Preprint: Sheath thickness measurements with the biased plasma impedance probe, Agreement with Child Langmuir scaling

TL;DR

This work demonstrates that biased plasma impedance probes (PIP) yield sheath-thickness measurements in close agreement with Child–Langmuir scaling when corrected by a single empirical factor $α \approx 0.74$, validating the PIP as a direct diagnostic of sheath structure without severely perturbing the plasma. By coupling this scaling with a comprehensive PIP model, the authors extend the floating PIP to infer electron temperature $T_{e}$ and plasma potential $V_{plasma}$ without bias, achieving results that align with Langmuir-probe measurements. The study also shows that biased PIP can reliably measure density and electron damping, supporting the PIP as a complementary tool to Langmuir probes for cross-validation and broader plasma characterization. Together, these findings imply that combining biased and floating PIP diagnostics with LP measurements can reduce model dependence while expanding accessible parameters in plasma environments relevant to processing, propulsion, and space-plasma interactions.

Abstract

Plasma sheaths play a central role in plasma-surface interactions, yet their thickness remains challenging to measure experimentally. Although classical analytical models such as the Child-Langmuir (CL) sheath model provide clear predictions for sheath thickness, experimental validation has been limited because most diagnostics either rely on indirect, multi-step inference (e.g., Langmuir probes) or require invasive and technically demanding techniques. In this work, we demonstrate that the plasma impedance probe (PIP), when operated with a controlled DC bias, enables relatively direct, model-informed measurements of sheath thickness that are reasonably straightforward to implement experimentally. Across a range of discharge conditions, biased-PIP sheath thickness measurements are found to follow CL scaling closely, requiring a single, consistent empirical correction factor of $α\approx 0.74$ to reconcile the measured thickness with CL predictions. Concurrent measurements of plasma density and electron damping show that probe biasing does not significantly perturb the bulk plasma density, supporting the validity of the biased-PIP approach. Building on this validation, we leverage the empirically determined $α$ factor to extend the floating (unbiased) PIP analysis to obtain model-dependent estimates of electron temperature and plasma potential without electrical biasing. A side-by-side comparison demonstrates close agreement between floating-PIP results and those obtained from a biased Langmuir probe. Taken together, these results establish the PIP as a complementary diagnostic to the Langmuir probe, expanding the range of accessible plasma measurements while providing experimental support for classical sheath models.

Figures (5)

• Figure 1: Conceptual comparison of electron density profiles near a probe surface. The PIP model assumes a step-function electron density profile corresponding to a vacuum sheath of thickness $t_{\mathrm{pip}}$. In contrast, more complete classical descriptions include a continuous sheath profile and an extended quasi-neutral presheath region Lieberman2005. The figure is intended to provide physical intuition regarding the differing levels of model abstraction rather than a quantitative comparison.
• Figure 2: (a) Photograph of the 1-inch spherical probe and adjacent electrical hardware used in the hybrid PIP--LP diagnostic. (b) Schematic of the electrical system, consisting of three circuit regions: RF-only (green), DC-only (blue), and a superimposed DC+RF (purple). These circuits enable operation of the same probe in three modes: LP, floating PIP, and biased PIP. Calibration planes used in the RF de-embedding procedure are indicated.
• Figure 3: Calibrated PIP measurements for the $I_D = 10$ A discharge case at select probe biases, showing the real (top) and imaginary (bottom) components of the reflection coefficient, $\Gamma$, along with corresponding fits from the PIP model. The lower, sheath resonance ( $\omega_-$) is shown to be a function of $\Delta V$. In contrast, the upper, damped-plasma resonance ( $\omega_+$) is not.
• Figure 4: Comparison of biased-PIP sheath thickness measurements, $t_{\mathrm{pip}}$, with Child--Langmuir sheath thicknesses, $t_{\mathrm{CL}}$, calculated from Langmuir-probe measurements of density and temperature and scaled by $\alpha = [0.71, 0.74, 0.74, 0.74, 0.78]$ for the five discharge currents, respectively. Empty markers denote biased-PIP measurements, while filled markers denote floating-PIP measurements ($\Delta V \approx -15$ V). The results show consistent agreement between $t_{\mathrm{pip}}$ and $\alpha t_{\mathrm{CL}}$ across all discharge conditions.
• Figure 5: (a) Comparison of plasma density measurements obtained from the plasma impedance probe (PIP) and the Langmuir probe (LP) as a function of probe bias. Horizontal lines indicate the LP density measured at each discharge current, while symbols show PIP-derived densities. (b) Electron damping rate $\nu$ extracted from PIP fits as a function of probe bias. Empty markers denote biased-PIP measurements, and filled markers indicate floating-PIP measurements ($\Delta V \approx -15$ V).