Amount of electrons in nuclei

[snorkack]

TL;DR Summary

Order of magnitude estimates of electrons in nuclei

How much electrons do nuclei contain? Are there any numeric estimates around?
I am quite aware that while estimating the numbers might be somewhat simple exercise for quantum mechanics, it is still complicated numeric computation which is not very closely related to observables, so somewhat limited point in performing it. Nevertheless, it seems like a nice illustration to question "why don´t electrons fall into the nucleus".
Simple search yields no quantitative estimates. First answers are two types: one is that "electrons cannot enter nucleus because of uncertainty principle", the other is "electrons do enter nucleus and have a cusp there", but without any estimates about actual quantity of electrons in the cusp.
But someone might have done and presented estimates - just that they don´t show up in simple search. Has anyone actually encountered any numeric estimates? At least order of magnitude?
Like:
How much electrons does the proton contain in ground state of hydrogen?
How much electrons total does an U atom nucleus contain?
What kinds of electrons does the U nucleus contain, total? Like, 1s has cusp in the U nucleus, but U has a total of 14 s electrons, and all s orbitals have a cusp in nucleus. How much of the electrons in U nucleus are 2s...6s electrons, how much are 7s (and therefore valence) electrons?
How much 1s electrons does a Be-7 nucleus contain, how much of the electrons in Be-7 are instead 2s electrons or molecular orbital or Fermi sea electrons?
Does the numeric amount of electrons of various orbitals inside the nucleus have tight tie to the observable probability/branching ratio that electron will be captured from that orbital?

 

[Nugatory]

First answers are two types: one is that "electrons cannot enter nucleus because of uncertainty principle", the other is "electrons do enter nucleus and have a cusp there", but without any estimates about actual quantity of electrons in the cusp.

It is likely that both are the result of you misunderstanding something. You’ve been around long enough to know that you should cite a source for this sort of thing.

But for an order of magnitude estimate of the “amount of electrons in nuclei” two somewhat sensible answers are “zero” and “the question makes no sense; you’re seeing those non-zero amplitudes in the wave function of an electron that is not localized to the nucleus”.

 

[snorkack]

It is likely that both are the result of you misunderstanding something. You’ve been around long enough to know that you should cite a source for this sort of thing.

Well, there is the question of "what kind of source".
One example which I found is:
https://www.wtamu.edu/~cbaird/sq/2013/08/08/why-dont-electrons-in-the-atom-enter-the-nucleus/
But it concentrates on electron capture; while it states "Electrons are always partially in the nucleus", it does not give numbers.

But for an order of magnitude estimate of the “amount of electrons in nuclei” two somewhat sensible answers are “zero” and “the question makes no sense; you’re seeing those non-zero amplitudes in the wave function of an electron that is not localized to the nucleus”.

Zero is minus infinite order of magnitude. Which means that the difference between "zero" and "almost zero" is an infinite order of magnitude.
A real catch here is that the "surface of a nucleus" is also fuzzy because of Heisenberg uncertainty relation. But that one is less than order of magnitude effect for most nuclei.
So, posing the question as conceptual one poorly related to observables would be "How much of the total probability of finding some electron in the atom, which integrates to the integer total count of electrons in the atom, is distributed inside the charge distribution of the nucleus?".
A related question about observables vs. forecasts might be "How much error do we get in precision of observables by handling the nucleus as a point charge, and neglecting to treat the nucleus as an extended distribution of charge?".

 

[Ammaniya]

Quantum mechanics suggests aspects like the cusp concept for electron density, clear quantitative data, such as the number of electrons in various orbitals, remains elusive. It seems researchers have explored this, but definitive order of magnitude estimates are rarely accessible in simple searches. Further specialized inquiry may yield better insights into this topic.