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Space charge strongly influences electron transport in high-field vacuum gaps, as electrons emitted from a cathode modify the electrostatic field governing their subsequent motion. The space-charge-limited current density was originally derived by Child and Langmuir for a one-dimensional planar diode [1,2], and analytical expressions have subsequently been obtained for selected non-planar geometries — for example, in Ref. [3] the limiting current density for an idealised one-dimensional tip-to-tip geometry is derived. Such analytical approaches provide the fully space-charge-limited current under specific geometrical assumptions, but do not generally resolve the self-consistent multidimensional electric-field distribution for arbitrary imposed currents, including partially space-charge-limited conditions. General geometries are therefore commonly studied using particle-in-cell methods, which may be computationally demanding and subject to statistical noise.
This work presents a self-consistent continuum method for calculating the redistribution of the electric field caused by space charge in complex vacuum geometries. The approach is applicable when the emitted electrons form a smooth, steady and non-crossing continuous flow. Electron trajectories are calculated in material coordinates, and the electron density is obtained directly from flux conservation using the Jacobian of the mapping between material and spatial coordinates. This formulation accounts for density changes caused by electron acceleration and transverse beam spreading. The resulting electron density is introduced as a source term in Poisson’s equation, and the potential, electric field, trajectories and charge distribution are computed iteratively until a self-consistent steady-state solution is obtained.
The model is validated using one-dimensional plane-parallel electrodes and a two-dimensional spreading electron beam, for which analytical solutions exist. The numerical results reproduce the expected density distributions and the reduction of the cathode field caused by space-charge shielding. The method is subsequently applied to axisymmetric tip-to-tip electrodes, for several cathode half-angles, and the predicted space-charge-limited currents follow the overall trends of analytical charge-free field-based estimates and particle-in-cell results reported in Ref. [4]. Sharper cathode tips produce more localised emission and space charge near the apex, leading to stronger deviation from idealised one-dimensional transport. For the geometries considered, self-consistent solutions were typically obtained within approximately ten minutes on a standard desktop computer.
The method therefore provides a physically consistent numerical approach for calculating both partial electrostatic shielding and fully space-charge-limited flow in complex vacuum devices.
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