$β$-delayed proton pandemonium: A first detailed $^{31}$Cl($βp γ$)$^{30}$P decay scheme

Positron decays of proton-rich nuclides exhibit large $Q$ values, producing complex cascades which frequently involve various radiations, including protons and $γ$ rays. Often, only one of the two is measured in a single experiment, limiting the accuracy and completeness of the decay scheme. An example is $^{31}$Cl, for which protons and $γ$ rays have been measured separately in detail but never with substantial sensitivity to proton-$γ$ coincidences. We provide detailed measurements of $^{31}$Cl $β$-delayed proton decay including $β$-$p$-$γ$ sequences, extract spectroscopic information on $^{31}$S excited states as well as their $β^+$ feedings, and compare to shell-model calculations. A fast fragmented beam of $^{31}$Cl provided by the National Superconducting Cyclotron Laboratory (NSCL) was deposited in the Gaseous Detector with Germanium Tagging (GADGET) system. GADGET's gas-filled Proton Detector was used to detect $β$-delayed protons, and the Segmented Germanium Array (SeGA) was used to detect $β$-delayed $γ$ rays. As many as 20 previously unobserved $β$-delayed proton transitions are reported, most of which populate excited states of $^{30}$P. The first detailed $^{31}$Cl($βp γ$)$^{30}$P decay scheme is presented, including updated $β$-delayed proton energies and intensities, as well as several new $^{31}$S levels. Improved agreement is found with theoretical calculations of the Gamow-Teller strengths $B(\text{GT})$ for $^{31}$S excitation energies $7.5 < E_x < 9.5$ MeV. The present work demonstrates that the ability to detect $β$-delayed protons and $γ$ rays in coincidence is essential for accurate positron decay schemes to compare with nuclear structure theory. This phenomenon for $β$-delayed protons resembles the pandemonium effect originally introduced for $β$-delayed $γ$ rays.