The electron and photon transport processes in spectroscopy techniques described by the invariant embedding theory is here revisited. We report a convergence method to obtain closed analytical solutions to the 3D integro-differential equations. This method was successfully used in calculating the dependence of the electron backscattered fraction on the atomic number and on the energy. Also the fraction of absorbed electron as a function of incident angles was calculated. This method, using a states ladder model for the electron energies, provides a tool for testing physical parameters involved in the transport theory, such as the elastic and inelastic cross sections. The outstanding feature of the invariant embedding differential equations of considering observable quantities (such as the emergent flux of particles) as independent variables makes them a suitable tool to describe experimental situations.