Is it possible to perform NMR/EPR spin alignment with an oscillating electric field instead of a magnetic field (so with a sample inside the RF electric field of capacitive plates rather than a RF magnetic field of a coil)? In other words, can an electric field align nuclei or does it just polarize the atoms without having that effect?
Nuclear Magnetic Resonance – NMR/EPR with an RF Electric Field
dipole-momentelectromagnetismnuclear-magnetic-resonancequantum-spin
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There is a difference between your two cases. When you are talking about a charge passing between magnets you are thinking of it as a uniform magnetic field. But it is not uniform, it gets stronger as you approach the magnets. if it were a uniform field the magnetic dipole you put in the middle would not feel a force along the line of the magnets, although it would still experience a torque.
To see where the force along the magnets comes from, imagine a single charge moving in a circle with velocity always perpendicular to the line of the magnets. If it were a uniform magnetic field the Lorentz force would act perpendicular to the line of the magnets producing an outward radial force. This is what you realized.
But really the field is getting weaker as you go away from the magnets, and since the magnetic field has no divergence that means the field lines must expand outward as you go away from the magnets. You see this on any diagram of the magnetic field of a dipole. So if you look at what the Lorentz force on the charge moving in a circle is now the the magnetic field has a (possibly small) component outwards you will see the charge picks up a force along the line of the magnets (in addition to the original force outward).
The electron magnetic moment is about 660 times larger than that of the proton, and the proton's magnetic moment is the largest of all the nuclei. Although most electrons occur in pairs, unpaired electrons, as they occur in radicals, give rise to electron paramagnetic resonance (EPR) signals.
Signal frequencies in magnetic resonance are, to a very good approximation, proportional to the magnetic moment (unless the external magnetic field becomes very weak or in the case of large quadrupolar splittings).
In a typical nuclear magnetic resonance (NMR) experiment one would thus observe either the proton or the $^{13}$C carbon NMR signal (much like listening to different FM radios). For a 10 Tesla magnet, these would have frequencies of approximately 400 MHz and 100 MHz, respectively. It is possible to excite proton or carbon NMR simultaneously, but this requires two channels, tuned to the respective (radio) frequencies.
On the other hand, an electron spin would precess at a (microwave) frequency approaching 300 GHz, requiring different excitation and detection pathways (waveguides rather than coaxial cables, and cavity resonators rather than $LC$-resonators).
However, the presence of free electron spins may manifest itself in the NMR detection via reducing the relaxation time $T_1$, a phenomenon known as paramagnetic relaxation enhancement (PRE).
Best Answer
Spin vs. electric dipole moment
NMR end EPR couple to spin, i.e., to the magnetic moment of nuclear and electrons respectively. Direct transposition of this mechanism to electric field would require manipulating electric dipolar moment of some particles. While such situations can be artificially designed, they are rather rare in nature (as far as I can judge).
Stark spectroscopy
Interaction between light and atoms is often described in terms of electric dipolar moment of electronic transitions (see, e.g., the model Hamiltonian discussed in this answer). In this sense atomic (or solid state) absorption and emission can be thought of as manipulating atoms using electric field. The reason why one usually does not use the same time-resolved techniques as with NMA/EPR is because the required frequencies are much higher than radio or microwave. However, the approach is valid when applied to the levels split due to Stark effect, in the context of Stark spectroscopy. In this case the electric moment is however artificially induced.
EDSR
Another relevant technique is the electric dipole spin resonance (EDSR) - here one manupiles the electron spin, but by coupling it to electric field via a spin-orbit coupling, which can be artificially engineered (Rashba coupling).