Why an atom is the size it is

Classical physics has a definite prediction about the size of an atom, and the prediction is zero. An electron circling a proton is accelerating, a charge that turns must glow, and radiating away the energy shrinks the orbit — from a tenth of a nanometre to nothing in about sixteen picoseconds. Nothing in the classical account supplies a length at all: Coulomb’s law has no scale in it, and neither does Newton’s second law.

So the size of an atom is a quantum quantity, and the argument that produces it is short enough to give in full.

Two powers, one minimum #

The whole argument is that the two terms have different powers.

The attraction goes as $1/r$ and the confinement cost goes as $1/r^2$. Far out, the attraction dominates and the electron is pulled in. Close in, the confinement cost dominates and pushes it back out. Between them there is exactly one radius where the sum is least, and that radius is the atom.

The confinement cost is the only quantum input, and it is sharpness has to be paid for applied to a bound state: an electron localised within $r$ has a momentum spread of at least $\hbar/r$, and a momentum spread is kinetic energy whether or not the electron is going anywhere. It is the same energy that appears as a zero-point motion in the motion that cannot be stopped, doing a different job.

Setting the derivative of the sum to zero gives $r = 4\pi\varepsilon_0\hbar^2/me^2$, which is 52.9 picometres, and the energy there is $-13.6$ eV. Neither number was put in. No orbit was quantised, no angular momentum was assigned, no boundary condition was imposed. The figure locates the minimum numerically and checks it against the accepted values rather than evaluating the formula it is meant to be demonstrating.

That is worth insisting on because the Bohr model gets the same two numbers by an argument that is wrong in its details — a circular orbit with quantised angular momentum, which the ground state does not have, since its angular momentum is zero. The balance argument gets them from a statement that survives into the full theory.

What the balance says when things change #

Because the answer is a minimisation of an expression with two terms, every scaling in the problem can be read off by inspection.

Multiply the charge by $Z$ and the attraction term multiplies by $Z$; the minimum moves in by $Z$ and the energy deepens by $Z^2$. That is the hydrogenic series, and it is why $\mathrm{He}^+$ has the same spectrum as hydrogen with every wavelength divided by four — the fact the spectrum is a subtraction uses to identify one-electron ions in stars.

Multiply the mass by $\mu$ and the confinement term divides by it; the minimum moves in by $\mu$ and the energy deepens by $\mu$. That scaling is easier to test than it sounds, because there are bound states available with masses quite unlike an electron’s.

The reduced mass is the mass that matters #

Positronium is the cleanest case. An electron bound to a positron has a reduced mass of exactly half an electron’s, so it is exactly twice the size of hydrogen and exactly half as tightly bound — and both are measured, to many figures, and both agree.

Muonic hydrogen is the case that has done the most work. A muon is 207 times an electron’s mass, so its atom is 186 times smaller once the proton’s motion is accounted for, and the muon spends a correspondingly larger fraction of its time inside the proton itself. That makes its energy levels far more sensitive to the proton’s charge radius than an electron’s are — the shift is larger by the cube of the mass ratio — which is why measuring a transition in muonic hydrogen became the most precise determination of the proton’s size, and why its disagreement with the electron-based value was a serious problem for several years.

None of this required a new theory. It is one expression with two adjustable inputs, and the four orders of magnitude in that figure come from changing them.

The two constants that are not there #

Something is conspicuously absent from the expression, and its absence is as informative as the terms that are present.

The speed of light does not appear. The Bohr radius is built from $\hbar$, the electron mass and the charge, and nothing about the atom’s size knows that electromagnetism has a finite propagation speed. That is why the non-relativistic treatment works at all for light atoms, and it is also why it stops working for heavy ones: the inner electrons of a heavy atom move at an appreciable fraction of $c$, their effective mass rises, and the balance moves in. Gold’s colour comes from that shift.

Newton’s constant does not appear either, and it is worth seeing how far away it is. Repeating the whole calculation with the gravitational attraction between an electron and a proton instead of the electrical one gives a bound state about $10^{39}$ times larger than an atom — some $10^{29}$ metres, several billion times the size of the observable universe, bound by about $10^{-78}$ electronvolts. Gravity is not weak in the atom; it is absent from it, and that ratio is the cleanest statement of how absent.

