Consistent solution for space-charge-limited current in the relativistic regime for monoenergetic initial velocities

Abstract

The solution for space-charge-limited (SCL) currents in electron vacuum diodes with monoenergetic initial velocity is extended to the relativistic regime. Two types of solutions are found: Type I corresponds to zero steady state surface electric field (field emission with high enhancement factor), and Type II corresponds to a finite steady state surface electric field (other emission mechanisms). Our solution compares well to the classical space-charge-limited currents with initial energy and relativistic space-charge limited currents without initial energy in the appropriate limits. The scaling law between the true SCL and the applied voltage is discussed and the two solution types are verified by particle-in-cell simulation.

Ion Accelerator Currents Beyond the Child-Langmuir Limit

“…One dimensional geometries are perhaps only reasonable approximations in ion thrusters where the accelerator region can only be designed "to first order" using the Child-Langmuir law but with suitably interpreted dimensional gaps between accelerating electrodes 2 . However, for analysis, the steady one-dimensional (Cartesian) description of space charge flows is very helpful for understanding the relevant non-linear phenomena, and this has been the subject of many publications (focusing primarily on the flow of electrons) for the last 100 years [3][4][5][6][7][8][9][10][11][12][13][14][15][16] .…”

Space-charge effect of the time-varying electron injection in a diode: Classical and relativistic regimes

“…8 Let the anode voltage / g ¼ 900 and U ¼ / g =ĉ 2 ¼ 1 so that the relativistic effect becomes noticeable. When the injection is time-invariant with v 0 ¼ĉ=60, the transmittance goes below the known SC limit [see …”

“…For a diode with a gap spacing of d and a diode voltage of / g , this is known as the well-known Child-Langmuir law. [1][2][3] This law extends to the more general case of nonzero initial velocity v 0 4,5 and the relativistic regime [6][7][8][9][10] when the voltage potential becomes relativistic. In recent years, the efforts to achieve a higher electron flow have been very active, in both classical and relativistic regimes.…”

“…According to our simulation, it seems unlikely that one can transmit a time-varying injection j(t) across the diode that is time-averagely higher than the conventional SC limited current density, in both classical 4 and relativistic regimes. 8,23 In our paper, the electrons are injected into the diode gap following the prescribed time profiles. It is assumed that the SC limit is reached when reflection occurs.…”

“…The increasing profiles (8)(9)(10) are normalised so that within the emission pulses they give the same account of charge Ð sp 0 j m ðtÞdt ¼ Ð sp 0 1dt ¼ s p and the other profiles are chosen to be as smoothly varying as possible. Note that the proper time s is operated in full differential, instead of partial differential because s is frame-independent for a certain particle.…”