Determining electron temperature for small spherical probes from network analyzer measurements of complex impedance

Abstract

In earlier work, using a network analyzer, it was shown that collisionless resistance (CR) exists in the sheath of a spherical probe when driven by a small rf signal. The CR is inversely proportional to the plasma density gradient at the location where the applied angular frequency equals the plasma frequency ωpe. Recently, efforts have concentrated on a study of the low-to-intermediate frequency response of the probe to the rf signal. At sufficiently low frequencies, the CR is beyond cutoff, i.e., below the plasma frequency at the surface of the probe. Since the electron density at the probe surface decreases as a function of applied (negative) bias, the CR will extend to lower frequencies as the magnitude of negative bias increases. Therefore to eliminate both CR and ion current contributions, the frequencies presently being considered are much greater than the ion plasma frequency, ωpi, but less than the plasma frequency, ωpe(r0), where r0 is the probe radius. It is shown that, in this frequency regime, the complex impedance measurements made with a network analyzer can be used to determine electron temperature. An overview of the theory is presented along with comparisons to data sets made using three stainless steel spherical probes of different sizes in different experimental environments and different plasma parameter regimes. The temperature measurements made by this method are compared to those made by conventional Langmuir probe sweeps; the method shown here requires no curve fitting as is the usual procedure with Langmuir probes when a Maxwell-Boltzmann electron distribution is assumed. The new method requires, however, a solution of the Poisson equation to determine the approximate sheath dimensions and integrals to determine approximate plasma and sheath inductances. The solution relies on the calculation of impedance for a spherical probe immersed in a collisionless plasma and is based on a simple circuit analogy for the plasma. Finally, the temperatures obtained using this method show reasonable agreement with those obtained using a conventional Langmuir sweep analysis of the spheres.

Using rf impedance probe measurements to determine plasma potential and the electron energy distribution

“…We briefly describe here the experimental basis for our measurements, leaving further description in references to earlier works 11,12,15 for the interested reader. Most experiments in these series were performed in the Space Physics experimental facility.…”

“…12 In this work, we extend the analysis to construct an algorithm for determining the plasma potential p from measurements of Re͑Z ac ͒ as a function of the applied dc probe bias V p at low frequency. Here, Z ac is the small-signal ac impedance of the probe and Re͑Z ac ͒ is the real ͑resistive͒ component.…”

Cutoff probe using Fourier analysis for electron density measurement

“…In addition, by virtue of this method, the cutoff frequency point can be measured without a network analyzer and the speed of cutoff probe measurement was enhanced more than a million times without loss of measurement accuracy compared to the previous one. This new method is expected to have applications not only in the fast measurement of absolute electron density with a cutoff probe, but also to other previously developed diagnostics where a network analyzer is used, specifically a hairpin probe, 21,22 and an impedance probe, [23][24][25] by replacing the network analyzer with a nanosecond impulse generator and an oscilloscope. …”

Determining Plasma Potential from RF Measurements Using an Impedance Probe

“…These equations arise from a combination of physical theory along with a circuit representation of the probe plasma interaction. 13 For the lower bound to this frequency range, we avoid any ion contributions to the total current as ions are unable to respond on the timescale of the electrons. For the upper bound, there will be no contribution from collisionless resistance (CR) (or resonance effects) covered in the original work in this series.…”

“…In earlier works using rf techniques with plasma probes in laboratory experiments we have demonstrated the existence of collisionless resistance in the sheath of a spherical probe 11 , shown that this leads to a method of finding the electron sheath density profile 12 , and proposed a method of measuring electron temperature using the rf results 13 . Most recently we have been able to determine plasma potential from these measurements.…”