Multiplets at zero magnetic field: the geometry of zero-field NMR

For liquid samples at Earth's field or below, nuclear-spin motion within scalar-coupled networks yields multiplets as a spectroscopic signature. In weak fields, the structure of the multiplets depends on the magnitude of the Zeeman interaction relative to the scalar couplings; in Earth's field, for example, heteronuclear couplings are truncated by fast precession at distinct Larmor frequencies. At zero field, weak scalar couplings are truncated by the relatively fast evolution associated with strong scalar couplings, and the truncated interactions can be described geometrically. When the spin system contains a strongly coupled subsystem A, an average over the fast evolution occurring within the subsystem projects each strongly coupled spin onto FA, the summed angular momentum of the spins in A. Weakly coupled spins effectively interact with FA, and the coupling constants for the truncated interactions are found by evaluating projections. We provide a formal description of zero-field spin systems with truncated scalar couplings while also emphasizing visualization based on a geometric model. The theoretical results are in good agreement with experimental spectra that exhibit second-order shifts and splittings.