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We report a determination of the n-$^3$He scattering length difference$\Delta b^{\prime} = b_{1}^{\prime}-b_{0}^{\prime} = $ ($-5.411$ $\pm$ $0.031$(statistical) $\pm$ $0.039$ (systematic)) fm between the triplet and singletstates using a neutron interferometer. This revises our previous result $\Deltab^{\prime} = $ (-5.610 $\pm$ $0.027$ (statistical) $\pm$ $0.032$ (systematic)fm obtained using the same technique in 2008. This revision is due to are-analysis of the 2008 experiment that includes a more robust treatment of thephase shift caused by magnetic field gradients near the $^3$He cell.Furthermore, we more than doubled our original data set from 2008 by acquiringsix months of additional data in 2013. Both the new data set and a re-analysisof the older data are in good agreement. Scattering lengths of low Z isotopesare valued for use in few-body nuclear effective field theories, provideimportant tests of modern nuclear potential models and in the case of $^3$Heaid in the interpretation of neutron scattering from quantum liquids. Thedifference $\Delta b^{\prime}$ was determined by measuring the relative phaseshift between two incident neutron polarizations caused by the spin-dependentinteraction with a polarized $^3$He target. The target $^3$He gas was sealedinside a small, flat windowed glass cell that was placed in one beam path ofthe interferometer. The relaxation of $^3$He polarization was monitoredcontinuously with neutron transmission measurements. The neutron polarizationand spin flipper efficiency were determined separately using $^3$He analyzersand two different polarimetry analysis methods. A summary of the measuredscattering lengths for n-$^3$He with a comparison to nucleon interaction modelsis given.

Neutron interferometric measurement of the scattering length difference between the triplet and singlet states of n-$^3$He