$^{25}$Si $\beta$-decay spectroscopy using the Gaseous Detector with Germanium Tagging (GADGET) system*

The protons and $\gamma$ rays emitted in $^{25}$Si $\beta$ decay were measured using the GADGET system. Three $^{24}$Mg $\gamma$-ray lines, eight $^{25}$Al $\gamma$-ray lines, and a 719-keV proton branch were observed for the first time in $^{25}$Si decay. A Monte Carlo method was used to model the Doppler broadening of $^{24}$Mg $\gamma$-ray lines caused by proton emissions. All the proton-bound states of $^{25}$Al are observed to be populated in the $\beta$ decay of $^{25}$Si. We have reported the first measurement of the $^{25}$Si $\beta$-delayed $\gamma$-ray intensities through the $^{25}$Al unbound states. An enhanced decay scheme of $^{25}$Si has been constructed and compared to the mirror decay of $^{25}$Na and the shell-model calculations using two newly-developed $sd$-shell Hamiltonians, USDC and USDI. This work offers insights into the fine nuclear structure of $^{25}$Al.