Plasma Diagnostic by Probes

NATIONAL RESEARCH TOMSK POLYTECHNIC UNIVERSITY





M. Tichy, V.F. Myshkin


PLASMA DIAGNOSTIC BY PROBES

Recommended for publishing as a study aid by the Editorial Board of Tomsk Polytechnic University








Tomsk Polytechnic University Publishing House
2015

     UDC 533.9.082.7
     BBC В333.481.2
          Т81


    Tichy M.

T81       Plasma diagnostic by probes : educational tool / M. Tichy, V.F. Mysh     kin ; Tomsk Polytechnic University. - Tomsk : TPU Publishing House, 2015. - 126 p.

          Scientific basis of low-temperature plasma probe diagnostics is described in the textbook. Overview of probe methods of plasma diagnostics and a wide range of issues in their practical use is shown. Theoretical study of these methods, detailed description of the design features and the necessary formulas are given. Estimates of possible errors are made, a variety of tests are described. Using the Langmuir probe allows to obtain reliable data on the concentration of free electrons with high spatial and temporal resolution and to obtain estimates of the ratio of the concentrations of negative ions and free electrons in the plasma.
          Designed for students studying in the direction 140800 «Nuclear physics and technology».


UDC 533.9.082.7
BBC В333.481.2




Reviewers
Leading Researcher of V.E. Zuev Institute of Atmospheric Optics
V.A. Khan

Leading Researcher of V.E. Zuev Institute of Atmospheric Optics
Z.T. Dmitrieva



For the Tomsk University compiled from the cited sources by Milan Tichy Charles University in Prague, Faculty of Mathematics and Physics, Department of Surface and Plasma Science, VHolesovickach 2, 180 00 Praha 8, Czech Republic




                                         © FSAEI HE TPU, 2015
                                         © Tichy M., Myshkin V.F., 2015
                                         © Design. Tomsk Polytechnic University

Publishing House, 2015

CONTENTS


FOREWORD......................................................5
1. INTRODUCTION ..............................................8
 1.1. Probe shapes and probe types ...........................8
 1.2. Measuring the probe characteristic .....................9
 1.3. Probe characteristic and its interpretation ........... 14
 1.4. The working regimes of the Langmuir probe ............. 18
 1.5. Advantages and disadvantages of the Langmuir probe diagnostics method....................21

2. THE LANGMUIR SINGLE PROBE METHOD ........................23
 2.1. Theoretical foundations of the Langmuir probe method .23
  2.2.1. Probe current at q ᵥ Up > 0..................................24
  2.2.2. Probe current at q ᵥ Up < 0..................................25
3. GENERAL THEORIES OF THE CURRENT TO A LANGMUIR PROBE ..................................28
 3.1. Starting system of equations .........................28
 3.2. The „cold ion“ model by Allen, Boyd and Reynolds (Ti / Tk « 0, spherical probe)........................28
4. THE DRUYVESTEYN METHOD FOR ESTIMATION
  OF THE ELECTRON ENERGY DISTRIBUTION FUNCTION (EEDF) ...................................................32
5. PROBE DIAGNOSTICS OF ANISOTROPIC PLASMAS ..........................38
6. LANGMUIR PROBE IN THE NON COLLISION-FREE CONDITIONS (TRANSITION REGIME OF PRESSURES) ...........45
7. LANGMUIR PROBE IN A MAGNETISED PLASMA ...................53
8. SPACE AND TIME RESOLVED LANGMUIR PROBE METHOD.................................59
 8.1. Space resolved Langmuir probe measurements ...........59
 8.2. Time resolved Langmuir probe measurements ............59
  8.2.1. Time resolved probe measurements in periodically changing plasmas at ® < coₚi..................................62
  8.2.2. Probe measurements of time-averaged plasma parameters at cop i < ® << ®ₚₑ..................................62
  8.2.3. Time resolved probe measurements in single shot experiments .67
9. PROBE DIAGNOSTIC OF CHEMICALLY ACTIVE PLASMAS ...68
10. DOUBLE PROBE TECHNIQUE ...........................................70

