Each electron shell can hold up to 2n² electrons. Is there a geometric reason for this?

It seems such a simple formula, that each electron shell can hold in principle two times a square number. I imagine there must be a geometrical reason for this, that's to do with the way electron orbitals "fit" around the nucleus. If this is so, what is it?




A good insight!


The [number of electrons in each orbital](https://en.wikipedia.org/wiki/Atomic_orbital) are double the odd numbers. The s orbital can hold 2, p holds 6, d holds 10 and f has 14: double 1, 3, 5 and 7. If you add the odd numbers, you get a square number:


1 = 1


1 + 3 = 4


1 + 3 + 5 = 9


1 + 3 + 5 + 7 = 16


The *doubling* is due to the [Pauli exclusion principle](https://en.wikipedia.org/wiki/Pauli_exclusion_principle); in each orbital, an electron of a specific energy can be in one of two spin states.


It is interesting to see spectral lines split in a magnetic field ([Zeeman effect](http://www.daviddarling.info/encyclopedia/Z/Zeeman_effect.html)) or an electrostatic field ([Stark effect](http://www.agkador.phy.uni-bayreuth.de/en/projects_web/Single-Molecule_Spectroscopy/index.html)): one line may become 3, 5 or 7 lines. It clearly shows how orbital theory was derived.


 8