Determining electron temperature for small spherical probes from network analyzer measurements of complex impedance

In earlier work, using a network analyzer, it was shown that collisionless resistance (CR) exists in the sheath of a spherical probe when driven by a small rf signal. The CR is inversely proportional to the plasma density gradient at the location where the applied angular frequency equals the plasma frequency omega(pe). Recently, efforts have concentrated on a study of the low-to-intermediate frequency response of the probe to the rf signal. At sufficiently low frequencies, the CR is beyond cutoff, i.e., below the plasma frequency at the surface of the probe. Since the electron density at the probe surface decreases as a function of applied (negative) bias, the CR will extend to lower frequencies as the magnitude of negative bias increases. Therefore to eliminate both CR and ion current contributions, the frequencies presently being considered are much greater than the ion plasma frequency, omega(pi), but less than the plasma frequency, omega(pe)(r(0)), where r(0) is the probe radius. It is shown that, in this frequency regime, the complex impedance measurements made with a network analyzer can be used to determine electron temperature. An overview of the theory is presented along with comparisons to data sets made using three stainless steel spherical probes of different sizes in different experimental environments and different plasma parameter regimes. The temperature measurements made by this method are compared to those made by conventional Langmuir probe sweeps; the method shown here requires no curve fitting as is the usual procedure with Langmuir probes when a Maxwell-Boltzmann electron distribution is assumed. The new method requires, however, a solution of the Poisson equation to determine the approximate sheath dimensions and integrals to determine approximate plasma and sheath inductances. The solution relies on the calculation of impedance for a spherical probe immersed in a collisionless plasma and is based on a simple circuit analogy for the plasma. Finally, the temperatures obtained using this method show reasonable agreement with those obtained using a conventional Langmuir sweep analysis of the spheres.