Dynamic nuclear polarization via the cross effect and thermal mixing: A. The role of triple spin flips

[Display omitted] •Generalized treatment of triple spin flips in dynamic nuclear polarization.•Scrambled states approach and fluctuating field approach are different limiting cases.•Two flows of energy to the nuclear spins identified.•Flow of electron Zeeman energy to nuclear spins (cross effect).•Flow of electron interaction energy to nuclear spins (thermal mixing). In dynamic nuclear polarization (DNP) via the cross effect (CE) and thermal mixing (TM) a microwave field first reduces the polarization of some electron spins, so the electron spin system deviates from thermal equilibrium with the lattice. Next, the mutual interactions combine with their interaction with the nuclear spins to transform this deviation into nuclear spin polarization. Two approaches were introduced to describe the latter process. The fluctuating field approach considers the electron spins to fluctuate between their up and down states due to their mutual interactions. This results in a classical fluctuating field at the position of the nuclear spins, and the component of this field at the NMR frequency induces flips of the nuclear spins. The scrambled states approach considers the electron and nuclear spin states to be mixed by the hyperfine and super-hyperfine interaction. Next the mutual interaction between the electron spins induces transitions between these mixed states and thus flips nuclear spins. Some authors considered the fluctuating field approach and the scrambled states approach to be just two equivalent methods to describe exactly the same process. Other authors considered the two approaches to describe two separate processes, the former exchanging electron interaction energy, the latter transferring differences of electron spin polarization to the nuclear spins. The present work introduces a generalized approach that first calculates the mixing of electron spin states exactly. Next it considers the hyperfine or super-hyperfine interaction to induce transitions involving these mixed states and the nuclear spin states. It is found that the scrambled states approach and the fluctuating field approach are neither fully equivalent, nor completely independent processes, but rather represent two distinct limits of a single process. The former corresponding to very weak mutual interactions between electron spins and the latter to very strong mutual interactions. It extends the treatment to the whole range of mutual interactions and shows that this single process simultaneous