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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?

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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).

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No this will not work. The electron does not have an electrical dipole or higher moment. Note that an oscillating electric field implies an oscillating magnetic field not far away.

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