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 4 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 $The innermost electron of an atom with Z = 79, with and without relativity. The probability of finding the innermost electron at each distance from a nucleus of charge 79 (Au), with the distance in units of a₀/Z so that the non-relativistic curve is the same for every element. The dashed curve is Schrödinger's, r² e^(−2Zr/a₀). The solid curve is Dirac's exact ground state, r^(2γ) e^(−2Zr/a₀) with γ = √(1 − (Zα)²) = 0.817. It is pulled in towards the nucleus: its mean radius is 0.878 of Schrödinger's, a contraction of 12 per cent, and it has a sharper peak close in. Nothing about the nucleus has changed; the electron, moving at an appreciable fraction of the speed of light near it, simply behaves as a heavier particle, and a heavier particle is bound more tightly and more closely.$

The metal that is yellow because it is heavy

Gold sits directly below silver in the periodic table, with the same arrangement of outer electrons, and by every non-relativistic calculation it should look like silver. It is yellow because its innermost electrons move at more than half the speed of light. The mass that motion adds pulls the s orbitals in, the tightened s level drops towards the d band beneath it, and the gap an absorbed photon must bridge shrinks from the ultraviolet into the blue. The same contraction is why mercury is a liquid.

quantum · Atomic structure

The square root of an X-ray's energy, one step per element. The square root of the energy of each element's Kα X-ray line, in units of the Rydberg energy, against its atomic number, for 24 elements from aluminium to lead (dots), with Moseley's law, √(3/4)·(Z − 1), drawn as a line. The points march up in equal steps, one per element, because each step adds one proton to the nucleus and one unit to the charge the falling electron feels. When Moseley first did this, in 1913–14, the line had gaps where no known element fell: the numbers marked, 43, 61, 72 and 75, were found later as technetium, promethium, hafnium and rhenium. At high atomic number the points rise above the line, by 6 per cent for lead, because the innermost electrons there move fast enough for relativity to deepen their binding.

The X-ray line that counts the protons

Strike a metal with fast electrons and it gives off X-rays at a few sharp energies characteristic of the metal. In 1913 Henry Moseley measured the strongest of them for element after element and found that the square root of its energy rose in equal steps, one step per element — as if something inside each atom were being counted. It was the charge on the nucleus. The periodic table had been ordered by weight; Moseley's line ordered it by protons, fixed the elements it was missing, and showed that an electron deep inside a heavy atom behaves like the electron in hydrogen, with the charge turned up.

quantum · Atomic spectra

Named alongside it

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

Atomic structure Screening Energy levels Absorption Accidental degeneracy Atomic number Bohr model Bohr radius Centrifugal barrier Degeneracy pressure Effective nuclear charge Exclusion principle

All concepts