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11. TRIPLE PROBE TECHNIQUE ...................................73
12. EMISSIVE PROBE ..........................................76
13. SPECIAL PROBES ..........................................83
 13.1. Plasma impedance probe ...............................83
 13.2. Plasma oscillation probe .............................85
 13.3. Ion flux pulsed and RF probe .........................87
 13.4. Hairpin probe ........................................92
14. PROBES IN (MAGNETIZED) FUSION-RELATED PLASMAS ............95
 14.1. Langmuir probe in fusion-related plasma................96
    14.1.1. Rake probe.......................................96
    14.1.2. Poloidal ring of electric and magnetic probes ...98
    14.1.3. Langmuir probes in JET divertor .................98
 14.2. Mach probe, Gundestrup probe .........................99
    14.2.1. Mach probe ......................................99
    14.2.2. Gundestrup probe ...............................101
 14.3. Retarding field analyzer.............................102
 14.4. Ion sensitive probes, Katsumata probe ...............104
 14.5. Tunnel probe ........................................108
 14.6. Ball-pen probe.......................................109
REFERENCES..................................................112

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                FOREWORD





      The Nobel laureate Irving Langmuir made outstanding contributions in different fields of Physics during the past century. He coined the term plasma in relation to the physics of partially ionized gases and also invented the Langmuir probes to measure the electron plasma density ne, the space potential Vs and the electron temperature kBTe in cold low density plasmas.
      In the year 2016 we shall celebrate 90th anniversary of the classical paper by I. Langmuir and H.M. Mott-Smith describing the theory of collectors in gaseous discharges [1]. It was not the first article by I. Langmuir and his colleagues, describing the probe diagnostics, but it summarized the effort that lasted several years [2-5]. At that time it was «a new method of studying electrical discharges through gases at rather low pressures». Since that time the probes were used at wide range of pressures including the atmospheric one, in plasma generated by direct current, radio-frequency, laser beam, electron beam, in tokamaks, in plasmas creating conducting or non-conducting layers and in many other applications. The original idea presented by I. Langmuir in [1] was broadened and several new probe designs were introduced that do not work on the Langmuir probe principle. Let me, please, use in this foreword for description of the Langmuir probe method, the exact words used by I. Langmuir in [1]:
      «The method consists in the determination of the complete volt-ampere characteristic of a small auxiliary electrode or collector of standard shape placed in the path of the discharge, and in the interpretation of this characteristic according to a new theory. In this way it is found possible to obtain an accurate value for the potential of the space near the collector, and much information concerning the nature, velocity and space density of the ions... Our problem is to calculate the current to the electrode contributed by
each kind of ion as a function of the applied potential, in terms of the assumed velocity distribution functions.
      For example, let us consider the collector to be a wire of small diameter, whose potential is made negative with respect to the region about it. The wire then repels negative ions and electrons, but attracts positive ions, and so becomes surrounded by a cylindrical positive «sheath» or region of positive space charge. This sheath is of such dimensions that the total positive charge in it equals the negative charge on the wire, so that the field of the wire does not extend beyond the edge of the sheath. The current taken by the wire therefore cannot exceed the rate at which ions arrive at the sheath edge in consequence of their proper motions.
      If we suppose the negative potential of the collector to be large compared with the voltage equivalent of the ion velocities, then the sheath may be divided roughly into two parts. In the center will be a region in which

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is concentrated most of the drop of potential between the gas and the collector, so that in this region there will be present only positive ions and possibly a few electrons or negative ions which had exceptionally high velocity. Outside of this is a region in which both negative and positive ions are present in comparable quantities, but in which the normal conditions existing in the discharge are modified through the withdrawal of positive ions by the collector. The two regions merge into each other more or less gradually in a way depending upon the distribution of velocities among the ions. In the outer region, as a rough calculation shows, the potential approaches the space value asymptotically but never reaches it in finite distance. Actually, therefore, the sheath does not have a sharply defined edge. Since, however, the whole drop of potential in the outer region is small compared with the total, it will be convenient to take as sheath boundary the surface at which the sharp drop of potential begins, and to regard the distribution of density and of velocity of the ions as known at this surface. This convention simplifies the discussion, and as we shall see, does not restrict the validity of the equations derived in the present article. Accordingly, we shall speak of the sheath as though it had a sharp edge; the potential at this boundary we shall speak of as the space potential.
     We shall further assume that the gas pressure is so low that there are only a negligible number of collisions in the sheath between ions or electrons and gas molecules. The ions in the sheath then describe free orbits, some of which end on the surface of the collector. Now if the sheath has axial symmetry so that the equipotentials are coaxial circular cylinders, it is found from simple mechanical principles that the condition for a positive or a negative ion to reach the collector depends not upon the nature of the field of force along the whole orbit, but only upon the initial and final potentials and the initial velocity of the ion on entering the sheath. The total current to the collector can thus be found by summing the contributions of the ions of different signs and initial velocities, and this current will be a function of the drop of potential in the sheath and of the sheath radius only. Another relation between these three variables can only be found by actually calculating the distribution of potential in the sheath through the use of Poisson's equation. It is thus seen that the problem of calculating the volt-ampere characteristic of the collector in general divides itself into two parts, the first of which is the purely „mechanical" problem outlined above, while the second is a „space charge" problem. From the solution of these two problems there are obtained two independent equations relating to the three variables just mentioned, and by elimination of the sheath radius between these two equations a single equation can be found expressing the current in terms of the potential.»
     Well, we see that already I. Langmuir was aware of the fact that the sheath does not have «... sharply defined edge ...», i.e. that the sheath merges

