Comparison of low frequency ac theory to Langmuir characteristics on the same probe

The small-signal, ac impedance $Z_{ac}(V_{dc})$ of spherical probes, offers a direct means of determining plasma potential $\varphi_{p}$ when the ac frequency $\omega $ is low and the probe radius $r_{p}$ is much larger than the electron Debye length.$^{\mathrm{a,b}}$ In particular, Re($Z_{ac})$ is a minimum at low f when the dc probe bias $V_{dc}$ equals $\varphi_{p}$. Here we compare low-frequency results for Re($Z_{ac})$ given by a network analyzer to the dc slope, (d$I_{dc}$/d$V_{dc})^{\mathrm{-1}}$, obtained by using the same sphere as a Langmuir probe. The two results agree well at moderate bias over a frequency range $\omega_{\mathrm{pi}}$\textless \textless $\omega $\textless 0.3$\omega_{\mathrm{pe}}(r_{p})$, where $\omega_{\mathrm{pi}}$ and $\omega_{\mathrm{pe}}$ are the ion and electron plasma frequencies, respectively.$^{\mathrm{a}}$ However, the agreement worsens as the bias voltage becomes increasingly negative. Furthermore, because $\omega $\textgreater \textgreater $\omega _{\mathrm{pi}}$, we expect the ions to contribute only weakly to the ac current, which suggests that Re($Z_{ac})$ should be no less than (d$I_{dc}$/d$V_{dc})^{\mathrm{-1}}$. Nevertheless, just the opposite behavior was seen experimentally. We will show results both from earlier work along with recent data in which we use frequencies in the range $\omega $\textless \textless $\omega_{\mathrm{pi.}}$. $_{\mathrm{.\thinspace }}^{\mathrm{a\thinspace }}$\textit{Phys. Plasmas }\textbf{17}, 113503 (2010); $^{\mathrm{b\thinspace }}$\textit{US Patent }\textbf{8,175,827 B2}$,$(5/2012) .