Single Langmuir Probe

The uncompensated single Langmuir probe (SLP) is one of the most fundamental diagnostics in plasma physics. Originally designed in 1924 by Langmuir[107,108], at its most basic, the simple Langmuir probe consists of a biased electrode inserted directly into a plasma. By varying the probe bias, electrons and ions of different energies will be collected depending on the EEPF and the sheath parameters. By careful consideration of the sheath theory and the electrode geometry, it is then possible to reconstruct the EEPF and associated plasma properties directly from a current-voltage (IV) characteristic collected by the probe. This allows us to

measure electron and ion density, electron temperature, floating potential, and plasma potential by sweeping the bias potential of the electrode.

While the uncompensated SLP is relatively straightforward to implement, it is susceptible to a broad array of interferences and deviations from ideal behaviour which are detailed in Section 2.3.1.3. For this reason, there have been a variety of modified probe designs developed based on the principles of SLP. These include rf- compensated probes[109], the double Langmuir probe (DLP, discussed in Section 2.3.4.3)[110], the triple Langmuir probe[111], the ball-pen probe[112], and the emissive probe[113], each of which takes advantage of various aspects of Langmuir probe theory to achieve improved reliability of certain measurements, or otherwise improved functionality.

The probe system must operate in a high-density, magnetised, rf plasma envi- ronment. The system must be sensitive to plasma densities between 5×1016 _and

5×1019_m−3_{, which requires sensitivity from 1×10}−4 _{to upwards of 1 A of current.}

It also needs to be able to resolve electron temperatures from ∼0.5 to above 10 eV, and be consistent between magnetised and unmagnetised conditions. Due to the temporal dynamics of the pulsed high power discharge, the entire range of the above conditions may be present within a single pulse at a single location. The voltage and collection area must therefore be large enough to ensure appropriate sensitivity to low density conditions, and, at the same time, not be large enough for the enormous thermal flux of the high density conditions to damage the probe or cause thermionic emission.

In order to meet the above requirement of broad applicability the SLP was chosen over more advanced probe designs. While this sacrifices the accuracy and reliability of more specific probe designs during individual measurements, it allows for improved access to a broader parameter space. It is also convenient to use the SLP for measurements of density and temperature, as the photodetachment

technique necessarily requires the use of a SLP for measurements of the plasma electronegativity. Furthermore, an uncompensated probe system is chosen due to the high currents and high thermal fluxes which would damage the sensitive electronic components typically used in rf compensating systems.

stainless steel chassis copper wire 32 mm borosilicate sheath tungsten tip

Figure 2.7: Schematic of the single Langmuir probe used in this work.

In this work we utilise a dog-legged single Langmuir probe with a stainless steel probe chassis, a dog-legged tungsten wire electrode insulated within a protruding glass tube and connected via an enamelled copper wire along the length of the probe chassis (Figure 2.7). The stainless steel chassis extends 920 mm beyond the vacuum seal, and is centred 32 mm off-axis. The tungsten wire electrode is frequently replaced, but typically protrudes ∼1 mm from the glass tube, with a primary length of ∼ 5 mm aligned with the axis of MAGPIE and a diameter of 0.1 mm. The bias potential is swept across a 30 V sinusoidal voltage at 10 kHz using a high-current linear amplifier allowing for the continuous collection of IV characteristics with 100µs temporal resolution. The probe signal is digitised with a Tektronix DPO-3014 at a data sampling rate of 2.5 GS/s, which allows for 8000

samples per characteristic.

2.3.1.1 Probe Theory

All Langmuir probe theory is based on an understanding of the transport of col- lected ions and electrons across the electrode’s sheath, and how it varies with probe bias. The simplest formulation of probe theory involves the calculation of flux, Γ, across a planar sheath to a biased electrode (Figure 2.8).

Γ

sheath

biased electrode

Figure 2.8: Plasma flux to a biased planar surface across a plasma sheath.

