Internal field of homogeneously magnetized toroid sensor for proton free precession magnetometer

The shift of the NMR spectral line frequency in a proton free precession absolute scalar magnetometer using the omni-directional toroid container for a proton-rich liquid depends on the magnetic susceptibility of the liquid and on the direction of the external field relative to the axis of the toroid. The theoretical shift is estimated for water by computing the additional magnetic field from the magnetization of the liquid and comparing it to the theoretical field in a spherical container. Along the axis the estimated average shift is -0.08 nT and perpendicular to the axis the shift is +0.08 nT relative to that of a spherical sensor. The field inhomogeneity introduced by the toroid shape amounts to 0.32 nT over the volume of the sensor and is not expected to significantly affect the signal decay time, when considering the typical water line width of about 2.5 nT.