Langmuir Probe

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Description of the Experiment

Figure 1: Photo of the experimental setup

Plasmas have different characteristics from other states of matter and in order to measure their parameters many diagnostic tools have been developed. This experiment allows the measurement of some of these characteristics using one of the most simple methods, an electrical probe, the Langmuir probe. It consists of a thin filament made of conductive material placed inside the plasma which either attracts or repels the electrons in the plasma according to its polarization. Measuring the probe I-V characteristic, that is the relationship between the polarization voltage and the respective current going trough the it, one can determine the electron temperature and density of the plasma.


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Experimental Apparatus

The main chamber consists of a pirex cylinder on which a modified Filament Head is placed to function as a Langmuir probe.

The central filament of 0.2mm diameter has been covered with alumina tube leaving only a 1cm tip exposed to the plasma. The two outer filaments are connected to an "of the shelf" RF generator which ionizes the gas and generates the plasma. This is a simple 3.5W generator which outputs a 50kHz AC signal with 1kV amplitude. To improve stability and homogeneity of the electrical field two molybdenum plaques have been attached using low vapour pressure epoxy, to each outer filament.


Determination of the electron temperature and density

Figure 2: Raw Data from the experiment
Figure 3: Linear fit to the ion saturation region
Figure 4: Fit to probe characteristic equation

The Langmuir probe uses one or more electrodes which are placed inside the plasma, in this case one of the filaments of the head. Measuring the current collected by the probe while applying a triangular wave, will provide the I-V characteristic that will be analysed later.

When the probe is electrically isolated, a plasma sheath is formed in the interface between the plasma and the probe. To compensate for the higher mobility of the electrons, the probe will attain a floating potential,\(V_f\), negative with respect to the plasma potential, \(V_p\). The density at the sheath entrance is roughly half of the density away from the probe.

The probe voltage, \(V_s\), has been changed with respect to the ground set by the winding filament using a variable voltage source. If the polarization of the probe, compared to the plasma, is negative enough, all the electrons will be repelled nd the ion flux to the probe is independent of the tension applied. In a totally ionized plasma, this ion saturation current is described by the following expression: \[ i^+_{sat} = j^+_{sat} A_s = e \, n \, c_s \, A_s \]

Where \(j^+_{sat}\) is the current density, \(A_s\) is the contact surface of the probe, \(e\) is the electron charge, \(n\) is the ion density in the plasma, \(c_s\) is the ion sound speed.

If we polarise the probe positively, the voltage drop in the sheath is reduced and electrons will be able to reach the probe. Taking a Maxwell distribution for the speed of the electrons, the relation between current and tension will be: \[ i = i^+_{sat} \left( 1- e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]

Where \(T_e\) is the electron temperature. This expression assumes that there is only one probe and that it is non-perturbative.

From the obtainend it is possible to extract an estimate for the floating potential, \(V_f\). This is done by taking the value at which the characteristic crosses zero.

It will also be easily seen that the data does not follow a the regular characteristic on the ion saturation side, it should be constant instead of having a slope. This has to do with the fact that the probe is cylindrical. To correct for this, a linear function is fitted on this side and the slope is subtracted (as seen in Figure 3).

But, in doing this we impose a condition of cold ions, which means that there will be no ion saturation current, since the ions aren't moving. However, we want to get the density, so we have to make another correction, which corresponds to add the value of current in point where we know the current is completely due to ions, namely to values voltage much lower than the floating potential (in this case we use two times the floating potential, hence the need for our initial estimate).

After that, the experimental data has to be adjusted to the equation presented in above (as seen in Figure 4). From the fit the \(T_e\) can be extracted as well as the ion saturation current, \(i^+_{sat}\) and the floating potential \(V_f\) which should have a value close to the one we determined empirically.

By knowing the area of the probe and the speed of sound in the ions of the plasma, one can determine the electron density in the plasma. Since \(c_s = \sqrt{\dfrac{k T_e}{M}}\), and the fact that the probe is \(1cm\) and has a diameter of \(0.2mm\) we can use the \(i^+_{sat}\) to determine the density.