A time dependent solution to the relativistic one-dimensional diode

Work in understanding electron flow across a simple potential gap, a diode, has a long history: The Child-Langmuir law [C. Child, Phys. Rev. 32 (1911) and I. Langmuir, Phys. Rev. 2 (1913) 450] describes the current density achievable in a one dimensional diode in the steady state. Pierce [Theory and Design of Electron Beams (Van Nostrand, Princeton and London, 1949)] extended the work by demonstrating that a beam of charged particles of finite transverse extent will exactly obey the one dimensional law as long as appropriate boundary conditions exterior to the beam are satisfied. Jory and Trivelpiece extended the solution to relativistic energies [J. Appl. Phys. 40 (1969) 3924], and Lampel and Tiefenback presented a solution for a time dependent non-relativistic step function current [Appl. Phys. Lett. 43 (1983) 57]. This paper presents an analytic solution for a voltage waveform across a fully relativistic one-dimensional diode which produces a constant electron current exiting the diode after a time equal to an electron transit time under steady state space charge limited operation. This result is compared to the previous result obtained for the non-relativistic case. Relevance of this solution to single cell RF-guns is discussed.