Non-commutative perturbation theory for spin dynamics explains the factorization properties of RIDME background

The intermolecular hyperfine relaxation-induced dipolar modulation enhancement (ih-RIDME) experiment has a promising potential to quantitatively characterize the nuclear environment in the 0.8-3 nm range around an electron spin. Such information about the spatial arrangement of nuclei is of great interest for structural biology as well as for dynamic nuclear polarization (DNP) methods. In order to develop a reliable and sensitive spectroscopic tool, a solid data model needs to be established. Here, we attempt to provide a theoretical explanation for the experimentally observed properties of the ih-RIDME signal. Our main approach uses a perturbation expansion of the Baker-Campbell-Hausdorff formula during the transverse evolution of the electron spin, treating the nuclear dipolar Hamiltonian as a perturbation. We show that a product structure of the ih-RIDME signal follows directly from the statistical independence of the perturbation terms and the multinuclear hyperfine coupling, and that this signal composition is expected when the mixing time exceeds the 95% decay of the Hahn echo.