Dynamic nuclear polarization via the cross effect and thermal mixing: B. Energy transport

The fundamental process of dynamic nuclear polarization (DNP) via the cross effect (CE) and thermal mixing (TM) is a triple spin flip, in which two interacting electron spins and a nuclear spin interacting with one of these electron spins flip together. In the previous article (Wenckebach, 2018) these triple spin flips were treated by first determining the eigenstates of the two interacting electron spins exactly and next investigating transitions involving these exact eigenstates and the nuclear spin states. It was found that two previously developed approaches-the scrambled states approach and the fluctuating field approach-are just two distinct limiting cases of this more general approach. It was also shown that triple spin flips constitute a single process causing two flows of energy: a flow originating in the electron Zeeman energy and a flow originating in the mutual interactions between the electron spins. In order to render their definitions more precise, the former flow was denoted as the CE and the latter as TM. In this article the treatment is extended to a glass containing N_I equivalent nuclear spins I=12 and N_S randomly distributed and oriented electron spins S=12. Rate equations are derived for the two flows of energy to the nuclear spins. The flow originating in the electron Zeeman energy-i.e. the CE-is found to lead to the same stationary state as was previously predicted by the scrambled states approach, though the rate may be smaller due to limitations imposed by conservation of energy. The flow originating in the mutual interactions between the electron spins-i.e. TM-is found to involve the full spectrum of the mutual interactions between the electron spins, while the fluctuating field approach only accounts for the component of this spectrum at the nuclear magnetic resonance (NMR) frequency. Still, TM is found to induce equal spin temperature for different nuclear spin species during nuclear spin-lattice relaxation and, at least in some cases also during polarization. It is also confirmed that TM couples local nuclear spins near the electron spins so strongly to the mutual interactions between electron spins, that they may constitute a single energy reservoir (Cox et al., 1973). Hence such local nuclear spins may have to be included in treatments of the dynamics of the electron spins.