It can readily be shown from the Bohm sheath criterion that, for an unmag- netised, electropositive, DC, Maxwellian plasma, the collected probe current, Ipr,

at a given bias, Vpr, below the plasma potential is given by

Ipr =Isat 1−exp qe(Vpr−Vf) kTe (2.4) where Isat = 0.61qeniAef f r kTe mi (2.5) andAef f is the effective collection area of the probe, accounting for a finite sheath

thickness. The detailed flux calculation can be found in most basic plasma physics texts[52, 53,102]. For high density discharges in high mass gases (such as argon) the sheath thickness is often small compared to the probe dimensions and can typ- ically be ignored, however, in light gases such as hydrogen the sheath is observed to expand with increasingly negative bias voltages[102]. For a cylindrical probe,

this expansion causes an approximately linear increase in collection area with in- creasingly negative bias, and hence a linearly increasing ion saturation current (Figure 2.9).

probe probe

low voltage

sheath high voltagesheath

A0 A0 Aeff Aeff -25 -20 -15 -10 -5 0 -0.15 -0.1 -0.05 0 0.05 0.1 Vf I0

Figure 2.9: The effect of sheath expansion on effective probe collection area and saturation current.

To account for this expanding sheath, we can define the effective area as

Aef f =A0[1 +α(Vf −Vpr)], Vpr < Vf (2.6) giving Isat,0 = 0.61qeniA0 r kTe mi (2.7) whereαis determined directly from the slope of the ion saturation region of the IV characteristic. By fitting the above form ofIpr to a measured IV characteristic, we

can directly obtain Isat,0, α, kTe, and Vf from a four parameter fit. Rearranging

Isat,0, we obtain ni = 1 0.61qeA0 r mi kTe (2.8) The plasma potential is indicated by the value at which the IV characteristic deviates from ideal exponential behaviour. This can be identified by the zero crossing of the second derivative of the IV characteristic[102].

By equating the ion and electron fluxes at the floating potential we obtain the following expression[46] Vf =Vp− kTe 2q ln mi 2πm (2.9)

which for a proton dominated hydrogen plasma gives us

kTe

qe

≈ Vp −Vf

2.8 (2.10)

Each of these values have already been determined using the ion saturation fitting method, which in principle gives us an overdetermined system, however, many anomalous behaviours will affect each of these measured values independently. In scenarios where one of these parameters in the ion saturation fitting procedure is strongly affected by anomalous behaviours, the remaining two parameters can be used to recover the true value of the affected parameter using Equation 2.10. This is especially valuable when evaluating IV characteristics from the extremes of the parameter space discussed above, where the ion saturation fitting procedure can become prone to numerical issues due to noise or limited sampling range.

While the above probe theory is extremely simplistic, ignoring a broad range of non-ideal plasma behaviours and approximating a cylindrical probe as a pro- jected planar probe, this simple analysis turns out to be surprising robust for moderate to high density plasmas when compared to absolutely calibrated mi- crowave interferometry (discussed in Section 2.3.4.2). In their initial investigation of this method, Chen et al.[114] identified that, despite its simplicity, this method was found to outperform more advanced probe theory (for a discussion of vari- ous advanced probe theories, see the lecture notes by Chen[46]). It is suspected that this is due to multiple ignored behaviours acting in opposition to each other. Measurements with the uncompensated SLP and basic analysis used in this work agree remarkably well with measurements previously made in MAGPIE using a compensated DLP calibrated against microwave interferometry measurements[57], considering the simplicity of this system.

2.3.1.2 Analysis Procedure

Having defined the theory for analysing individual IV characteristics, we now discuss the practical aspects of determining time resolved plasma properties from the measured probe signal.

Beginning with the raw voltage and current signals (Figure 2.10) we subdivide the data into each individual 100 µs period. For a 40 ms pulse, this produces a total of 400 individual IV characteristics which must each be analysed to determine plasma properties at that time in the pulse.

1.98 2 2.02 2.04 2.06 2.08 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Probe Current (A)

-30 -20 -10 0 10 20 30 Probe Voltage (V) 0 1 2 3 4 5

Figure 2.10: Current and voltage traces from a typical probe sweep.

