Doppler broadening in the $\beta $-proton-$\gamma $ decay sequence

We report the first observation of Doppler-broadening in $\beta $ delayed proton- $\gamma $ decay. The broadening occurs because the daughter nucleus $\gamma $ decays while recoiling from proton emission. A method to analyze $\beta $ delayed nucleon emission was applied to two Doppler-broadened $^{25}$Al peaks from the $^{26}$P($\beta $p$\gamma)^{25}$Al decay. The method was first tested on the broad 1613 keV $\gamma $-ray peak using known center-of-mass proton energies as constraints. The method was then applied to the 1776 keV $\gamma $-ray peak from the 2720 keV excited state of $^{25}$Al. The broadening was used to determine a $^{26}$Si excitation energy of 13.3 $\pm$ 1.0 (stat.) $\pm$ 0.7 (syst.) MeV. This energy is consistent with proton emission from the known T$=$2 isobaric analog state of $^{26}$P in $^{26}$Si.