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A description of the underlying physics governing the operation of pulsed-power driven, gas-filled, paraxial diodes is presented. Gas-filled focusing cells are routinely used to transport high energy density electron beams for use in flash X-ray radiography experiments. The paraxial diode acts as a 1/4 betatron focusing element with a focal length F proportional to the square root of the beam energy and inversely proportional to the square root of the net current in the gas cell F∝(γ/Inet)12/. Particle-in-cell simulations demonstrate that the time integrated radiation spot is determined both by focal sweeping due to time varying net currents and by finite beam emittance. The calculated radiation focal plane, spot, and dose are compared to data obtained by the Atomic Weapons Establishment, U.K., from a variety of experimental configurations and demonstrate good agreement between simulation and experiment. Suggestions to improve the focal properties of the diode are presented.