in the region close to a probe as the voltage on it is varied. we are able to draw useful conclusions concerning the validity of the standard method described above for analysis of the magnetized probe characteristic. In addition, our experiments demonstrate how net electron collection can be reasonably well described by theory and indicate that it may be possible to derive usable information from this region of the characteristic.

2 . T H E P O T E N T I A L RISE N E A R A BIASED PROBE In the standard picture (STANGEBY, 1982 and Fig. 1), the probe is assumed floating

(V, = V i ) , of square shape (width d ) and normal to the magnetic field. A region of disturbance termed the ambipolar collection length ( L,amb z d2c,/8D";"b. with c, the ion sound speed, DImb the ambipolar cross-field diffusion coefficient), develops along the field into which ions and electrons diffuse across field lines to replenish parallel particle loss to the solid surface. During net ion collection, the large ratio of ion to electron mass prevents the drifting ions from losing significant momentum by collision with the electrons. Thus, although the bias may influence the cross-field transport rate and hence L:mb [and there is experimental evidence that it does (MATTHEWS and STANGEBY, 1989)], conditions should remain much the same as in the floating probe case. Until now; there has been no direct experimental confirmation of this assumption.

When the probe is biased into net electron collection, the electrons suffer significant momentum loss by collision as they are drawn to the probe through the comparatively stationary ion fluid and it is proposed (BOHM et al., 1949; COHEN, 1978; STANGEBY, 1982) that an electric field develops along the probe flux tube [now of length L;, where L; >> Limb (MATTHEWS and STANGEBY, 1989)] to aid the electrons in over- coming this friction. As a consequence, the ions find themselves in a retarding field and a density depression appears somewhere in front of the probe (approximately one electron/ion collision mean-free-path in front). Figure 1 illustrates schematically

Tests of Langmuir probe theory 1239

FLUX TUBE

d

in1 I .P I :r P resh ea t h

--] Sheath

FIG. 1 .-Qualitative variation of the sheath and presheath potential distribution along the magnetic collection tube of a probe under floating conditions and biased into net ion and

electron collection.

the behaviour of the potential in the presheath of a probe biased into net electron collection (but not electron saturation).

Use of the Boltzmann relation (based on TI) for the ions in this presheath retarding field provides a connection between the density at the crest of the hill potential and the magnitude of the potential. Combining this with a formulation for the parallel electron current leads to an expression (STANGEBY, 1982) relating the hill potential to the probe potential for all V, > Vf,

where yp = VJT , and qH = V,/T, are respectively the normalized probe and hill potentials (T , in eV) relative to the distant plasma potential, V,,,,,,, defined as zero, and z = Tl/Te. The reduction factor is given by (STANGEBY. 1982; TAGLE et al., 1987)

with ;.el the electronjion collision mean-free-path (mfp) and CY = D ; / D ; the ratio of electron diffusion coefficients across and along field lines (classically. D = ;%,&/4 with C, the Maxwellian mean speed). The transcendental relationship of equation (1) can be solved for the unknown q H and the result used in an expression for the net current in order to reproduce the observed characteristic for given r and 5 .

Using the pin and plate probe to be described in the next section, we are able to

1240 R. A. PITTS and P. C. STAKGEBY

measure the magnitude of the potential hill, V,, directly and hence use equation (1) to check the validity of the model. It is important to note; however, that the theory incorporates a number of debatable assumptions. For example, part of the electron cross-field transport is attributed to anomalous cross-field mobility, p y , which is assumed to satisfy the traditional Einstein relation between mobility and diffusion, D i p = - kT,je. Since, for classical, ambipolar cross-field transport in a fully ionized plasma, there is no cross-field mobility (CHEN, 1984), the concept of an anomalous, net electron 1.1- is clearly questionable. In the highly turbulent edge plasma, however, the existence of mobility-like cross-field transport processes driven by radial electric fields may play an important role. Indeed, in a recent proposal to protect edge structures such as ICRH antennas through the creation of an electrostatic-barrier scrape-off layer using an externally imposed positive bias, LABOMBARD (1 990) includes cross-field mobility in a non-ambipolar, two-fluid transport model of the boundary plasma.

