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A new method has been employed for the design of solid-beam electron guns of high perveance and area convergence. This has resulted in designs with perveance 2.2Ã10-6 a/v3/2 and convergence ratio 300; and perveance 5Ã10-6 a/V3/2 and area convergence 6. Using conventional methods, a design with perveance 0.1Ã10-6a/v3/2 and area convergence ratio 1000 has been obtained. Each of these guns has yielded over 95 per cent transmission through a drift tube, in most cases with less than 1.5 times theoretical Brillouin focusing magnetic field. The design method consists of 1) paper design following earlier workers; 2) construction of a model using the design cathode and anode, but with the focus electrode replaced by a series of annular disk electrodes; 3) measurement, in pulsed bell-jar beam tester, of the beam leaving this gun, by means of a pinhole aperture followed by a split collector, yielding data on current density and trajectory angle as a function of radius and axial position; 4) modifications of annular disk potentials and cathode surface shape to improve beam quality; 5) electrolytic tank determination of the shape of a single electrode to replace the annular disks; and 6) test of the final design in a sealed-off, shielded-cathode, pulsed beam tester in which the beam flows through a drift tube in a uniform magnetic field. These methods are relatively exact and rapid. Drawings are presented for some specific designs.