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Limits of plasma focusing of high current electron beams

Limits of the electron beam density in the focus are evaluated. As it shown, the maximal beam density is achievable in the relativistic vacuum diode with the needle anode. In such a device, the initial emittance of a beam is rather small. The electron beam gains emittance while being focused with positive ions into the anode due to collisions with the ions. Based on the paraxial beam model, the beam density in the focus is estimated. As is shown, the lowest limit of focus length is proportional to the square root from the ratio of the beam particle energy to the initial beam density. The maximal coefficient of the beam initial density multiplication in the focus is proportional to the initial transverse dimension of the beam and square root from the beam current.