Electron and ion sheath effects on a microwave “hairpin” probe

A microwave measurement of electron density in low-pressure plasmas can be based on a hairpin probe. The hairpin forms a transmission line that supports a quarter-wavelength standing wave. The resonance is related to the relative permittivity of the surroundings, and hence, in a plasma, electron density can be evaluated. For improved fidelity, a general model is developed to include the effects of positive and negative space-charge sheaths formed around the hairpin wires. The former tends to lower the resonance, whereas the latter tends to raise it initially. This is qualitatively in agreement with experiments in dc argon plasmas.