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into the undisturbed plasma «... more or less gradually in a way depending upon the distribution of velocities among the ions ...». In other words that there exists the «pre-sheath» that accelerates the positive ions to almost the electron temperature; the fact that has been treated by more than twenty years later and that is know as «Bohm sheath criterion» [20]. However, to make the theory simpler, he used the assumption of a sharp sheath edge. I. Langmuir also used the assumption of a low pressure, i.e. of a negligible number of collisions of charged particles with neutrals inside the probe sheath.
     The fig. 1 shows the characteristics of probes of different form assuming the distribution of velocities of uniform magnitude, but with directions

distributed at random in space. The space potential is at the position of the ordinate axis. Note that the ordinate axis is in this case plotted in linear scale. In fact, in case of the Maxwellian distribution of charged particles energies the figure would look similar, only the part for the retarding probe voltages would have the exponential form and for the cylindrical probe the curve for accelerating voltages would be described by a square root, see fig. 2. In other words by plotting the square of the electron current versus the probe potential with respect to the space (plasma) potential we would obtain the straight line from the slope of which we could estimate the electron density. Hence, the so-called I² vs Vp method described later in this textbook chapter, is already also postulated in the original I. Langmuir publication [1].
     Let us extract from these few words that I used for the foreword the general truth, namely that one should first look at what the predecessors achieved before making his/her own research. It is often surprising, what everything has been done already, and what can serve as a solid basis for future work.

Fig. 1. Current to the planar, cylindrical and spherical probe assuming distribution of velocities of uniform magnitude, but with directions distributed at random in space [1]

Fig. 2. Currents to the planar, cylindrical and spherical probe (probe characteristics) assuming
Maxwellian distribution of velocities [1]

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                1. INTRODUCTION





     Diagnostics of plasmas fall roughly into three categories: passive remote sensing, active noncontact methods, and contact methods. Passive sensing of incoming radiation is all that is available for plasma astrophysics studying the Sun or plasma phenomena related to objects beyond the Solar system. Active noncontact methods are widely applied to geophysical plasmas such as the ionosphere (radar scattering) and to hot laboratory plasmas (scattering of electromagnetic radiation or particle beams). Contact methods are applied to interplanetary, magnetospheric and ionospheric plasmas, and to cold laboratory plasmas. Among the contact methods, electric probes are without doubt the most widely used.
     The Langmuir probes are just one of the different electric probe diagnostics that are employed today. In a broader sense, the electric probes measure the local plasma parameters by using stationary or slow time varying electric (and/or magnetic) fields to emit or to collect charged particles to or from the plasma. These measuring techniques still constitute an active field of research and are particularly well suited for low density cold plasmas, as low pressure electric discharges, ionospheric and space plasmas.