3 . E X P E R I M E N T Measurements have been made on the DITE Tokamak using the pin and plate

probe (Fig. 2), also described in an earlier paper (STAKGEBY et al., 1984). A small tungsten plate of dimension 5 x 10 mm and a cylindrical pin length 5 mm and diameter 1 mm are housed within a graphite probe head which is electrically connected to a stainless steel probe tube. The pin and plate are rigidly fixed into a machineable-glass ceramic base and are thus isolated from each other and the probe tube. Their separation is 2.5 mm, measured from the front face of the plate to the pin axis. Bias voltages may be applied and currents to each element measured simultaneously with all bias potentials referenced to the torus. In all cases, the plate is used to determine the local T, and n, since uncertainties in the projected area of the pin during ion collection can introduce errors in the calculated density (STANGEBY et al., 1984).

The probe was inserted from the top of DITE through a standard drive assembly

GRAPHITE

CERAMIC

PROBE TUBE

FIG. 2.-Cut-away isometric of the pin and plate probe head


along a vertical chord displaced - 4 cm outwards of the magnetic axis (R = 1.20 m). Experiments were performed in deuterium for I , = 100 kA and B, = 2.0 T with the pin and plate at relatively large minor radii (rprobe - 28-29 cm) facing the ion drift direction. Under these conditions, the shortest connection length to the plate, L,,, -v 3.9 m, is from the DITE pumped limiter located 180" away and centred on the outside midplane defining a plasma minor radius of a = 0.24 m. This should be compared with the collection length defined by the probe housing, L:mb - 3.0 m, for the edge conditions pertinent to this study.

4 RESULTS A N D DISCUSSION

4.1. General remarks Figure 3 gives an example of the typically observed behaviour. The data have been

obtained during the steady-state phase of the discharge with 2, = 2.2 x 10" m- 3,

r'probe 2: 28 cm. A ramped voltage with period 50 ms is applied to the plate so that the usual Langmuir characteristic is obtained (Fig. 3a). At the same time, the voltage appearing on the pin is monitored when the pin itself is floating (Fig. 3b), or the pin ion saturation current (Fig. 3c) is measured for a large negative bias applied to the pin (the data in Figs 3b and c were obtained in consecutive identical discharges and have been smoothed slightly for clarity). Included in Fig. 3a is a list of the standard parameters derived from the normal non-linear least-squares fit to the characteristic using only data for V,,,,, d Vglate. The fitted line is also shown. As an illustration of the point made earlier, if the characteristic in Fig. 3a is re-analysed including all data up to the point just before electron saturation is reached, a much poorer fit results and the fitted temperature at T, = 13.8 f2.S eV is more than a factor of two greater.

Note that the plate voltage scan reaches positive values high enough to push the plate well into electron saturation and that the ratio ZG~/Z& - 8 is much smaller than the field free value (i.e. - 60 for a pure D- plasma). The variation of VLln with V,,,,, is a striking illustration of the marked difference between the physics of net ion and electron collection. If the pin voltage really is a direct indication of the plasma potential near the plate, then the theory predicts little change in its value for V,,,,, d Beyond this point, net electron collection occurs and the behaviour of V,,,,,, should be governed by equation (1) if the theory is correct. Inspection of Fig. 3b shows that the pin voltage is indeed constant for most of the negative part of the voltage sweep. This observation rules out any possibility of a (constant) resistive path for leakage current between pin and plate. In addition, for these conditions, the local Debye length, A,, = (EoTe/n,e)' * z 25 pm and, since the sheath potential fall occurs over a distance of order IO& (ALLEN, 1974), the piniplate separation (= 2 mm) should be sufficient to ensure that the pin potential is not directly affected by the plate sheath.