Each individual period then consists of an ‘up sweep’ and a ‘down sweep’ with voltage increasing and decreasing over time respectively. Electrostatic coupling between the probe’s main lead and its chassis (as well as other smaller contri- butions from the rest of the probe system) leads to a stray capacitance. Due to the rapid variation in bias voltage this stray capacitance produces a noticeable current on top of the plasma signal. Fortunately due to symmetry, the capacitive component of the up sweep is equal and opposite to the capacitive component of the down sweep. If the plasma is slowly varying compared to the sweep frequency, then the capacitive current can be removed by simply averaging the up and the down sweeps. The voltage and current signals can then be combined to obtain the IV characteristic.

The plasma potential is determined by the zero crossing of the second deriva- tive, however the second derivative is extremely sensitive to signal noise, which makes this impractical to be applied to the raw IV characteristic. Perversely, this problem is worsened by increased resolution, due to more rapid variations in the noise. A common approach to obtaining the zero crossing of the second derivative under these conditions is to apply strong smoothing to the raw characteristic, however this has the potential to distort the underlying characteristic and is not guaranteed to produce a sufficiently clean signal for an automated analysis code to reliably determine the correct value of the zero crossing. To solve this problem, we fit an analytical expression to the raw IV and determine the zero crossing of the analytical second derivative of the fitted expression, removing the effect of noise. By applying the fit to the combined up and down slopes, this also acts to effectively average the two signals.

-20 -10 0 10 20 -0.2 0 0.2 0.4 0.6 0.8 1 Data Fit -20 -10 0 10 20 -20 -15 -10 -5 0 5 10 15 20

Figure 2.11: A fitted IV curve displaying both up and down sweeps. The plasma potential is determined from the zero crossing of the second deriva- tive of the analytic fit.

The ion saturation fit relies on the ideal exponential behaviour of the electron component of the probe current in order to determine the correct electron temper- ature, however this is necessarily only valid up to the plasma potential where the current deviates from exponential behaviour. The ion saturation fit is therefore typically performed only on the section of the IV characteristic below the plasma

potential. Close to and above the plasma potential, the IV characteristic is sub- ject to distortion; from rf interference, magnetic fields, and other non-ideal effects. This can cause the fitted electron temperature to appear artificially high. It is therefore recommended that the fitting be confined to a region of the IV charac- teristic limited by some maximum potential,Vmax, less than the plasma potential.

The choice of Vmax is not immediately obvious a priori. If the value chosen is too

low, then only the high energy electrons will be sampled and it is possible that a small population of high energy electrons could also artificially raise the fitted electron temperature. Knowing that the electron temperature is overestimated if the chosen Vmax is either too high or too low, we can conclude that the most

appropriate choice of Vmax will be that which produces a minimum value for the

fitted electron temperature. The four parameter ion saturation fitting procedure is therefore performed for a series of different values of Vmax between Vf and Vp,

with the fit whose value of electron temperature is minimal being taken as the canonical fit. -15 -10 -5 0 5 10 -0.2 0 0.2 0.4 0.6 0.8 Vf Vp fit range

Figure 2.12: The fitting range is varied betweenVf and Vp to identify the

minimal apparent electron temperature.

This procedure is then repeated for each of the IV characteristics in the time series, producing time resolved measurements of ni, Vp,Vf, andTe. The values of

the values calculated from the floating and plasma potentials via Equation 2.10. It is found that the potentials method is more consistent and less susceptible to noise, but is typically inaccurate by a scaling factor due to the effect of the magnetic field (discussed in Section 2.3.1.3). It is therefore often useful to obtain a scaling coefficient from the electron temperature obtained via the fitting process and use this in combination with the electron temperature calculated from the potentials. This is especially useful in scenarios where the electron temperature is at either extreme, or the probe signal is particularly noisy, when the fitting procedure can be highly susceptible to inaccuracies and errors (Figure 2.13). It is therefore at the discretion of the analyst to decide which value is most reliable and most consistent with other observations and expectations from the operating conditions. 0 10 20 30 40 0 2 4 6 8 10 12

Scaled Potential Method Ion Sat. Fit Method

Figure 2.13: Electron temperatures as determined from both the scaled potential method and the ion saturation fitting method. The fitting pro- cedure encounters difficulties after 10 ms in this example, but the scaled potential method remains reliable, tracing out the upper range of fitted values.

