Space-charge limited density of consecutively injected electron pulses with uniform separation

A one-dimensional model to study the space-charge limited density injection of a train of multiple electron pulse into a diode is presented. It is found that there is a maximal value of charge density per pulse for N number of pulse of equal time separation that can be injected. By comparing with numerical solutions, we obtain an analytical formula, which can quickly provide such upper limit of charge density injection once the values of gap spacing, gap voltage and the initial time separation between the pulses are provided. The model has been verified with the numerical solutions of the equation of motion up to a few MeV of beam energy for which the relativistic effect is included. This work should be useful in the design of multiple pulses of electron guns in the application of ultra-fast electron microscopy or free electron lasers.