Allowed $\beta$-decay spectrum with numerical electron wave functions

Using numerical electron wave functions and state-of-the-art nuclear many-body methods, I evaluate the $\beta$-decay spectra for typical decay channels of spherical nuclei. I check errors brought by various approximations used for deriving the analytical shape factors (the so-called Fermi Function) of allowed decay. I estimate the errors brought by different electric charge distributions and give a way of estimation of $\beta$-spectra with available decay data of specific nuclei. I found that the traditional ways of approximating the electron wave functions by Fermi Function could be a very severe source of error for spectra simulation.