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 NMR signals. For a spin with a large g-anisotropy and long T1 relative to the rotor period, sample rotation distributes saturation to the whole powder sample spectrum. Analytical expressions for the saturation and frequency profile are obtained. For a pair of coupled electrons such as bis-nitroxides commonly used for MAS DNP, an EL-ER model (el and er 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 el and er 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.