Theory of the negative anode drop in low-pressure discharges

A kinetic model of the anode region with a negative anode potential drop is proposed, which explicitly takes into account the ratio of the directed velocity of electrons in plasma $v_0$ to their thermal speed $v_T$ as a parameter of the electron velocity distribution function. A transcendent equation is derived for determination of the negative anode drop as a function of the ratio $v_0/v_T$. It is shown that, unlike the known Langmuir formula, the anode drop remains negative for any value of $v_0/v_T$. In the case of small values ($v_0/v_T\ll1$), the derived expression asymptotically reduces to the Langmuir formula. It is shown that the dependence of electron concentration on the potential differs from the Boltzmann law. The anode regions of the short high-current vacuum-arc discharge and the classical low-pressure discharge are considered as examples. For the vacuum-arc discharge, the current density distribution and the anode drop distribution over the anode surface under strong current contraction are calculated. For the low-pressure discharge, potential drops are calculated across the region of the inhomogeneous near-anode plasma and the space charge layer.