in dynamic nuclear polarization by microwave induced cross-polarization ( polarization transfer from one spin to another, for example, from to ), the condition is called Hartmann-Hahn matching.

The Hamiltonian of the 2 kind of spin in lab frame is:

where A and B are scalar coupling when the 2 spin in contact and the dipolar interaction.

the angle in dipolar interaction is relative to the direction of the external B-field. when the angle is at:

the dipolar interaction disappear and the spectrum line get thinner, and higher resolution. this is called the Magic Angle.

when a transverse oscillating field applied, and only affect the S spin, then the total Hamiltonian is :

where U is the rotation operator. a standard method is switch to the rotating frame along with the transverse field. the Hamiltonian in the rotating frame is:

we can see that, during the Spin-Lock, i.e. , the longitudinal component of S spin gone. But in general, it is not the case, thus, we have the rotation axis of S spin is titled. we can simplify the Hamiltonian by transform it in a titled coordinate, by another unitary transform which rotate on Sy axis.

in this tilted axis, the rotating axis is on the axis with magnitude:

the tilted Hamiltonian is :

For the interaction terms are small, the energy level is just like ordinary 2 spin system. but when

which is the Hartmann-Hahn matching, the flip-flop exchange of the spin no need any energy and then the spin transfer. on the other hand, if:

the flip-flip forbidden transition happen.

In the case of electron spin to proton spin, if we apply a ESR freqeuncy, which is GHz order, so it is microwave, the power of the microwave have to be matched to the proton Larmor frequency.

here i used the microwave wave B-field strength is proportional to the voltage applied, and power is proportional to the square of voltage.