Practical Electron Microscopy and Database

Assuming no external force is applied to an electron beam in vacuum, the electrons at the edge of the electron beam suffer two forces: outward force from the electrical field due to space charge and inward force due to the azimuthal magnetic field formed by the current of the electron beam. The net positive outward force F_r acting on an electron is given by [1],

          F_r = e²n_e(1-v_z²/c²)/2πε₀r_b ------------------------------- [4249]

where,
          e – Electron charge
          n_e – Electron density in the beam (electrons per unit axial length)
          v_z – Electron velocity
          c – Speed of light
          ε₀ -- Permittivity of of vacuum
          r_b -- Beam radius

Therefore, the beam tends to expand as it propagates in vacuum due to the net positive outward force induced by the space-charge effect. However, electron beams can travel with constant beam diameters when the velocity of the electrons becomes equal to that of light (v_z= c in Equation 4249).

[1] Hidenori Matsuzawa, Novel magnetic applications of high - Tc bulk superconductors: Lenses for electron beams, J. Appl. Phys. 74 (12), R111 (1993).