Theoretical Limitations of Cathode-Ray Tubes

The current density in a focused beam of cathode rays is shown to have an upper limit defined by I = I0(Ee/kT+1) sin²φ, where I is the maximum current density obtainable in the focused spot, I0is the current density at the cathode, E is the voltage at the focus relative to the cathode, T is the absolute temperature of the cathode, e is the electronic charge, k is Boltzmann\´s constant, and φ is the half angle subtended by the cone of electrons which converge on the focused spot. The cases in which the focused spot is an image of the cathode, and in which it is a pupil, or "crossover", are considered separately, and the above formula is shown to apply to both. The necessary initial assumptions are (1) that electrons leave the cathode with a Maxwellian distribution of velocities, and (2) that the focusing system is free from aberrations and obeys the law of sines. Aberrations may reduce the current density, but nothing can raise it above the value defined. In the Appendix the focusing properties of a uniform accelerating field are calculated. The virtual image of a plane cathode formed by such a field suffers from spherical aberration. The diameter of the circle of least confusion formed by electrons from a single point is approximately equal to the distance the electrons can travel against the field by virtue of their initial velocities.