Editing Dynamic nuclear polarization (section)

===Solid effect===
The simplest spin system exhibiting the SE DNP mechanism is an electron-nucleus spin pair. The Hamiltonian of the system can be written as:

:<math>H_0=\omega_eS_z+\omega_{\rm n}I_z+AS_zI_z+B\ S_zI_x</math>

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 {{frac|1|2}} considered for simplicity), ''ω<sub>e</sub>'' and ''ω''<sub>n</sub> are the electron and nuclear [[Larmor precession|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''|<<|''ω''<sub>n</sub>|. 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''<sub>MW</sub> and intensity ''ω''<sub>1</sub>, resulting in a rotating frame Hamiltonian given by
:<math>H=\Delta\omega_e\;S_z+\omega_{\rm n}I_z+AS_zI_z+B\ S_zI_x+\omega_1 S_x</math>where <math>\Delta\omega_e=\omega_e-\omega_{\rm MW}</math>

The MW irradiation can excite the electron single quantum transitions ("allowed transitions") when ''ω''<sub>MW</sub> is close to ''ω''<sub>e</sub>, 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''<sub>MW</sub> = ''ω''<sub>e</sub> ± ''ω''<sub>n</sub>, resulting in polarization transfer between the electrons and the nuclei. The effective MW irradiation on these transitions is approximately given by ''Bω''<sub>1</sub>/2''ω''<sub>n</sub>.

====Static sample case====
In a simple picture of an electron-nucleus two-spin system, the solid effect occurs when a transition involving an electron-nucleus mutual flip (called zero quantum or double quantum) is excited by a microwave irradiation, in the presence of relaxation. This kind of transition is in general weakly allowed, meaning that the transition moment for the above microwave excitation results from a second-order effect of the electron-nuclear interactions and thus requires stronger microwave power to be significant, and its intensity is decreased by an increase of the external magnetic field B<sub>0</sub>. As a result, the DNP enhancement from the solid effect scales as B<sub>0</sub><sup>−2</sup> when all the relaxation parameters are kept constant. Once this transition is excited and the relaxation is acting, the magnetization is spread over the "bulk" nuclei (the major part of the detected nuclei in an NMR experiment) via the nuclear dipole network.
This polarizing mechanism is optimal when the exciting microwave frequency shifts up or down by the nuclear Larmor frequency from the electron Larmor frequency in the discussed two-spin system. The direction of frequency shifts corresponds to the sign of DNP enhancements.
Solid effect exist in most cases but is more easily observed if the linewidth of the EPR spectrum of involved [[unpaired electron]]s is smaller than the nuclear Larmor frequency of the corresponding nuclei.

====Magic angle spinning case====
In the case of magic angle spinning DNP (MAS-DNP), the mechanism is different but to understand it, a two spins system can still be used. The polarization process of the nucleus still occurs when the microwave irradiation excites the double quantum or zero quantum transition, but due to the fact that the sample is spinning, this condition is only met for a short time at each rotor cycle (which makes it periodical). The DNP process in that case happens step by step and not continuously as in the static case.<ref name="Mentink-Vigier, F. Akbey, U. Hovav, Y. Vega, S. Oschkinat, H. Feintuch, A. 2012 13–21">{{cite journal
|author = Mentink-Vigier, F.
|author2 = Akbey, U.
|author3 = Hovav, Y.
|author4 = Vega, S.
|author5 = Oschkinat, H.
|author6 = Feintuch, A.
|title = Fast passage dynamic nuclear polarization on rotating solids
|journal = [[J. Mag. Reson.]]
|volume = 224 |pages = 13–21
|year = 2012
|doi = 10.1016/j.jmr.2012.08.013
|pmid = 23000976
|bibcode = 2012JMagR.224...13M
}}</ref>