DavidPace.com - FAQ → Example of Langmuir Probe Analysis

The more detailed answer: it is not wise for me to make specific suggestions about a plasma system that is described through email. There are many potential difficulties that might not be explained well enough either to me in the initial contact or by me in response. I enjoy discussing experimental plasma physics and will continue to do so. The limit is when someone sends me an outline of their device and asks for a design that will allow them to measure the plasma properties.

There are plenty of companies that would be happy to serve as the consultant for plasma related engineering and research projects. They all charge for the service.

Is the reported floating potential determined automatically by an analysis code, or did you determine it manually?

The floating potential is calculated automatically by finding the bias at which the probe current is zero. In cases for which there is not a lot of data it is perfectly fine to find it manually by looking at a graph.

Does the data provided in this article have noise?

The data provided in the download contains the raw values from the original acquisition. This includes noise from the experiment and measurement scheme. There is no artificially added noise.

Do you do your computations without filtering this data?

In the example I do not perform any digital filtering of the data.

There are many situations in which some type of filtering is appropriate. In the LAPD it is possible to acquire many individual plasma discharges and build up statistics. Those plasmas are highly reproducible and the IV characteristics I obtain are averaged over many different shots. Generating such an ensemble average data set is a form of filtering. Digital filtering to remove background noise is another form.

I do not have an analytic expression for the value of I_probe, so how do I determine the various plasma parameters?

If you are doing probe measurements, then you have an analytic expression for the relationship between the bias voltage and the collected probe current based on Langmuir probe current collection theory. The behavior of the curve is predicted by theory and you have the relationship before ever making a measurement. Now, whether that theory applies to your experiment is a question for consideration.

If you are interested in references, there is a very good search function for the APS journals at Physics Review Online Search - APS. Try narrowing your search to older papers (pre 1960) and you should find some of the first work with Langmuir probe interpretation.

Also, Ian Hutchinson is a good author for plasma diagnostics. He has written many papers on probe usage and his textbook, Principles of Plasma Diagnostics, is widely used.

With respect to parameters, the floating potential for example, make a first pass by plotting the current (y axis) versus bias (x axis) and then drawing a line to connect all the points. The intersection of this plot with the bias (x) axis is the floating potential. It is a rough estimate and you can refine it by calibrating your data, accounting for errors, and further processing.

Is there a way to automate the analysis of the swept Langmuir probe trace?

There are many ways to implement such an analysis tool. The specifics depend on what type of analysis software you intend to use (a programming language like C++ or an analysis kit like IDL).

The difficulty in such automation is providing a way for the routine to pinpoint the exponential region of the IV characteristic. The behavior of this region allows for the determination of the electron temperature, but it is surrounded on both ends by saturation currents. Any error in finding this region can have significant effects on the final electron temperature value. I do this by performing multiple linear fits of the natural log of the current and then looking for the best fitting parameters. This is another area in which there have been many papers published, some by programming experts with much more advanced ideas than mine.

I know where the I_sat region is, but how do I determine a single value for the I_sat?

The I_sat current does not necessarily require fitting a line within the I_sat region. Looking at figure 3, the I_sat is just the flat part of the curve at the low end of the bias potential. Theoretically, this current should be constant and that value is the I_sat. If it is not constant (which can happen for multiple reasons, look up sheath expansion if you are interested in one example), then it is up to you to determine whether its value should be taken as the mean across some part of the IV plot. There is no way to determine a correct bias at which the current will be the I_sat value.

Floating potential does not appear to be necessary in order to calculate the electron temperature or the plasma potential, so why do we bother to determine the floating potential?

One reason floating potential is tossed around so much is that it is experimentally difficult to measure plasma potential. Many experiments cannot make use of swept measurements because the sweep rate is slower than the physics behavior they are trying to study. If you perform a swept measurement, then you get one T_e, Φ_pl, n_e, and V_float for each period of the sweep. If instead, you connect a probe and measure floating potential, then you can collect a value instantaneously (better time resolution).

The argument is that while floating potential and plasma potential are not equivalent, the floating potential profile will be similar to that of the plasma potential. So, if you measure the floating potential everywhere in space and then calculate the gradient, that would give you the electric field profile. Electric field is always difficult to measure and this is one way of determining it.

