Influence of space charge on the Langmuir limit for focused electron beams

A theoretical investigation of the validity of Langmuir´s expression for the limiting current density in a focused electron beam when space-charge forces are significant was carried out by computing the performance of a series of simple electron tubes, each consisting of a Pierce gun coupled by a thin lens to an equipotential drift space. The analyses show that, while the cathode current density and the peak current density in the spot are related, they are not directly proportional as stated in Langmuir´s expression which, therefore, does not give an accurate prediction of the current density that can be obtained in the spot. The dimensions of the tubes were chosen such that this conclusion is also valid in the case of television picture tubes. It is also shown that thermal velocities have only a minor influence on the lens strengths required to focus beams, and that the focused spots havea Gaussian distribution of current density even though the emission density in a Pierce gun is uniform.