This paper discusses the extension to different electron beam aspect ratio of the Child-Langmuir law for the maximum achievable current density in electron guns. Using a simple model, we derive quantitative formulas in good agreement with simulation codes. The new scaling laws for the peak current density of temporally long and transversely narrow initial beam distributions can be used to estimate the maximum beam brightness and suggest new paths for injector optimization.
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Filippetto et al. (2014) studied this question.
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