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Can the value of Newton's Gravitational Constant $G$ be measured within a stably bound atom?

PLEASE NOTE: Since scattering experiments do not involve stably bound systems, their results are not germane to the specific question asked here.

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If you're asking whether we can measure the effect on atomic structure of gravitational forces between the nucleus and the electrons, then the answer is that not only have we never measured such effects but it's unlikely we'll ever be able to measure them as they would be many orders of magnitude below the electrostatic forces that hold the atom together.

Outside the atom (you didn't ask this but I thought I'd mention it anyway :-) it is extraordinarily difficult to measure gravitational effects on subatomic particles. For a long time it wasn't even known for sure whether anti-particles experienced the same gravity as particles. This is mainly because most subatomic particles have very short lifetimes and don't fall any measurable distance.

However a few years back an experiment was done to measure the effect of gravity on neutrons. This not only measured the gravitational force on neutrons but managed to see quantised states in the gravitational potential.

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But if we cannot measure the strength of G inside an atom, then how can we know that it is "many orders of magnitude below the electrostatic forces..."??? Is this value of G inside the atom a pure assumption? – Robert L. Oldershaw Feb 1 at 16:11
We can't measure gravitational effects in an atom because they're far too small. However we can measure effects like hyperfine splitting to great precision, and we would see if there was an contribution from gravity - needless to say no such effects are seen, which is exactly what we would expect if $G$ inside the atom is the same as $G$ at the large scale. – John Rennie Feb 1 at 17:16
There is theoretical justification for $G$ becoming large at very short distances/energies, but we're talking about the Planck scale and there is no currently foreseeable way we'll ever be able to probe that scale directly. – John Rennie Feb 1 at 17:18

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