### Re: Why aren't the full amount of electrons in certain shells complete?

Date: Sat Mar 31 22:48:28 2001
Posted By: Vladimir Escalante-Ramírez, Faculty, Institute of Astronomy, National University of Mexico
Area of science: Physics
ID: 985043745.Ph
Message:
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It is not exactly true that a shell must be complete before
the next shell starts to form in an atom. The reason is that the
energy of an electron depends both on the shell and on a quantity
called "angular momentum". Electrons with low angular momentum in
a shell tend to be closer to the nucleus and have lower energies
than electrons with high angular momentum in the same
shell. This happens because an electron that tends to be far
from the nucleus feels more strongly the repulsive force
of the electrons that are closer to the nucleus and this
"pumps up" its energy, so to speak.

Like energy, angular momentum in an atom can take
only discrete values. Angular momentum values can be
labeled with a number l, which takes integer values
starting with 0, and increases with increasing angular
momentum. Electrons with l=0 are called "s" electrons,
electrons with l=1 are called "p" electrons, with
l=2, 3, 4, 5, ... are called "d", "f", "g", "h" ...
electrons.

As we add electrons to an atom, they take the shell
and the angular momentum that gives them the
lowest energy. According to the Pauli
exclusion principle, in any given shell there can
be up to 2*(2*l+1) (that's two times l+l+1) electrons
in a shell with the same l value. Therefore we can have
only 2 s electrons, 6 p electrons, 10 d electrons,
and so forth until the shell is complete. Within the
SAME shell s electrons have lower energy than p electrons, and
p electrons have lower energy than d electrons and so on
as we explained above.
Thus s electrons start to fill the M shell and d electrons
complete the shell, but how do energies of those d electrons
compare to the energy of the s electrons of the NEXT shell?

It so happens that the d electrons of the M shell can have
slightly higher energies than the s electrons of the N shell.
Actually their energies depend on how many electrons the
atom has, therefore the available places in the M shell compete with
the N shell for the lowest energy to offer to the electrons.
An authoritative book on the subject, "The Theory of Atomic
Structure and Spectra" by Robert D. Cowan, p. 115, says that
this fact can only be demonstrated quantitatively with a complex
calculation, and only in some cases. Calculations of many-electron
structures are very difficult and its accuracy is very limited.
What happens in reality can be deduced from experiment though.
Table 4-3 of that book shows that the N shell starts to fill
with s electrons in potassium when the M shell only has 2 s electrons
and 6 p electrons. The missing d electrons of the M shell start to
appear in scandium. As we add electrons, it can happen that the M
shell can gain or loose an extra electron from the N shell. This
happens for example when we go from vanadium to chromium and
manganese or when we go from nickel to cooper and zinc. The same
happens when the O shell starts to form before the N shell is
complete in rubidium. This situation only happens
with neutral and singly ionized (i.e. those with an electron
missing) atoms. More ionized atoms (those missing two or more
electrons) don't show this behavior
because the energies of their electrons don't depend so strongly
on angular momentum.

```

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