Sheath size and Child–Langmuir law in one dimensional bounded plasma system in the presence of an oblique magnetic field: PIC results

Abstract

The sheath properties are studied by using 1d3V particle-in-cell simulations in a plasma bounded by two conductive electrodes between which is applied a constant voltage, Vw. A magnetic field tilted by θ with respect to the wall is considered in the simulations. Elastic collisions with neutrals are modeled by an operator that randomly reorients ions and electrons in the velocity space. The ratio between the ion mean-free-path and the Larmor radius is chosen such that λciR>1, ≃0.5 or < 1, whereas the same ratio for electrons is always ≫1. For low ion collisionality ( λciR>1) and incidences such that sin θ>Rλci, the sheath size is shown to scale with sin θ and depends on Vw according to the Child–Langmuir law, with a 3/4 exponent. For larger collisionalities, the sin θ dependence of the sheath size disappears because ions are demagnetized by collisions. Then, for incidences θ>5°, the sheath size varies with a 3/5 exponent of the wall potential, as expected in moderate collisional sheaths. More interestingly, for grazing incidences and λciR≃0.5, an inverse sheath, i.e., an electro-negative space charge, arises at the wall vicinity in order to screen the positive wall potential (instead of the negative one). Its size, comparable to a classical Debye sheath, is shown to vary with a 2/3 exponent of the wall potential. Finally, our simulation results show that the Child–Langmuir law is a good way to evaluate the sheath size for a large range of collisionality at any magnetic field incidence as long as the exponent is chosen accordingly.