The recent developments in nanosecond pulsed power supplies facilitate the emission of high density electron bursts but their safe operation demands avoiding breakdowns. Using the theoretical and numerical modeling of the electron emission phenomena from a tip (micro-protrusion), the breakdown threshold (pre-breakdown) is analyzed considering it as the highest value of the voltage preserving the system out of the thermo-emission instability regime. However, the space charge that builds up in front of the tip limits the performance of these electron sources by decreasing the local electric field and consequently the thermo-field emission as well as the temperature of the emissive surface. Hence, it is found that the system can safely hold higher voltages (without breakdown) in the presence of dense space charge. In direct current, for a titanium elliptic tip, the highest operation voltage increases by about 15%, whereas for a tungsten hyperbolic tip, it increases by 70%. Remarkably, the emitted current close to the pre-breakdown voltage stays unchanged with or without taking into account the space charge. Surprisingly, when very short pulses (3 ns) are applied to a tungsten hyperbolic tip, the pre-breakdown voltage additionally increases by 30%, and the Coulomb screening, very effective in front of the tip apex, enlarges the electron emission area by 60%, releasing about 1.3 times more electrons compared to vacuum emission (without the space charge). Moreover, the ring effect, experimentally discovered by Dyke and Trolan [Phys. Rev. 89, 799 (1953)] on the radial electron density distribution, can be microscopically observed and understood with your model.
This paper presents solutions for the classical one-dimensional (1D 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 terms of 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.
Properties of bound states for Schrödinger operators are reviewed. These include: bounds on the number of bound states and on the moments of the energy levels, existence and nonexistence of bound states, phase space bounds and semi-classical results, the special case of central potentials, and applications of these bounds in quantum mechanics of many particle systems and dynamical systems. For the phase space bounds relevant to these applications we improve the explicit constants.
We compute the Pauli–Jordan, Hadamard, and Feynman propagators for the massive metrical perturbations on de Sitter space. They are expressed both in terms of mode sums and in invariant forms.