Two-fluid solutions for Langmuir probes in collisionless and isothermal plasma, over all space and bias potential

Abstract

This paper presents solutions for the classical one-dimensional (radial and Cartesian) problem of Langmuir probes in a collisionless, isothermal plasma. The method is based on two-fluid equations derived from the first two moments of Vlasov's equation. In contrast to commonly-used approximations, electron inertia and ion temperature are not neglected so that the fluid equations are symmetric in the terms for electrons and ions. The fluid equations are reduced analytically so that the electric potential is the only remaining spatial function, which is numerically determined using Poisson's equation. The single radial solution applies continuously over the whole region from the probe up to the unperturbed plasma, in contrast to theories which separate the probe boundary region into a charged sheath and a quasi-neutral pre-sheath, and is valid for all values of probe bias potential. Current-voltage characteristics are computed for cylindrical and spherical probes, which exhibit non-saturation of the ion and electron currents. The 1D Cartesian case is also analysed, and the Bohm criterion is recovered only in the limit of large radius probes.

Two-fluid plasma model for radial Langmuir probes as a converging nozzle with sonic choked flow, and sonic passage to supersonic flow

“…6 In this Brief Communication, it is shown that the Lambert function is in fact a general solution for onedimensional, isothermal compressible fluid equations in classical fluid dynamics. The analogy with fluid dynamics gives physical insight into the Lambert solution, and confirms the intuitive assumptions for the fluid boundary conditions in [5].…”

“…Langmuir probes are one of the most fundamental diagnostics of plasma physics. [1][2][3][4] A recent paper 5 presented a mathematical treatment for radial Langmuir probes in collisionless, isothermal plasma by using the Lambert function. 6 In this Brief Communication, it is shown that the Lambert function is in fact a general solution for onedimensional, isothermal compressible fluid equations in classical fluid dynamics.…”

“…An analogy with classical fluid dynamics could be expected because [5] solves the Euler compressible fluid equations for mass continuity and momentum conservation, along with the ideal gas law, treating ions and electrons as two separate fluids. To illustrate this, Fig.…”

“…To illustrate this, Fig. 1(b) for radial probes in [5] is redrawn here in Fig. 1 using fluid dynamics terminology: The equivalence of the terms in classical fluid dynamics and in [5] is given in Table I, where the textbooks by Shapiro, 7 and Landau and Lifshitz, 8 are followed throughout.…”

“…1(b) for radial probes in [5] is redrawn here in Fig. 1 using fluid dynamics terminology: The equivalence of the terms in classical fluid dynamics and in [5] is given in Table I, where the textbooks by Shapiro, 7 and Landau and Lifshitz, 8 are followed throughout. It can be seen that the flow towards the radial probe (cylindrical or spherical) is equivalent to one-dimensional (1D) radial flow in a converging nozzle.…”