Concept

Periodic table — where it appears

The arrangement of the elements by atomic number in rows whose lengths are set by which subshells are filling. The block widths come from the exclusion principle's 2(2ℓ+1) states and the sequence from penetration — 4s falling below 3d is why potassium is an alkali metal rather than the first transition metal.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

No two in the same state, and why matter has volume

Nothing in the energy levels of an atom says how many electrons may occupy each one. The answer is one per state, it is not derived from any force, and it is the reason a table holds a cup up.

quantum · Exclusion

Where 4s goes below 3d. The energies of five orbitals of a nucleus of charge 19, solved in a screened Coulomb potential, against how far out the screening extends. Nothing about the ordering is assumed: each level is found by integrating the radial equation outward and bisecting on the energy until the solution has the number of nodes that state is supposed to have, and the same solver returns hydrogen's −1/2n² to 7.3e-12 hartree when the screening is switched off. With no screening the three n = 3 levels would lie on top of one another. With screening they separate, always in the same order — s lowest, then p, then d — because a low angular momentum has no centrifugal barrier keeping it out of the core, so it spends part of its time inside the other electrons where the nuclear charge is unscreened. And at a screening length of 0.41 bohr the 4s level crosses below 3d, which is the fourth row of the periodic table: potassium and calcium put their electrons in 4s before anything goes into 3d, so they are an alkali metal and an alkaline earth rather than the first two transition metals. The model is a caricature — one screening length for every electron, no self-consistency, and no exchange — and it gets the ordering right anyway, which is the argument that the ordering is about penetration and nothing subtler.

The order the shells fill

In hydrogen every state with the same principal number has the same energy, and 4s and 3d differ by nothing. In every other atom they do not, and 4s is below 3d — which is why potassium is an alkali metal rather than the first transition metal. The difference is a small piece of probability that an s orbital has inside the innermost shell and a d orbital does not.

quantum · Atomic structure

Named alongside it

The objects these essays reach for when they reach for this one.

Atomic structure Accidental degeneracy Centrifugal barrier Degeneracy pressure Effective nuclear charge Energy levels Exclusion principle Penetration Quantisation Quantum numbers Screening Spin

All concepts