The applicability of this process depends on the plasma parameters, in particular the electron temperature profile. If you can measure the plasma potential, then go ahead and ignore the floating potential (or compare them for fun).

Is V_bias applied as an AC or DC voltage?

The theory behind the IV trace is that it represents the current collected by a conducting probe over a range of probe biases. For some applied bias to the probe (with respect to the plasma, though experimentally the probe is often biased with respect to the vacuum chamber or plasma source), a certain amount of current is collected through the probe. This means the bias has to be DC.

It may seem as though this is an AC procedure because we change the value of the applied bias, but this bias sweep is always much slower than the response of the plasma. The full range of probe biases may be completed with a frequency in the kilohertz range, but that behavior would seem very slow, DC-like even, to a plasma in which the native frequencies (gyrofrequencies and plasma frequencies for example) are in the megahertz and gigahertz ranges.

Now, experimentally you can use an AC bias for this same reason. A sinusoidal probe bias is fine as long as this waveform is recorded. Your IV trace will be plotted versus the bias, so the shape of the bias does not matter. Some people use AC bias because it is easier to control a sinusoidal bias than it is to generate a linear ramp. Most frequencies of AC voltage that can be easily generated will still be much slower than the plasma response and therefore not significantly perturbative.

If the bias is DC, then how do you perform a sweep?

Now we are dealing with technicalities. Since the probe is swept through a range of values, it may be considered AC. The point is that we want to sweep it slowly enough that it is equivalent to DC. Since the plasma response is so fast we can generally perform sweeps within the range 0 - 100 kHz. Going beyond this range is difficult (one group built a sweeper that performed at 400 kHz and published a paper on it).

Think of it this way, we want to vary the probe bias through a range that covers the saturation and exponential (electron temperature) regions, but if we sweep this bias too quickly then we will have made an antenna that disturbs the plasma and possibly alters the density and temperature values.

Can I use a standard 60 Hz oscillating bias as the sweep waveform?

Yes, you can use as slow a sweep as you want.

The problems occur at the other end, very high sweep frequencies. If you sweep the probe bias at a high enough frequency that the probe excites a wave, then you will have significantly affected the plasma properties. Slow bias sweeps appear as DC effects to the plasma.

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The difference is part of the uncertainty in measuring density with Langmuir probes. A rule of thumb is that density measurements from probes are never accurate to better than a factor of two.

Some of the more common differences between the expression I have used and those found in other references are extra factors of two and π. The extra π term might be from their choice of normalization. Their factor of two in the square root is probably because they define the thermal velocities using two degrees of freedom, resulting in a 2kT term instead of just kT.

There are also issues concerning sheath expansion and probe geometry. Various correction factors might be used to better account for the shape of the collecting region due to sheath expansion and the design of the probe head (e.g., a spherical head, planar, or cylindrical). Once again, however, the improvement in the measured density value is probably not worth the effort of these corrections. The density will just about always be the least certain result acquired.

The Hutchinson book, Principles of Plasma Diagnostics, does not have these additional factors. You will be fine using either expression (the one on my website or your present one). It is funny that something as old as probe usage is still difficult enough that there are different theoretical results. As long as you remember the large uncertainty in density from probe measurements you will return as accurate a result as possible from any of the prominent derivations.

I have found equations for Langmuir analysis that do not include Boltzmann's constant. Are these different from the ones you use?

In the case of Boltzmann's constant, kB, some authors never include it in the relevant expressions because they express temperature in energy units. The Boltzmann factor is in units of energy per temperature (e.g., the SI Units value is kB = 1.3807 × 10-23 Joules per Kelvin). A factor of kBT represents an amount of energy.

There are multiple reasons for only writing temperature instead of the kBT factor. One reason is that the expressions are always considered more elegant if they are simpler, i.e., if they have fewer variables. Boltzmann's constant is absorbed by the temperature and everything looks a little nicer. Another reason is that temperature is often used interchangeably with energy in plasma physics. Plasma temperature relates to particle velocities, which is a measure of energy. In most cases a distinction between temperature and energy is not necessary. This might lead to some problems for people that study plasma turbulence in the form of temperature fluctuations, but that is another topic entirely.

In my Langmuir analysis item I try to write out the complete expression at all times. While this may make some items longer than they could be, it hopefully reduces confusion through all the different areas of the analysis.

Last Updated ( Monday, 31 March 2008 )

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