Beam generation in foil-less diodes

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Abstract
A study is made of the generation of intense relativistic electron beams in rectangular and cylindrical foil‐less diodes. The diode space charge is treated self‐consistently. The electron emission from the cathode is assumed to be space‐charge‐limited. A strong axial magnetic field is assumed to prevent the electrons from reaching the anode surface(s) directly, and to constrain the electron motion to be approximately one‐dimensional. A useful dimensionless measure of the diode potential φc is ε≈[mec2/(eφc)]1/2, with me and e the electron rest mass and charge, and c the speed of light. Properties of the diodes are first analyzed in the ultra‐relativistic limit, ε=0, where the condition for space‐charge‐limited emission gives rise to a linear singular integral equation. This equation is solved for rectangular diode geometry, and the solutions are studied in detail. In particular, the diode impedance is independent of φc, and the beams are, in general, hollow. The beam particle kinetic energy flux, Γp, decreases as the beam width, b, increases; for b=a, Γp=0, where a is the diode width. A treatment of the diodes is then given for small but nonzero values of ε. For finite ε, there is a nonrelativistic Child–Langmuir sheath of thickness ≈εa at the cathode surface. For b=a, Γp is shown to be proportional to φc3/2.