Accurate Determination of the Electron Spin Polarization In Magnetized Iron and Nickel Foils for M{o}ller Polarimetry

The M{\o}ller polarimeter in Hall A at Jefferson Lab in Newport News, VA, has provided reliable measurements of electron beam polarization for the past two decades reaching the typically required $\pm$1\% level of absolute uncertainty. However, the upcoming proposed experimental program including MOLLER and SoLID have stringent requirements on beam polarimetry precision at the level of 0.4\% \cite{MOLLER2014, SoLID2019}, requiring a systematic re-examination of all the contributing uncertainties. M{\o}ller polarimetry uses the double polarized scattering asymmetry of a polarized electron beam on a target with polarized atomic electrons. The target is a ferromagnetic material magnetized to align the spins in a given direction. In Hall A, the target is a pure iron foil aligned perpendicular to the beam and magnetized out of plane parallel or antiparallel to the beam direction. The acceptance of the detector is engineered to collect scattered electrons close to 90$^{\circ}$ in the center of mass frame where the analyzing power is a maximum (-7/9). One of the leading systematic errors comes from determination of the target foil polarization. Polarization of a magnetically saturated target foil requires knowledge of both the saturation magnetization and $g^\prime$, the electron $g$-factor which includes components from both spin and orbital angular momentum from which the spin fraction of magnetization is determined. This paper utilizes the existing world data to provide a best estimate for target polarization for both nickel and iron foils including uncertainties in magnetization, high-field and temperature dependence, and fractional contribution to magnetization from orbital effects. We determine the foil electron spin polarization at 294~K to be 0.08020$\pm$0.00018 (@4~T applied field) for iron and 0.018845$\pm0.000053$ (@2~T applied field) for nickel.