What does appear, in combination, is the fine-structure constant. The Bohr radius can be written as the Compton wavelength of the electron divided by $2\pi\alpha$, and the binding energy as $\alpha^2/2$ times the electron’s rest energy. So the atom is large compared with the electron’s own quantum scale, and weakly bound compared with its rest energy, both by powers of the same dimensionless number — and both facts are why atomic physics is a non-relativistic subject with relativistic corrections rather than the other way round.

Why atoms are not all different sizes #

Applying the hydrogenic scaling to real atoms produces an immediate and instructive failure.

If the size went as $1/Z$, lead would be eighty times smaller than hydrogen. It is seven times larger. The prediction does not merely miss by a factor; it gets the sign of the trend wrong, and the figure reports a correlation of $-0.50$ between it and the data.

What went wrong is that a bigger nucleus never arrives alone. Each additional proton comes with an additional electron, and the electrons already there stand between the outer one and the charge. What the outermost electron responds to is not $Z$ but the part of $Z$ that is left over.

The sawtooth, and what makes it #

The screened estimate reproduces the shape of the periodic table, and the mechanism is worth stating because it is entirely contained in the two numbers the estimate uses.

Along a row, the nuclear charge grows by one at each step while the added electrons all go into the same shell — and electrons in the same shell screen each other poorly, since they are at the same distance and spend as much time outside one another as inside. So the effective charge grows nearly as fast as the real one, and the atom contracts steadily from left to right.

At the end of a row the next electron has to start a new shell. It sits outside everything that came before, so it is screened almost completely: the effective charge drops back to near one, and the radius jumps. That jump is the step from a halogen to the next alkali, and it is the largest single change in the whole table.

The principal quantum number is the other half. A new shell has a larger $n$, and the estimate carries $n^2$ in the numerator, so a new row is further out for two reasons at once. Both effects come from the same two-term balance, with the charge and the shell number as its only inputs, and neither the periodicity nor its amplitude was supplied.

The correspondence with the order the shells fill is exact and worth noticing: that essay explains which orbital an electron enters, and this one shows that the same bookkeeping — how many electrons are inside, and in which shell — predicts how large the atom ends up.

Why matter takes up room #

There is a further question the balance answers almost as a by-product, and it is the one that matters for everything larger than an atom.

Solid matter resists compression, and the resistance is not electrostatic — a solid is electrically neutral, and Coulomb’s law alone would let neutral matter collapse. What stops it is that compressing matter means confining its electrons, and confinement costs energy at a rate that rises as the inverse square of the spacing. That is the same term as in the first figure, applied to a lattice rather than to a single atom.

So the reason a table is solid and the reason an atom is 0.1 nanometres across are the same reason, and it is the same reason a white dwarf resists gravity until the electrons turn relativistic. The exclusion principle enters when the electrons are many — no two in the same state is what forces them into successively higher momenta rather than all sitting in the lowest — but the mechanism by which momentum resists compression is already visible in one atom.

Matter’s incompressibility is the uncertainty relation at macroscopic scale, and the numbers work out: the bulk modulus of an ordinary solid, estimated as a Rydberg per cubic Bohr radius, comes out within a factor of a few of measured values across most of the periodic table.

What would happen if the numbers were different #

Since the size came out of three constants, it is worth asking what it would be if they were other than they are — not as speculation, but because it is the quickest way to see which features of matter are load-bearing.

Halve Planck’s constant and every atom shrinks by a factor of four while every binding energy rises by the same factor. Chemistry would be qualitatively similar and quantitatively unrecognisable: bond energies of tens of electronvolts, room temperature effectively absolute zero, and no molecule breaking without ultraviolet light.

Increase the electron’s mass and the same thing happens, which is the muonic case already drawn — muonic chemistry exists, is a hundred and eighty-six times smaller, and lasts as long as the muon does.

