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Question / light thought experiment on atomic structure in the presence of a large gravitation force.

Background:

This is a general, conceptual question, not looking to solve any particular problem, but still one that should have an answer, albeit a general one. I will use the phrase "distance from the nucleus" when describing electrons in an atom in this question. Please understand that I realize that electrons exist in a field of probability and when I say "distance from the nucleus" I am referring to the distance from the nucleus that describes a field in which there is a 90% chance of finding an electron of given energy level within it. I understand that electrons do not orbit the atom at a fixed distance.

This question is about how matter is compressed under gravity. Initially, I was thinking about the inside of black holes. I know matter is likely completely destroyed and turned into energy inside a black hole, however, so let's say at some level of gravity less than that of the inside of a black hole, but still very strong. Maybe a level experienced on approaching a black hole at some distance. The fundamental question is in what way is the matter compressed? Is it simply atoms are squeezed closer to each other until some point where they can no longer take it and eventually collapse (the scenario in the linked question below) or are the atoms themselves fundamentally, internally changed more gradually as they approach?

I did find a similar question here but my question is a more general, less extreme case and I want to know what understanding we currently have.

Setup

To illustrate in more detail, here's the setup I imagine (with lots of simplifications to try to get to the fundamental point):

Let's say I have a "magical" device that can somehow create a very strong, localized, gravitational field without a source of mass. Further I can use the device to restrict that gravitation field such that it only affects the things I want it to (i.e. it does not have an infinite range, I can restrict its range to a specific area regardless of how strong the field is). Again, this is a "magical" device as I have to violate some rules of physics to simplify the problem statement.

Let's say I also have "magical" instruments which can measure things inside and outside of this gravitational field from the same frame of reference (i.e. the instruments themselves are not affected by the gravitational field machine even if measuring things inside of it).

Let's say I have two atoms of Xenon. I place one of these atoms of Xenon inside of my magical gravity device and turn the gravity up very high. It is extremely strong gravity, but not strong enough gravity to destroy the atom. Let's also assume it's otherwise a vacuum inside of this device and the Xenon atom centers itself inside of this gravitational field - so, if there's any "spaghettification" effect, it would be the same at a given distance from the nucleus.

EDIT: Please assume that this gravitational field is the equivalent of a very large point mass centered in the center of the atom, though "magically" without that mass existing. So, it would be strongest at that point and weakening as it goes out. Assume the weakening is uniform in all directions.

Question

If I were to compare measurements of electron "orbitals" between these two Xenon atoms, what would I see? Would the orbitals look the same, or would they be "squeezed" in the atom that is inside the gravitational field?

For example, if I took measurements of shell K (1), would I find the typical 140 pm distance from the nucleus, or would that value now be decreased?

Big picture question: Assuming the field is not strong enough to tear the atom apart, do the effects of a strong gravitational field affect basic atomic structure the way we know it?

Trashman
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2 Answers2

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I think the answer is "nothing". An atom in a strong gravitational field is going to do one thing: fall. As in free-fall, aka "zero gee". Hence: nothing.

Spaghettification occurs in a strong field with a strong gradient (aka, tidal forces). The field can be Taylor expanded:

$$ \vec g(z') = -(g_0 - \frac 1 2 g' z')\hat z $$

where $g' > 0$ is the rate of increase of the field in the direction of the field. Now we're in free fall, so we subtract $-g_0\hat z$ and work un primed coordinates:

$$ \vec g(z) = \vec g(z') + g_0\hat z = \frac 1 2 g' z $$

which can be integrated to get a perturbing potential:

$$ \Phi_g(z) = g'z^2 $$

for a perturbation:

$$ H_g(z) = \mu g'z^2 $$

where $\mu$ is the reduced mass. Ofc, I am going to use a hydrogen atom because the is zero reason use xenon with 54 separate coordinates.

The Hamiltonian is:

$$ H(\vec r) = H_{\rm Coulomb}(r) + H_g(z) $$

$$ H(\vec r) = \frac{\hat p^2}{2\mu} - \frac{e^2} r + \mu g'z^2 $$

Note that:

$$ \frac{z^2}{r^2} = \sqrt{\frac 2 3} Y_2^0(\theta, \phi) + \sqrt{\frac 1 3} Y_0^0(\theta, \phi) $$

so the perturbation is a constant plus a quadrapole operator, so you can use standard perturbation theory to calculate the energy shifts or the 1st order wave functions. If the energy shift gets close to 13.8 eV, then you need to worry about ripping the atom apart.

JEB
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The time-independent Schrodinger equation which describes the energy levels of a atom is solved by asuming that coordinate and proper length are the same.But under strong gravitational effects that is not the case so you can correlate the coordinate length to the proper length according to the metric and solve that equation from there.This may not be entirely correct but it is a useful approximation.