Space-charge sheath with ions accelerated into the plasma

The conventional model of near-cathode space-charge sheath with ions entering the sheath from the quasi-neutral plasma may not be applicable to discharges burning in cathode vapour, e.g. vacuum arcs, where ionization of emitted atoms may occur inside the sheath with some of the produced ions returning to the cathode and others moving into the plasma. In this connection, a simple model is considered of a sheath formed by electrons and positive ions injected into the sheath with a very low velocity and moving from the sheath into the plasma. It is shown that such a sheath is possible provided that the sheath voltage is equal to or exceeds approximately 1.256kTe/e. This limitation is due to the space charge in the sheath and is in this sense analogous to the limitation of ion current in a vacuum diode expressed by the Child--Langmuir law. The ions leave the sheath and enter the plasma with a velocity equal to or exceeding approximately 1.585uB.