Change the fine-structure constant and both the size and the binding energy move, but the ratio of the binding energy to the electron’s rest energy moves faster, and past a certain point the inner electrons of ordinary atoms become relativistic and the non-relativistic account collapses. That threshold is why atomic number 137 is sometimes quoted as a limit; the real limit is softer and further out, but the mechanism is the one the balance points at.

The single most consequential fact in the list is that the electron is light and the proton is not. Because the reduced mass is nearly the electron’s own, the nucleus barely moves, which is what allows the whole of chemistry to be done with fixed nuclei and electrons arranged around them. A world in which the two masses were comparable would have no molecular structure in the sense the word is used, since there would be no frame in which the nuclei sit still to be bonded.

Where the model stops #

The confinement term is an order-of-magnitude statement dressed as an equality. Writing the kinetic energy as $\hbar^2/2mr^2$ chooses a particular convention for what “confined to $r$” means, and a different convention shifts the coefficient. That the answer comes out at exactly the Bohr radius is a consequence of the convention chosen, not a triumph; what the argument genuinely establishes is the combination $\hbar^2/me^2$ and the fact that a minimum exists.

Slater’s rules are an empirical fit. They were obtained by fitting to computed atomic energies in the 1930s, and while they capture the screening trend they get individual elements wrong by tens of per cent. The figure’s correlation of 0.94 is a statement about the pattern, not about any one atom.

Transition metals and lanthanides are omitted. Their filling order has exceptions, their d and f electrons screen unusually poorly, and the resulting contractions — the lanthanide contraction in particular — are exactly where the simple rules fail worst. Including them would lower the correlation and would need a different account.

And “the size of an atom” is not one quantity. The radius plotted here is an empirical one derived from bond lengths; a van der Waals radius, an ionic radius and a calculated orbital radius differ from it by tens of per cent and from each other in different directions. The trend survives all of them and the absolute values do not.

The measurement that says it is right #

An argument this cheap deserves an independent check, and there is one that does not go through spectroscopy at all.

X-ray diffraction from a crystal measures the spacing of its planes directly, in units of the X-ray wavelength, and the wavelength can be fixed against a ruled grating without knowing anything about atoms. The spacings that come back are a few hundred picometres for every solid ever measured — silicon at 543 picometres for its cubic cell, iron at 287, ice at 452 — which is a handful of Bohr radii, as the balance requires.

The same numbers arrive from an entirely different direction through the density of ordinary matter. A solid’s density is its atomic mass divided by the volume each atom occupies, and inverting that for any element in the table gives a per-atom volume of a few tens of cubic ångströms — again a few Bohr radii on a side. Nothing in that calculation touches quantum mechanics; it is a mass and a measured density.

Three independent routes to one length — a variational balance, a diffraction pattern, and a bucket of water weighed — is the kind of agreement that makes a scale believable, and it is worth more than any one of them measured precisely. It also shows why the failure of the classical account was so serious: classical physics does not merely get this length wrong, it has nothing at all to offer in its place.

What the pictures cannot show #

The balance figures draw an energy against a radius as though the electron had a radius, which it does not — it has a distribution, and the drawn curve is a function of a parameter in a variational argument rather than of anything observable. Where the electron probably is draws the honest object, and it has no single radius in it.

The screened estimate is drawn as a curve through discrete elements, and the line between two elements means nothing at all. There is no atom of atomic number 21.5, and the eye reads a continuous trend where the physics has only the points.

Where the ladder goes next #

The atomic-structure ladder began with where the electron probably is, replacing the orbit with a distribution, and continued with the order the shells fill, which is the bookkeeping that makes chemistry periodic. This rung asks what fixes the scale of the whole thing and finds a competition rather than a constant. The rungs after it: the ionisation energies, which are the same screened balance read as an energy instead of a length; the relativistic contraction of the inner shells of heavy atoms, which is why gold is not silver-coloured; and the transition from atoms to molecules, where two of these balances share their electrons.

The habit worth carrying away is that a length can be an outcome rather than an input. Nothing in the problem was 0.1 nanometres, and the number appeared from combining three constants that are not lengths at all — which is what it looks like when a theory explains a scale rather than accommodating it.