Beta-delayed-neutron ($beta$n) spectroscopy was performed using the Beta-decay Paul Trap and an array of radiation detectors. The $beta$n branching ratios and energy spectra for $^{135,136}$Sb and $^{140}$I were obtained by measuring the time of flight of recoil ions emerging from the trapped ion cloud. These nuclei are located at the edge of an isotopic region identified as having $beta$n branching ratios that impact the r-process abundance pattern around the A~130 peak. For $^{135,136}$Sb and $^{140}$I, $beta$n branching ratios of 14.6(11)%, 17.6(28)%, and 7.6(28)% were determined, respectively. The $beta$n energy spectra obtained for $^{135}$Sb and $^{140}$I are compared with results from direct neutron measurements, and the $beta$n energy spectrum for $^{136}$Sb has been measured for the first time.
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