A Complete Approach to Determine the $^3$He neutron incoherent scattering length $b_i$
We report the first results from a new approach for measuring the $^3$He neutron incoherent scattering length $b_{i}$. $b_{i}$ is directly proportional to the difference $Δb=b_{+}-b_{-}$ in the two low-energy s-wave neutron-nucleus scattering amplitudes $b_{+}$ and $b_{-}$, corresponding to the singlet $J=0$ and triplet $J=1$ states of the neutron-$^3$He interaction, respectively. An accurate measurement of $b_{i}$ can help distinguish among different models of three-nucleon interactions by comparison to {\it ab initio} nuclear theory calculations. The neutron birefringence caused by $Δb$ results in neutron spin rotation around the nuclear polarization. We measured $Δb$ using polarized neutron spin rotation and the transmission of neutrons through a $^3$He gas target polarized in situ by spin-exchange optical pumping. This brief test measurement, conducted at the FZ-Jülich neutron spin echo spectrometer at the Heinz Maier Leibnitz Zentrum (MLZ), yielded $Δb = [-5.27 \pm 0.05$ (stat.) $- 0.05$ (syst.)] fm. We argue that this method can be improved in precision to resolve the discrepancies between two prior measurements of $b_i$ which are dependent on the polarized absorption cross section $σ_p$. Further with absolute $^{3}$He polarization via NMR (in a properly-shaped cell) concurrent with accurate neutron transmission measurements, $σ_p$ can be measured to obtain independent values of $b_{+}$ and $b_{-}$.