4.2. Net ion collection Concentrating on the region of plate voltage V,,,,, < V,f,,,,, the pin ion saturation

current remains relatively constant and the ratio I,',, (plate)/Z& (pin) 2: 3.2 is much lower than the ratio of the projected areas (2 lo), in agreement with the findings and conclusions of STANGEBY et al. (1984) regarding the ambiguity of the effective ion collection area of small objects. Also, the pin floats at a fixed value of a few volts negative with respect to the plate floating potential throughout most of the net ion

1242 R. A. PITTS and P. C. STANGEBY

- 0.24 -

80 60 40 20 0 20 40 60 80

Plate Voltage (VI FIG. 3.-Typical examples of the experimental data obtained with the probe; (a) plate I-V characteristic, (b) behaviour of pin floating potential in response to plate bias, (c) behaviour

of pin ion saturation current with plate potential. Plasma conditions as in text.

collection region of the plate characteristic. This is direct experimental confirmation that most of the bias voltage change applied to the plate is indeed taken up by the plate sheath. Noting from Fig. 3a that VLlate = -2.3 V, it is clear: however, that Vi,,, begins the steep increase evident in Fig. 3b before Vplate = Vfplate is attained.

To illustrate this, Fig. 4 concentrates on the behaviour of Vil,, in the small region of Fig. 3b near the plate floating potential and shows that the increase begins near


C Y- '6 >

Vplate (V) FIG. 4.-Expanded region of V;," from Fig. 3b near the plate floating potential.

V,,,,, - - 15 V at which point VL,,, has decreased to a value of - -2.5 V relative to its value when V,,,,, = VLlare. Note that VLlate is also marked on this curve and that it is merely coincidental that Vbln N 0 at this point. It is generally the case, however, that in our data Viln reaches a constant magnitude some 2-3 V below its value when VPI,,, = VLlare. The increase in VLln occurs just in the voltage range most important for the extraction of T,, i.e. - - 3kT,/e < V,,,,, < Vilate (for Boltzmann electrons). In simple probe theory, such a difference would not be expected since, during net ion collection, the plasma potential is assumed to remain unchanged from its value at

f Vpldte = Vplare .

There are two possible explanations for this observation :

1 ) A small fraction of the bias voltage appears not across the plate sheath, but across magnetic field lines completing the biasing curcuit. If this is the case, Fig. 4 implies that the maximum error incurred in using data for VPidre < VgId,, to extract T, will, in this example, be - 2.5115, that is, an error of - 17%. A survey of all available data showed this result to remain approximately true although our measurements encompass a T, variation of only a few electron volts. As the plate voltage decreases below VLlare, electrons are repelled in increasing numbers from the plate and some will be collected by the nearby pin. Since, by definition, the floating pin must collect zero net current, its potential must decrease to compensate for the excess negative current and the decrease will continue until V,,,,, = - - 3kT,/e when the maximum possible number of electrons are being repelled.

( 2 )

Of course, if the effect is due to the second process, our results indicate that there will be a relatively small error in the electron temperatures deduced from the characteristic

1244 R. A. PITTS and P. C. STAXGEBY

using data in this region. However, since the nature of cross-field diffusion is not understood, we cannot rule out the possibility that a small fraction of the bias voltage is appearing elsewhere.

4.3. Net electron collection Above the plate floating potential, VLin increases monotonically with V,,,,, (Fig.

3b), indicating the formation of a potential hill in front of the plate as predicted. Recalling that the crest of the hill is expected to form -A,, away from the plate (STANGEBY, 1982) it is clear that, since Ael is in the range 10-30 cm for these conditions (depending on choice of ZeE, T, etc.), the pin will not be located at this position. Provided, however, that the piniplate separation is small in comparison with the distance between the hill crest and the plate, the pin should be a measure of the potential rise. This is illustrated in Fig. 5, where the potential distribution in the presheath has been approximated as linear for convenience. In the simple fluid theory of plasma flow to a surface (STANGEBY, 1986), a total presheath drop of -0.5kTJe exists to accelerate the ions to sonic speed by the time they reach the sheath edge. For a stable sheath to form in net electron collection, the same -0.5kTJe must also exist from the top of the potential hill to the plate sheath. Thus, as shown by Fig. 5, the magnitude of V, should be correctly monitored by the pin.

