Compact, bright neutron sources are opening up several emerging applications including detection of nuclear materials for national security applications. At Los Alamos National Laboratory, we have used a short-pulse laser to accelerate deuterons in the relativistic transparency regime. These deuterons impinge on a beryllium converter to generate neutrons. During the initial experiments where these neutrons were used for active interrogation of uranium and plutonium, we observed β-delayed neutron production from decay of 9Li, formed by the high-energy deuteron bombardment of the beryllium converter. Analysis of the delayed neutrons provides novel evidence of the divergence of the highest energy portion of the deuterons (i.e., above 10 MeV/nucleon) from the laser axis, a documented feature of the breakout afterburner laser-plasma ion acceleration mechanism. These delayed neutrons form the basis of non-intrusive diagnostics for determining the features of deuteron acceleration as well as monitoring neutron production for the next generation of laser-driven neutron sources.