Minimizing neutron population for one-sec lifetime measurement

The number of ultracold neutrons (UCN) within a certain volume can be measured by counting the number of electrons or protons from the neutron beta-decay, or by direct counting of UCN [Salvat et al., Phys. Rev. C. {\bf 89} (2014) 052501]. In both cases, the error of measured neutron lifetime increases with the counting error, decreases with the waiting time inbetween counting, and is ultimately limited by the neutron population experimentally. A one-percent neutron lifetime measurement requires at least a few times 10$^4$ neutrons. A one-sec neutron lifetime measurement requires close to 10$^7$ neutrons initially. Both estimates are for a waiting period of about 1000 sec and a total neutron counting efficiency of 50\%. We lay out experimental options for error reduction with an emphasis on optimizing UCN detection efficiency, and discuss the minimum number of neutrons required to achieve one-second neutron lifetime measurement.