The effective path length that an electron travels before being ejected into the vacuum is related to the depth below the surface at which it is generated and the exit angle by d = λcosθ , where λ is the inelastic mean free path and θ is the angle between the surface normal and the exit direction.

a. Justify this equation based on a sketch of the path that an electron travels before exiting into the vacuum.

b. The XPS signal from a thin layer on a solid surface is given by I = I₀ (1- e^{-d / (λ cosθ)}, where I₀ signal that would be obtained from an infinitely thick layer, and λ is defined in Problem P 22.14. Calculate the ratio I /I₀ at θ = 0 for λ = 2d. Calculate the exit angle required to increase I/I₀ to 0.50.