The Langmuir current limit in the presence of a magnetic field

The Langmuir current limit is the theoretical limit, imposed by the thermal velocity distribution, on the ratio of current density collected at a screen at potential V0to current density emitted by the cathode. This expression was first derived by D. B. Langmuir for the case of a beam in zero magnetic field and for a single beam temperature. However, realistic electron beams are Characterized by distinct axial and radial temperatures, Tzand Trrespectively, and are usually focussed by magnetic fields. In this paper an analogous current density limit is derived for a solid electron beam characterized by two temperatures. Both uniform and spatially varying magnetic fields are considered for unintercepted beams and for beams which pass through a limiting aperture. A typical result, for the case of a beam with no limiting aperture is: α is the ratio of magnetic field at the target to that at the gun, α is greater than one, and jois the emitted density.