On improving impedance probe plasma potential measurements in low density plasma

We have used impedance probes of various sizes and shapes in demonstrating a method of determining plasma potential, $\varphi_{p}$, when the probe radius is much larger than the Debye length. The method\footnote{\textit{Phys. Plasmas }\textbf{17}, 113503 (2010).}$^,$\footnote{\textit{NRL Memorandum Report 6750-12-9413}(2012).} relies on applying a small amplitude ac signal to a probe in a plasma and measuring the complex reflection coefficient, $\Gamma $, as a function of varying probe bias, $V_{b}$. Re($Z_{ac})$ (the real part of the ac plasma impedance determined from $\Gamma )$ is plotted versus $V_{b}$, and a minimum predicted by theory occurs at $\varphi_{p}$ for a large range of electron density, $n_{e}$.\footnote{\textit{Phys. Plasmas }\textbf{17}} However, the frequency range of the applied signal is restricted and as $n_{e}$ decreases it becomes even more restrictive. In addition, the minimum in Re($Z_{ac})$ ($\sim$ 1/$n_{e})$ becomes more difficult to discern. Here, we suggest additional means to isolate $\varphi_{p}$. These measures (1) incorporate $\Gamma $ to search for a minimum, (2) use not only the first derivative of Re($Z_{ac})$, but also that of Im($Z_{ac})$ with respect to $V_{b}$ and, (3) use the second derivatives of both. With the additional indicators, $\varphi_{p}$ is more easily detected in low density plasma. We present data for cylinders, spheres and a disk. \textit{Phys. Plasmas }\textbf{17}, 113503 (2010). \textit{NRL Memorandum Report 6750-12-9413}(2012).