When the plate is biased into electron saturation, the sheath no longer exists and the same conclusions regarding the presheath fall do not apply. Indeed, the theory (STANGEBY, 1982) applies only up to the point where electron saturation begins. Beyond this point, the plasma is unshielded from potentials applied to the plate and is presumably strongly perturbed. I t is evident from Fig. 3b, however, that no sharp break is observed in the behaviour of VLln at the value of V,,,,, corresponding to the onset of electron saturation. Such a change in gradient might be expected if. in the absence of a sheath, the plasma potential simply followed that applied to the plate,

VpIasma

0

'i / Location of pin

-cl Sheath

FIG. 5.--Schematic of the potential variation in the presheath showing how V;)" should be a direct measurement of V,.

Tests of Laiigmuir probe theory 1245

although this portion of the characteristic is difficult to interpret owing to the sub- stantial plasma disturbance caused by drawing large electron currents.

Proceeding with the assumption that Vkln gives V , directly [at least until I;, (plate) is reached], it is instructive to compare the experimental data with the variation of VH predicted by equation (1). To do this requires only input values of r and T, since T, is determined from the plate Langmuir characteristic. Once TI has been chosen, r may be calculated from equation (2) if Re, and D are known and D I is assumed classical.

Retarding field analyser measurements (PITTS et al., 1989) in deuterium at the same poloidal. toroidal and radial location and for similar global operating conditions as those relevant to these experiments, indicate values of TI in the range 20-30 eV. Choosing T, = 25 eV gives 7 - 4.5 for the example in Fig. 3. The electronlion collision mfp is given by (WESSON, 1987)

where T, is in eV, n, in m P 3 and with log, A the Coulomb logarithm ( w 13 for conditions in these experiments). Mass-spectrometric measurements of the charge state distribution in the DITE boundary (MATTHEWS, 1989) indicate the effective charge to be - 3 in deuterium discharges. The density may be calculated in the normal way (STAKGEBY, 1986) using the value of Is;, from the characteristic in Fig. 3a

where c, = ([e(T,+ T,)]/TN~)'~~ and Aplate is the plate area. This gives n, = 3.9 x 10'' m-3, leading to i,,, = 28.5 cm and 0 ; = 1.12 x 10' m2 S K I . Finally, our estimate of r requires knowledge of D I -an experimentally unknown quantity. Previous measurements of Dymb in the DITE edge (MATTHEWS et al.: 1987) indicated values comparable to the empirical Bohm expression (BOHM et al., 1949), i.e. DImb e 0.06TJB N 0.17 m2 s- ' for our example. Setting D; = Dymb = 0.2 m2 s-I gives r = 0.60.

A second estimate for the reduction factor can be made directly from the experimental data using an equation from the theory (STANGEBY, 1982) giving the expected electron saturation current in a strong magnetic field

Using the experimental values of I;,, T, and from Fig. 3a and the calculated density, equation ( 5 ) gives r = 0.80. We note that the relatively close agreement between these two estimates of r is perhaps fortuitous given the assumptions involved in deriving the theory (Section 2).


Theory and experiment are compared in Fig. 6, where data from Fig. 3b have again been used. this time concentrating on the behaviour of VLln for V,,,,, 3 VLlate. The theoretical prediction for T, = 25 eV, T, = 5.5 eV and the two values for the reduction factor are shown as dotted curves. The theoretical curves are generated by using the experimental values of Vfiln to calculate V,,,,, for the given Y and z. Since equation (1) is only valid for the case of electron collection, the point where V,,,,, = VLiate has been taken as the reference point for all voltages.

An evident feature of this comparison is the relatively close agreement between the theory and experiment in terms of the shape of the dependence on plate voltage. That the experimental data behave in approximately the predicted manner is encouraging and indicates again that the increase in Viin is not due to some simple instrumental effect such as leakage currents. On the basis of scale lengths, one would certainly expect collisional behaviour since the computed value of iel is very much lower than the connection or collection lengths given in Section 3. This helps to justify the use of equation (1) for comparison, since the theory assumes the potential fall along the flux tube to develop as a consequence of momentum transfer collisions between the drifting electrons and the quasi-stationary ions. Moreover, since the theory leading to equation (1) is derived on the assumption that the electron drift velocity toward the probe greatly exceeds that of the ions, it will not strictly apply in the voltage region very near VLldte where U; just begins to exceed E D . If instead, the theory and experiment were not really comparable until V,,,,, is N kT,/e above Vglatewhen the drift velocities begin to differ significantly, one could argue that shifting the theoretical curves up to

- 5 0 -LO -3G -20 -10 0 10 20 30 LO 50 50 70 80

" w e (")

FIG. 6.-Comparison of experimental data with the predictions of equation (1) for fixed T, = 25 eV, Z,, = 3.0 using two values of r calculated in the text and for a single value of r

using equation (6).


