Increases of velocity variance and energy spread in a charged particle beam, known as the ‘‘Boersch effect,’’ were treated statistically, based on energy conservation including kinetic energy and Coulomb potential energy between charged particles. The velocity variance in a drift space was represented by the total kinetic energy, total Coulomb potential energy between particles, and total momentum of particles. The velocity variance changes according to the change of total Coulomb potential energy between particles. The generation of anomalous energy spread is due to the conversion of the Coulomb potential energy into kinetic energy. The formulas for the ultimate increase of the velocity variance and the energy spread in a drift space were derived, and a Monte Carlo simulation was performed for their evaluation. The increase of position variance by Coulomb repulsion force was also formulated. The upper limit of the anomalous energy spread of an electron beam in a drift space was calculated for various acceleration voltages and beam currents.

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