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It had been found experimentally in the past that there was a partial reflection of an electron beam falling obliquely on a material medium. It had been shown as well theoretically that the average reflection angle of a beam depends on its incident angle only and does not depend on particle impulses. Based on this phenomenon and the appropriate theoretical explanation the method of a beam formation from an isotropic charged particles point source had been proposed. The method is similar to this used in light optics to transform light radiated by a lamp into unidirectional ray. Appropriate surface form had been calculated for the case of charged particles source.
In this paper, the calculations that are more detailed are presented. First, these concern the base properties of partial beam reflection. Namely, the surface distribution function for a point size beam falling at some angle on the plane separating vacuum and medium is derived, multiple Coulomb scattering being taken into account only. This function is the exact solution of appropriate boundary problem and is derived from the known solution of the appropriate partial differential equation for unlimited medium. As the result, one has the family of phase portraits on interface surface and similar portrait family of the particles on the plane, which is perpendicular to the reflected part of the beam in vacuum. Then the appropriate differential equation determining the profile of the surface that transfers any particle source ray into the image plane is derived. At the last step, the real beam emittance on the image plane is calculated.
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