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Accurate prediction of electron beam profile is one of the key objectives of electron optics, and the basis for design of the practical electron gun. In this paper, an improved model describing electron beam in Pierce gun with both space charge effects and emittance effects is proposed. The theory developed by Cutler and Hines is still applied for the accelerating region of the Pierce gun, while the motion equations of the electron beams in the anode aperture and drift tunnel are improved by modifying electron optics theory with emittance. As a result, a more universal and accurate formula of the focal length of the lens for the electron beam with both effects is derived for the anode aperture with finite dimension, and a modified universal spread curve considering beam emittance is introduced in drift tunnel region. Based on these improved motion equations of the electron beam, beam profiles with space charge effects and emittance effects can be theoretically predicted, which are subsequently approved to agree well with the experimentally measured ones. The developed model here is helpful to design more applicable Pierce guns at high frequencies.


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