Neutrons can't interact with electrons through coulombic force. But both have associated with spin magnetic moment. So is there any kind of interaction possible via these magnetic moment or some other kind of weak interactions?
There is a so called hyper fine interaction of magnetic moment of nuclei and electrons. This causes the HFS spectral lines. So in principle free neutrons and electrons can interact via their magnetic moment. The magnetic moment of neutrons has the value of mü(n) = -1,913042 x mü (kern), last is the so called nuclear magnetron.
NEUTRON DIFFRACTION is used to study the magnetic structure of crystals. It is based on the interaction between the neutron magnetic moment and the magnetic field produced by aligned electrons in the materials.
It must be taken into account that neutrons are not elementary but are rather composite particles whose inner scatterable components are 3 massive and charged particles, that is 2 down quarks, each with a negative charge 1/3 of that of an electron, and 1 up quark with positive charge 2/3 of that of a positron, which when added give the null charge of the neutron. This was experimentally clarified at the SLAC facility during experiments carried out from 1966 to 1968 when the inner scatterable structure of neutrons and protons was discovered.
This means that contrary to expectation, these inner components, are subject to Coulomb interaction with electrons, just like the inner components of protons.
Neutrons are composite particles made up of quarks which are charged, so the electromagnetic interaction is not out of the picture. Also, electrons and quarks can interact via the weak interaction. Both neutrons and electrons have mass so gravity will have an influence between the particles at the very least. Gravity is normally very weak compared to the electrostatic force, but if you had a very dense (massive) neutron star it seems reasonable that the gravitiational attraction between such a star and an electron can be quite large.
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