## Formation of focussed space-charge-limited electrons and ion beams |

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abscissa scale absolute value amp/cm2 analytic form angle axis beam is given beam whose minimum CATHODE Figure cathode wing center of curvature charge density due charge limited electron cm/sec constriction convenient convergence corresponds current per unit curve cylindrical differential equation dimensionless ratio electric field electrode shapes electron or ion emitting surface equation 27 equation 33 equation 34 ferent values finite flare focussed given by equation hence HYPERBOLIC TRAJECTORIES initial current density initial section Introduction and Sections ION BEAMS kilovolts large distances limited electron beam MDDC method for calculating minimum section move radially obtained parameter particular solution place of minimum plane parallel beam Poisson equation Poisson's equation polar coordinates potential becomes Potential function potential variation radius satisfy Laplace's equation slope solution of equation space charge limited SPACE-CHARGE-LIMITED spread supposed Table total current trajectories are given unit length units arbitrary upper edge values of vm velocity