Fig. 1: Cartoon of the frequency shift mechanism.

Considering the diffusion motion and boundary relaxation of Xe spins, a nonuniform magnetic field can shift the resonance frequency of Xe spins. The first order frequency correction slightly depends on the boundary condition λ, while the third order correction is proportional to the ratio of gyromagnetic ratio γ_Xe and diffusion constant D. Both λ and \({\gamma }_{{{{\rm{Xe}}}}}^{2}/{D}^{2}\) can be different for different Xe isotopes, thus will introduce systematic error in comagnetometer type experiments. As an example, the systematic errors introduce to NMR gyroscopes, a kind of ¹²⁹Xe-¹³¹Xe comagnetometer, are shown by \(\delta {{{\Omega }}}_{{{{\rm{rot}}}}}^{({G}_{2},1)}\) and \(\delta {{{\Omega }}}_{{{{\rm{rot}}}}}^{({G}_{2},3)}\). These systematic errors are generally not negligible, limiting the absolute accuracy of comagnetometers as well as the long time stability of NMR gyroscopes. On the other hand, the factor (λ₁₃₁ − λ₁₂₉) in \(\delta {{{\Omega }}}_{{{{\rm{rot}}}}}^{({G}_{2},1)}\) enables a new tool for boundary relaxation rate measurement, which should be much faster than previously reported methods. Refer to Eqs. (24), (25), (29) and (30) for more details.