Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameterless integral equation. For any multipolarity. the discrete part of the spectrum is found to consist of infinite number of modes with ... https://www.eleanorscholz.com/product-category/laptop-bag/
Longitudinal modes of bunched beams with weak space charge
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