  The dielectric constant of a sheath, whether ionic or electronic, formed
around the cylindrical limbs of a hairpin probe, is often considered the same
as that of a vacuum. However, this assumption does not hold true for electron
sheaths and electron-permeating ionic sheaths, resulting in a deviation of the
sheath dielectric constant from that of a vacuum. This deviation significantly
influences the effective dielectric between the cylindrical limbs. As a result,
it impacts the theoretically estimated resonance frequency characteristic curve
of a DC-biased hairpin probe. In this study, we investigate the influence of
electron temperature on the sheath dielectric and, consequently, on the
resonance frequency characteristic curve. The findings shows that electron
temperature primarily determines the resonance frequency characteristic curve.
With increasing electron temperature, the peak in the resonance frequency
characteristic curve shifts towards higher positive probe bias values and
exhibits a broadening near the maxima instead of a sharp peak. This broadening
near the maxima has also been validated with an experimentally measured
resonance frequency characteristic curve in a capacitively coupled argon
discharge.
