Measurement of the $^{14}$C spectrum with Silicon Drift Detectors: towards the study of forbidden $\beta$ transitions

The ASPECT-BET (An sdd-SPECTrometer for BETa decay studies) project aims to develop a novel technique for the precise measurement of forbidden $\beta$ spectra in the 10 keV - 1 MeV range. This technique uses a Silicon Drift Detector (SDD) as the main spectrometer, surrounded, if necessary, by a veto system to reject events with only partial energy deposition in the SDD. Accurate knowledge of the spectrometer's response to electrons is essential to reconstruct the theoretical shape of the $\beta$ spectrum. To compute this response, GEANT4 simulations optimized for low-energy electron interactions are used. In this article, we present the performance of these simulations in reconstructing the electron spectra, measured with SDDs, of a $^{109}$Cd monochromatic source, both in vacuum and in air. The allowed $\beta$ spectrum of a $^{14}$C source is also measured and analyzed, and it is shown that the experimental shape factor commonly used in the literature to reconstruct the measured spectrum is not necessary to explain the spectrum.