text: The simplest spin system exhibiting the SE DNP mechanism is an electron-nucleus spin pair. The Hamiltonian of the system can be written as:
  These terms are referring respectively to the electron and nucleus Zeeman interaction with the external magnetic field, and the hyperfine interaction. S and I are the electron and nuclear spin operators in the Zeeman basis (spin  considered for simplicity), ω and ω are the electron and nuclear Larmor frequencies, and A and B are the secular and pseudo-secular parts of the hyperfine interaction. For simplicity we will only consider the case of |A|,|B||. In such a case A has little effect on the evolution of the spin system. During DNP a MW irradiation is applied at a frequency ω and intensity ω, resulting in a rotating frame Hamiltonian given by
  :where 
  The MW irradiation can excite the electron single quantum transitions ("allowed transitions") when ω is close to ω, resulting in a loss of the electron polarization. In addition, due to the small state mixing caused by the B term of the hyperfine interaction, it is possible to irradiate on the electron-nucleus zero quantum or double quantum ("forbidden") transitions around ω = ω ± ω, resulting in polarization transfer between the electrons and the nuclei. The effective MW irradiation on these transitions is approximately given by Bω/2ω.