Focusing and guiding charged particles by a superconducting tube: An analytical nonlinear approach for the complete flux expulsion model

We give an analytical solution for the nonlinear equation of motion of an electron beam in the radial direction of a superconducting tube, for the case of electrons moving close to the tube axis, in the complete Bur expulsion model. Hence, we deduce a general expression for the focusing length that is dependent on the parameters of the superconducting tube and electron stream. Our focusing length is inversely proportional to x(0), the width of the electron beam, and is in good agreement with experimental data.