A method of computing X-ray photo-emission spectra in the wide range of energy losses and different sighting angles is presented. Photo-emission spectra for layers of finite thickness are investigated. Angular and energy spectra are analyzed using the invariant imbedding principle. They are computed using the small-angle approximation and the exact numerical solution of the multiple photo-electron scattering problem in solids. The presented methods of X-ray photo emission spectra analysis are compared regarding their efficiencies. The comparison of the exact numerical solution to those based on the straight line approximation and the small angle approximation reveals an error of the straight line approximation of about 50%. Numerical solutions are compared with the experimental data and Monte-Carlo simulations.

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