We consider the energy spectrum of emitted electrons in $beta$-decay. Exact Coulomb Dirac wave functions describing the $beta$-electron in the Coulomb field of the daughter nucleus are used. Further, the improved wave functions which include the scre
ening of the Coulomb field due to the atomic electron cloud are also used. Thus, the interaction between the $beta$-electron and the field due to the daughter atom is treated within a nonperturbative approach. Are shown the modifications due to the screening on the $beta$ spectra and shown that those effects are very important. In addition, are addressed the contributions to the $beta$ spectra due to the exchange terms and shown that the corresponding effects can be substantial. Higher orders arising from the multipole expansion are considered. A comparison of the theoretical results obtained in this work has been made with recent experimental data and a very good agreement was observed.
