An analytical treatment of electron spectral saturation for dynamic nuclear polarization NMR of rotating solids

Saturation of electron magnetization by microwave irradiation under magic-angle spinning (MAS) is studied theoretically. The saturation is essential for dynamic nuclear polarization (DNP) enhancement of nuclear magnetic resonance signals. For a spin with a large g-anisotropy and a long T₁ relative to the rotor period, the sample rotation distributes saturation to the whole powder sample spectrum. Analytical expressions for the saturation and frequency profiles are obtained. For a pair of coupled electrons such as those in bis-nitroxides, which are commonly used for MAS DNP, an e_l-e_r model (where e_l and e_r stand for electrons on the left and the right, respectively, in their spectral positions) is introduced to simplify the analysis of a coupled two-spin system under MAS. For such a system, strong electron couplings exchange magnetization during dipolar/J rotor events when the two electron frequencies cross each other. The exchange is equivalent to a swap of the e_l and e_r electrons. This allows for the treatment of a coupled spin pair as two independent spins such that an analytical solution can be obtained for the steady-state magnetization and the difference between the two electrons. The theoretical study with its analytical result provides a simple physical picture of electron saturation under MAS and of how radical properties and experimental parameters affect cross-effect DNP. The effects of depolarization and the extension to more coupled electron spins are also discussed using this approach.