Geometric Optimization of a Symmetric System of Neutron Flux Detectors for BL3

The recent measurement of the lifetime of the free neutron using the beam method has an 8.7 s (4$\sigma$) discrepancy with UCN measurements. The goal of the BL3 experiment is to improve the statistical error of this measurement and explore systematic uncertainties as an explanation for the discrepancy. A well-characterized neutron flux detector with flat response is important since the neutron flux enters linearly into the neutron lifetime. I will present a group theoretic formalism for the geometric optimization of $n$ neutron detectors for uniform acceptance as a function of the neutron position and describe how to use this symmetry to extract other parameters from the detector rates such as the beam centroid, ellipticity, and RMS width.