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Microwave driven dynamic nuclear polarization (DNP) is a process in which the large polarization present in an electron spin reservoir is transferred to nuclei, thereby enhancing NMR signal intensities. In solid dielectrics there are three mechanisms that mediate this transfer--the solid effect (SE), the cross effect (CE), and thermal mixing (TM). Historically these mechanisms have been discussed theoretically using thermodynamic parameters and average spin interactions. However, the SE and the CE can also be modeled quantum mechanically with a system consisting of a small number of spins and the results provide a foundation for the calculations involving TM. In the case of the SE, a single electron-nuclear spin pair is sufficient to explain the polarization mechanism, while the CE requires participation of two electrons and a nuclear spin, and can be used to understand the improved DNP enhancements observed using biradical polarizing agents. Calculations establish the relations among the electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR) frequencies and the microwave irradiation frequency that must be satisfied for polarization transfer via the SE or the CE. In particular, if δ, Δ ω(0I) > δ, the CE dominates the polarization transfer. This two-electron process is optimized when ω(0S(1))-ω(0S(2)) = ω(0I) and ω(M)~ω(0S(1)) or ω(0S(2)), where ω(0S(1)) and ω(0S(2)) are the EPR Larmor frequencies of the two electrons. Using these matching conditions, we calculate the evolution of the density operator from electron Zeeman order to nuclear Zeeman order for both the SE and the CE. The results provide insights into the influence of the microwave irradiation field, the external magnetic field, and the electron-electron and electron-nuclear interactions on DNP enhancements.
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