Accurate and general solutions to three-dimensional anisotropies: Applications to EPR spectra of triplets involving dipole-dipole, spin-orbit interactions and liquid crystals

Thanks to the squared Cartesian coordinates and its corresponding ternary diagram, several types of anisotropic physical quantities are expressed by a weighted mean of their principal values. Specifically, in terms of the electron paramagnetic resonance (EPR) spectra of triplet states under various conditions, the anisotropies in the resonance field (due to spin dipole-dipole interaction), in the spin polarization (due to spin-orbit intersystem crossing, SO-ISC), and in the distribution of molecular orientations (due to liquid crystal alignment) are all linearized. The spectral intensity becomes a path integral on the ternary diagram along the field isolines. These major simplifications afford, for the first time, analytical line shape formulas of arbitrarily polarized triplets as sums of elliptical integrals. Even with approximations applied, the analytical results agree almost perfectly with both simulated and experimental spectra and accurately capture the higher-order spectral effects such as peak shifts and net spin polarization. This Universal scheme is also promising for other spectroscopic techniques in which anisotropy plays a significant role.