Electron density dependence of impedance probe plasma potential measurements

In earlier works, we used spheres of various sizes as impedance probes in demonstrating a method of determining plasma potential, φp, when the probe radius is much larger than the Debye length, λD. The basis of the method in those works [Walker et al., Phys. Plasmas 13, 032108 (2006); ibid. 15, 123506 (2008); ibid. 17, 113503 (2010)] relies on applying a small amplitude signal of fixed frequency to a probe in a plasma and, through network analyzer-based measurements, determining the complex reflection coefficient, Γ, for varying probe bias, Vb. The frequency range of the applied signal is restricted to avoid sheath resonant effects and ion contributions such that ωpi ≪ ω ≪ ωpe, where ωpi is the ion plasma frequency and ωpe is the electron plasma frequency. For a given frequency and applied bias, both Re(Zac) and Im(Zac) are available from Γ. When Re(Zac) is plotted versus Vb, a minimum predicted by theory occurs at φp [Walker et al., Phys. Plasmas 17, 113503 (2010)]. In addition, Im(Zac) appears at, or ve...