Generalized Bohm's criterion and negative anode voltage fall in electric discharges

The value of the voltage fall across the anode sheath is found as a function of the current density. Analytic solutions are obtained in a wide range of the ratio of the directed velocity of plasma electrons v0 to their thermal velocity vT. It is shown that the voltage fall in a one-dimensional collisionless anode sheath is always negative. At the small values of v0/vT, the obtained expression asymptotically transforms into the Langmuir formula. Generalized Bohm's criterion for an electric discharge with allowance for the space charge density ρ(0), electric field E(0), ion velocity vi(0), and ratio v0/vT at the plasma-sheath interface is formulated. It is shown that the minimum value of the ion velocity vi*(0) corresponds to the vanishing of the electric field at one point inside the sheath. The dependence of vi* (0) on ρ(0), E(0), and v0/vT determines the boundary of the existence domain of stationary solutions in the sheath. Using this criterion, the maximum possible degree of contraction of the electron current at the anode is determined for a short high-current vacuum arc discharge