The Doppler broadening of $\gamma$-ray peaks due to nuclear recoil from $\beta$-delayed nucleon emission can be used to measure the energies of the nucleons. The purpose of this Thesis is to test and apply this Doppler broadening method using $\gamma$-ray peaks from the $^{26}$P($\beta$p$\gamma$)$^{25}$Al decay sequence. A fast beam of $^{26}$P was implanted into a planar Ge detector, which was used as a $^{26}$P $\beta$-decay trigger. The SeGA array of high-purity Ge detectors was used to detect $\gamma$ rays from the $^{26}$P($\beta$p$\gamma$)$^{25}$Al decay sequence. Radiative Doppler broadening in $\beta$-delayed proton-$\gamma$ decay was observed for the first time. The Doppler broadening analysis method was verified using the 1613 keV $\gamma$-ray line for which the proton energies were previously known. The 1776 keV $\gamma$ ray de-exciting the 2720 keV $^{25}$Al level was observed in $^{26}$P($\beta$p$\gamma$)$^{25}$Al decay for the first time and used to determine that the center-of-mass energy of the proton emission feeding the 2720-keV level is 5.1 $\pm$ 1.0 (\textit{stat.}) $\pm$ 0.6 (\textit{syst.}) MeV, corresponding to a $^{26}$Si excitation energy of 13.3 $\pm$ 1.0 (\textit{stat.}) $\pm$ 0.7 (\textit{syst.}) MeV for the proton-emitting level. The Doppler broadening method has been demonstrated to provide practical measurements of the energies for $\beta$-delayed nucleon emissions populating excited states of nuclear recoils at least as heavy as $A=25$.
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