            1.1. Probe shapes and probe types


     The measurements with electric probes belong to the oldest as well as to the most often used procedures of the low-temperature plasma diagnostics. The method has been developed by Langmuir and his co-workers in the twenties [1]. Since then it has been subject to many extensions and further development in order to extend its applicability to more general conditions in comparison to the one presumed by Langmuir. Such investigations proceeded continuously and the research on extension of applicability of Langmuir probe diagnostics continues also in the present time.
     The method of the Langmuir probe measurements is based on the estimation of the current-voltage characteristic - the so-called «probe characteristic» - of the circuit consisting of two metallic electrodes that are both immersed into the plasma under study. Two cases are of interest:
     •       the surface areas of both electrodes being in contact with plasma differ by several orders of magnitude and
     •       the surface areas of both electrodes being in contact with plasma are very small in comparison with the dimensions of the vessel containing plasma and approximately equal to each other.
     Case (a) is called «the single probe method» and case (b) «the double probe method». Most of this text is devoted, in accordance with the frequency

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of use of either method, to the single probe method; the double probe method is discussed in chapter 10.
     The Langmuir probe is usually constructed in simple geometric shapes: spherical probe, cylindrical probe and planar (flat) probe, see fig. 3. The probe is immersed into the plasma and polarized to the potential фp by an external circuit. This biases the probe to potential U with respect to the local space plasma potential ф^, U = фₚ -ф^. Then, the drained current for different probe potentials is monitored and the plasma parameters are calculated from the voltage - current (IV) characteristic curves.

Fig. 3. Typical shapes of Langmuir probes

     When constructing the probe we have to take into account that not only the active metallic part - the collecting surface of the probe - is small in comparison with the characteristic dimensions of the plasma vessel. Also the isolated parts of the probe must fulfil the same condition since often just these passive parts substantially influence the plasma around the probe.
     Apart from the «classic» Langmuir probe there exist many other types of probes, some of them will be discussed in the part «special probes». Let us now just give the names of those that are used in low temperature plasma: emissive probe, plasma oscillation/resonance probe, plasma impedance probe, ion flux RF probe, hairpin probe, triple probe, ball-pen probe. In the hot plasma devices there are used, apart from the Langmuir probes constructed either as a single probe or as a 1D/2D array of probes further the Mach probe, Gundestrup probe, Katsumata probe, ion sensitive probe, tunnel probe, ball-pen probe. In these devices the probes can either be mounted statically or they can be pushed to the hot plasma for a fraction of a second; these probes are called «reciprocating probes».


            1.2. Measuring the probe characteristic


     A simple experimental set-up for measurements of probe characteristics is shown in fig. 4. Apart from the DC high-voltage supply that feeds the glow discharge and the stabilising resistor Za (the DC supply can also work in the constant-current mode) the experimental system consists of the following parts:


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     1. DC voltage source for compensation of the potential difference

between the reference electrode (anode) and the probe.

      2. Sawtooth- or staircase-like voltage generator.

     3. Current measuring instrument.

    4. Computer that stores the measured current and voltage data, controls

the measurement procedure and processes the acquired data.

Za

Fig. 4. Typical probe circuit:
1 - DC bias voltage; 2 - sawtooth (staircase) Up generator; 3 - current voltage converter; 4 - Ip-Up data acquisition (computer)

     The scheme in fig. 4 is, of course, simplified. In practical measurements, especially when the probe is small, plasma less dense or the probe current is reduced by a magnetic field, it is necessary to measure probe currents that are substantially lower than a microampere. Nowadays there are available so-called source/measure units (SMUs) that can generate voltages up to around 200 V and measure currents down to picoamperes, e.g. by Keithley series 24xx or Keysight (formerly Agilent) series B2900A. However, these elaborate instruments have the disadvantage that their output cannot float at higher voltage, e.g. the Keysight B2901A’s output can float only up to the voltage ±210 volts. This feature

of the SMU’s is given mostly by the fact that they are computer-controlled, i.e. that their body and part of their electronics has to be grounded. The floating part of the SMU’s electronic can therefore bear only a limited voltage for the experiments where it is expected higher plasma potential than

±200 volts, e.g. in thermionic vacuum arc systems [6], it is necessary to construct the special probe measuring system.
     In most cases it is sufficient, in order to let the probe measuring circuit float at higher voltage, to separate the current measuring circuit from the computer control circuit using either isolation amplifiers, see e.g. [152] or opto-coupling of the computer control digital signals, see e.g. [173]. Moreover, probe circuit constructed in this manner increases the signal-to-noise ratio, since in such a way the ground of the computer where the noise level is comparatively high due to large currents necessary to power the processor and the adjacent chipsets is galvanically isolated from the probe current measuring circuit.
     Example of such a probe circuit is shown in fig. 5. The computer generates the bias voltage using a computer card provided by a D/A


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