A universal equation for computing the beam broadening of incident electrons in thin films

• A new equation for computing the beam broadening of incidents electrons in thin films.
• The new equation provides a universal curve for beam broadening using the power law.
• Excellent agreement between the model and Monte Carlo simulations of beam broadening.
• The Goldstein equation of beam broadening is a limiting case of the suggested equation.

A universal equation for computing the beam broadening of incident electrons in thin films is presented. This equation is based on the concepts of anomalous diffusion with the Hurst exponent H  . When the thickness to elastic mean free path ratio, t/λt/λ, is greater than 1, the Hurst exponent goes to 0.5 and this random walk behavior leads to the Goldstein et al. [1] beam broadening equation when non-relativistic screened Rutherford elastic cross-sections are used. When t/λ≪1t/λ≪1, the lack of elastic collisions for the electron trajectories gives an H exponent of 1 and a different beam broadening equation is obtained. A general equation to compute the beam broadening that takes into account the variation of H   with t/λt/λ is presented and this equation was fitted and validated with Monte Carlo simulations of electron trajectories in thin films.