Vplare axis by this amount would approximately allow for the effect. This would improve the agreement between experiment and theory.

For sufficiently large values of VHIT,, the logarithmic term in equation (1) is dominated by the exponential and the equation becomes

so that the theoretical curve in this approximation is simply linear with slope determined solely by the ratio z and with an intercept on the abcissa governed by T, and r . Using once more the experimental values of V , to calculate Vplate and choosing r = 0.80, T, = 25 eV, equation (6) has been used to generate the third theoretical curve in Fig. 6. The result is clearly a reasonable approximation throughout most of the plate voltage range. Equation (6) also demonstrates how future experiments encompassing a wider range of parameter variation might hope to observe significantly different behaviour of VH. For example, measurements of T, and T, in DITE (PITTS et al., 1989) have shown variations in z greater than a factor 2 for a 5-fold increase in 5,.

Finally, we note that the behaviour of I,+,,(pin) (Fig. 3c) during net electron collection is not predicted theoretically. The abrupt increase when V,,,,, = VLiate is repro- ducible and is in contrast to the density depression (STANGEBY, 1982) expected in this range of plate voltage. Given the negative bias on the pin ( - - 90 V), it may be that the large potential differences between pin and plate are responsible for complex two- dimensional effects occurring there (e.g. the deflection of ion trajectories from plate to pin) and the simple theory becomes invalid. Note, however, that IGt (pin) does begin to decrease around V,,,,, = 25 V corresponding to the onset of electron saturation when the space potential is reached, the sheath disappears and the plate begins to repel ions.

5 . CONCLUSIONS The behaviour of the plasma potential near a biased element in a Tokamak edge

plasma can be monitored directly by observing the changes in potential on a second, electrically floating element, located a short distance away. Such measurements pro- vide direct evidence for strong perturbations to the local plasma potential when a probe is above its floating potential. This implies that beyond this point the collected current no longer obeys the simple exponential electron collection assumption generally applied to the interpretation of Langmuir probes operating in strong magnetic fields.

During net ion collection, the plasma potential remains fixed until Vpiprobe N 3kTJe below the floating potential and then rises a few volts until Vp/probe = VLrobe. This may be due either to an extraneous effect or to some of the probe bias voltage appearing not across the sheath, but elsewhere in the biasing circuit. If this is the case, our data indicate that an error of - 15-20°/0 should be associated with any value of T, derived from analysis of the magnetized probe characteristic using only data for vpiprobe d VLrobe. This is in addition to any error generated in fitting the characteristic.

The experimental dependence of the local plasma potential on probe bias during


electron collection is in reasonable agreement with the predictions of simple theory. This is despite the assumptions inherent in the model and the lack of any good estimate of the electron diffusion coefficient across magnetic field lines. It does, however. help to increase confidence that the basic picture of the physics of electron collection (BOHM et al., 1949; COHEN, 1978; STANGEBY, 1982) is of some value. In turn this suggests that further related studies might improve our understanding and hence enable the Langmuir probe to be used as a valuable diagnostic tool not just for T,. but also for T,, Z,, and D; .

Acknowledgements-The authors would like to thank Drs G. M. MCCRACKEN and G. F. MATTHEWS for helpful comments on this work and the DITE team for their assistance in making these measurements possible. PCS acknowledges support from the Canadian Fusion Fuels Technology Project.

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BUDKY R. and MANOS D. M. (1984) J . ?hicl. Mater. 121, 41. CHEK F. F. (1984) Introduction to Plasma Physics and Controlled Fusion, 2nd Edn. Vol. 1. p. 188. Plenum

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MAKOS D. M. and MCCRACKEN G. M. (1986) Physics ofPlasma-Wall Interactions in Controlled Fusion

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