Illustrated Physics
physics, drawn
Series

Atomic structure — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Where the electron actually is, by radius. The radial probability density of the hydrogen 1s, 2s, 2p states — the chance of finding the electron in a thin shell at each radius, in units of the Bohr radius. 1s is most likely at 1.00 Bohr radii and averages 1.50; 2s is most likely at 5.24 Bohr radii and averages 6.00; 2p is most likely at 4.00 Bohr radii and averages 5.00. Each curve integrates to one, and each has n − l − 1 radial nodes where the electron is never found.

    Where the electron probably is

    The Bohr atom put the electron on a circle of definite radius. What replaced it keeps the radius as the most likely place to find the electron and gives up the circle, the speed and the trajectory entirely.

    part 1 · quantum
  2. 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.

    part 2 · quantum
  3. Why it stops falling in. The energy of an electron confined to a region of radius r around a nucleus, as the sum of two terms with different powers: a confinement energy ħ²/2mr² that rises without limit as the region shrinks, and a Coulomb attraction −Ze²/4πε₀r that falls. At Z = 1 the sum is least at 52.92 pm, where it is -13.61 eV. Both numbers are found by searching the drawn curve and both agree with the Bohr radius over Z and minus Z² Rydbergs to a part in a million. Nothing was quantised to get them. The only quantum input is that confining an electron to a region costs kinetic energy, which is the uncertainty relation and nothing more.

    Why an atom is the size it is

    A tenth of a nanometre is not a measured constant of nature but the outcome of a competition: confining an electron costs kinetic energy, and the nucleus pays for confinement with attraction. Minimising the sum gives the number, and changing the masses moves it by four orders of magnitude.

    part 3 · quantum
  4. 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.

    part 4 · quantum
  5. What correlation is worth, to a helium atom and to H⁻. The energy needed to remove the second electron, by three descriptions, as a fraction of the measured value, for the negative hydrogen ion and for helium. Helium: one shared orbital 23.07 eV, one electron in and one out 23.83 eV, measured 24.59 eV — the simple picture is 94 per cent right. H⁻: −0.74, 0.362 and 0.755 eV — the simple picture is not merely inaccurate but has the wrong sign. The energy the simple picture misses — the electrons avoiding each other — is about 1.5 eV in both; in helium it is a small correction to 24.6 eV, and in H⁻ it is everything.

    The ion that exists because its electrons keep apart

    Add a second electron to a hydrogen atom and it holds on, by three-quarters of an electronvolt. The simplest respectable calculation says it should not — that the two electrons, sharing one orbital, repel each other more than the proton can hold them, and the ion falls apart. The same calculation gets helium's energy right to two per cent. The difference is not a failure of the method but a measure of what it leaves out: the two electrons avoiding each other. In helium that is a small correction. In the negative hydrogen ion it is the whole of the binding, and that ion is what makes the Sun opaque.

    part 5 · quantum
  6. Two lifetimes a thousand times apart. The fraction of positronium atoms surviving in the ground state against time since formation, on a logarithmic time axis, for the spin singlet, para-positronium (red), and the spin triplet, ortho-positronium (blue), in vacuum. Para lives 125.1 ps and ortho 142.1 ns, a ratio of 1135. The lowest-order formulas give 124.5 ps, from α⁵mc²/2ħ, and 138.7 ns, from 2(π² − 9)/9π · α⁶mc²/ħ; the ratio is almost entirely one extra power of the fine-structure constant, the price of a third photon.

    The atom with no nucleus

    An electron and a positron can bind into an atom whose structure is hydrogen's with every level halved and every size doubled, because neither partner sits still at the centre. The atom then destroys itself, and how fast depends on one symmetry. With the two spins opposed it vanishes in 125 picoseconds as two photons; with them aligned it must make three, and lives 142 nanoseconds — over a thousand times longer — for no reason but that a rule about charge conjugation forbids the shorter route.

    part 6 · quantum
  7. A current that falls every 4.9 volts. The share of electrons reaching the anode of a mercury-vapour tube against the accelerating voltage, from a simulation of 3000 electrons at each voltage: the electrons must arrive at the grid with more than 1.5 eV to climb the last small retarding voltage, and lose 4.886 eV — the energy of mercury's first excited state — whenever they hit an atom with at least that much, after a random further distance averaging 0.03 of the gap. The current rises with voltage, then collapses each time the electrons have gained just enough to excite an atom and too little left to reach the anode: minima at 6.3, 11.2, 16.1, 21.0, 25.8 V, about 4.9 V apart. Energy was being handed to the atoms in lumps of 4.886 eV and in no other amount — the first direct evidence, in 1914, that an atom's internal energy is quantised.

    The electrons that pay in fixed amounts

    Fire electrons through mercury vapour and raise the voltage that drives them, and the current they carry should simply rise. In 1914 James Franck and Gustav Hertz found it rising, then collapsing, then rising again, over and over, every 4.9 volts. Below that energy an electron bounces off a mercury atom and keeps almost everything; above it, it can hand the atom exactly 4.9 electronvolts and no other amount. The atom will accept energy only in the size of the gap between its levels — shown here by collisions rather than by light, and misread by its discoverers as something else entirely.

    part 7 · quantum

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