2.3.1.3 Anomalous Behaviour

A large range of plasma conditions can affect the analysis of Langmuir probe mea- surements by either directly distorting the IV characteristic or by undermining the

of these effects could potentially be accounted for and corrected, either in situ or within the analysis, however as previously discussed, the extreme conditions do not lend themselves to in situ solutions, and the sheer breadth of different plasma conditions observed at different times and locations, even within the same pulse, limits the applicability of any corrections within the analysis. Any such corrections would need to be highly adaptive and run the risk of over-correcting or obfuscating the measurements. It is far more practical, therefore, to apply our understanding of anomalous behaviour to the interpretation of analysed results, using a naive analysis, rather than to attempt to correct the analysis itself. By taking the following effects into account we can gain a clearer understanding of the plasma’s behaviour when interpreting Langmuir probe measurements.

Magnetisation

As will be discussed in Section 3.4, magnetic fields strongly affect the transport properties of electrons and ions. This can affect the particles fluxes to the probe electrode and strongly distort the collected IV characteristic[102,115]. The degree to which the collected probe current will be affected by the magnetic field is determined by the ratio of the probe radius,rp, to the particle’s gyroradius, rg =

√

2mkT

qB . In the case where rp

rg 1, the plasma species is considered ‘unmagnetised’

and the collected probe current will be unaffected by the magnetic field. If rp

rg &1,

then the plasma species is considered ‘magnetised’ and the collected probe current will be significantly suppressed. Under the operational conditions of MAGPIE, we almost exclusively find that the ions remain unmagnetised, however the electrons can range from partially magnetised to strongly magnetised.

Under these conditions the IV characteristic will be affected in the following ways: the ion saturation current will remain unaffected; the electron saturation current will be suppressed; the apparent plasma potential will be suppressed and the floating potential will also be suppressed but to a lesser extent; the electron

current profile is distorted around the measured plasma potential, but remains un- affected near the floating potential. This has three important ramifications for the analysis of IV characteristics. Firstly, due to the fact that the ion saturation cur- rent is unaffected, we can still recover relatively reliable ion density measurements even with strong mirror field configurations. Secondly, due to the suppression of the plasma potential, the electron temperature estimated from Equation 2.10 will be too low by a scaling factor. Finally, the distortion of the current profile near to the measured plasma potential can artificially raise the fitted electron temper- ature, however this effect has been shown to be fairly minimal due to the profile remaining unaffected closer to the floating potential[46].

We can directly compare magnetised and unmagnetised probe collection un- der identical plasma conditions by exploiting the fact that the magnetic field only affects transport perpendicular to the the field direction. A surface whose normal is aligned with the direction of the magnetic field will therefore collect an unmagnetised current. By rotating the probe electrode such that it extends transversely across the magnetic field lines, the projected area of its sides will col- lect an unmagnetised current. If the plasma is approximately uniform along the length of the probe tip in both dimensions, then this serves as a direct comparison between magnetised and unmagnetised collection under otherwise identical condi- tions. This condition is generally true on-axis, where the length scale of significant variation in plasma properties is greater than the length of the probe tip. Off-axis however, plasma properties can vary by roughly an order of magnitude in the radial direction over the same length. For this reason the transverse configuration is used only for the sake of comparison. In Figure 2.14 we observe the differences between magnetised and unmagnetised probe traces, at 500 mm under standard operating conditions in the steady state.

30 30.05 30.1 30.15 30.2 30.25 30.3 0 0.5 1 1.5 2 2.5 Parallel Perpendicular

Figure 2.14: Current traces for perpendicular and parallel probe alignment positioned under the magnetic field peak in standard operating conditions. but the electron current, and plasma potential are strongly suppressed. In Figure