Larmor frequency in heterogeneous media.

High-precision signal phase measurements ignited an ongoing discussion of the microstructural correlates of the Larmor frequency shift of water in biological tissues. In a broader context, this is the question about the averaged precession frequency in magnetically heterogenous, in particular, porous media. In this study, the Larmor frequency shift is found analytically for water filling connected pore space between NMR-invisible magnetized inclusions with a constant magnetic susceptibility tensor. The magnetic microstructure that encompasses the inclusions' shape and their spatial arrangement is arbitrary as well as the inclusions' magnetic susceptibility tensor. The result is limited to the case of effectively fast diffusion of water molecules. In this limit, the effect of magnetic microstructure on the Larmor frequency shift is represented by only five relevant parameters in the general case and by a single parameter in the case of axially symmetric microstructure. This single parameter enters the dependence of the Larmor frequency on the sample orientation relative to the main magnetic field in the previously performed experiments. The result can help interpreting known experimental data and developing realistic models of biological tissues.