Abstract
Active Plasma Resonance Spectroscopy (APRS) is a well known diagnostic method, where a radio frequency probe is immersed into a plasma and excites plasma oscillations. The response of the plasma is recorded as frequency dependent spectrum, in which resonance peaks occur. By means of a mathematical model plasma parameters like the electron density or the electron temperature can be determined from the detected resonances.The majority of all APRS probes have in common, that they are immersed into the plasma and perturb the plasma due to the physical presence of the probe. Thus, they are invasive and can at least influence the homogeneity of the plasma. To overcome this problem, the planar Multipole Resonance Probe (pMRP) was invented, which can be integrated into the chamber wall of a plasma reactor.Within this paper, the first analytic model of the pMRP is presented, which is based on a cold plasma description of the electrons. The general admittance of the probe-plasma system is derived by means of functional analytic methods and a complete orthonormal set of basis functions. Explicit spectra for an approximated admittance including a convergence study are shown. The determined resonance frequencies are in good agreement with former simulation results.
Highlights
A plasma occupies the natural ability to resonate near the electron plasma frequency ωpe
To determine plasma parameters from the measured resonance peaks, a mathematical model is needed. In this manuscript the general description of active plasma resonance spectroscopy (APRS) in an electrostatic approximation will be applied to the geometry of the planar Multipole Resonance Probe (pMRP) and an analytic solution for the admittance of the probe-plasma system will be presented
The spectrum is not given by an infinite number of discrete resonance modes, like the spectrum of probes with a spherical probe tip [27], because the electrode geometry of the pMRP is not represented by Delta functions in the corresponding Fourier space
Summary
A plasma occupies the natural ability to resonate near the electron plasma frequency ωpe. In this manuscript the general description of APRS in an electrostatic approximation will be applied to the geometry of the pMRP and an analytic solution for the admittance of the probe-plasma system will be presented. The final converged spectra lead to a proportional relation between the resonance and the plasma frequency, which can be used as a simple formula to determine the electron density from a measured resonance.
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