The atom that does not grow with its electrons

Why an atom is the size it is explained the size of hydrogen: an electron confined near a proton pays a kinetic energy for its confinement, and the balance between that and the Coulomb attraction puts it about half an ångström out. Apply that thinking naively to a heavy atom, and an awkward fact appears. Hydrogen has radius of half an ångström. Radon has eighty-six electrons and a radius of a little over two ångströms. Atoms grow, if they grow at all, by a factor of four while their electrons increase by a factor of nearly ninety. Solids made of heavy atoms are not more voluminous than ones made of light atoms. Lead is dense because its atoms are heavy, not because they are small or big, and a table of atomic radii sits in a range of a factor of about three from one end to the other.

This is not a small fact. It determines the density of every material. It explains why a block of lead is much heavier than a block of aluminium of the same size. Yet nothing in the Schrödinger equation looks as though it would predict it, because an exact solution for even three bodies is out of reach. What can be done is the thing done by Llewellyn Thomas in 1927 and independently by Enrico Fermi soon after, before any serious calculation of heavy atoms was possible. They treated the electron cloud as a gas.

An atom as a gas held by its own nucleus #

The idea is this. Near any point in a heavy atom, there are many electrons, so treat them locally as a Fermi gas, the kind that no two in the same state introduced. A density of $n$ fills momentum states up to a Fermi momentum proportional to $n^{1/3}$, with a kinetic energy per electron that grows as $n^{2/3}$. Nothing else about the quantum mechanics is used: no orbitals, no angular momentum, no spin but for its factor of two. Then require that each electron at the top of the local Fermi sea has the same total energy everywhere. If it did not, electrons would flow from higher to lower.

The potential energy comes from the nucleus and from the other electrons, which are themselves described by the density. Poisson’s equation ties that potential to the density. Tidying up, with the potential written as the bare nuclear one times a screening function $\phi$, and the distance measured in a scaled unit $x = r/b$ with

$$b = 0.885\, a_0\, Z^{-1/3},$$

the entire atom reduces to one equation with one solution:

$$\frac{d^2\phi}{dx^2} = \frac{\phi^{3/2}}{\sqrt{x}}, \qquad \phi(0) = 1, \quad \phi(\infty) = 0.$$

The atomic number is gone. There is one function for every element, and the atom’s size enters solely as the scale in $b$. Before doing anything numerical, this already predicts that all atoms have the same shape, stretched by $Z^{-1/3}$.

The equation looks innocent but it is not. It has a singularity at the origin in the $\sqrt{x}$, it is nonlinear and the condition at infinity cannot be imposed directly. It has to be solved by shooting: guess the initial slope, integrate outward and see what happens. If the slope is too steep, $\phi$ crosses zero at a finite $x$; too shallow and it turns round and grows without limit. Between those two lies a single critical slope, and bisection finds it. Fermi did this on a hand-cranked calculator in about a week. The change of variable $t = \sqrt{x}$ turns the equation into one that is smooth at the origin. With that, the initial slope comes out at $-1.5881$, the published value, and the function reproduces the tabulated value at $x = 20$ to within a couple of per cent.

So here is a universal curve for all the atoms in the table, a single function of a single variable. The tail is an exact solution of the equation in its own right, as is easy to verify by substitution, and it contains no adjustable numbers. Sommerfeld noticed that in 1932. It is approached slowly, as the dashed line in the figure makes clear.

The heavier the atom, the smaller its middle #

Since $b$ falls as $Z^{-1/3}$, the radius within which any given share of the electrons lies shrinks as atoms get heavier.

The explanation in words is that the nucleus’s charge wins. Add a proton and an electron, and the extra nuclear attraction falls on every electron, while the extra electron’s repulsion falls on a share of them only, and it is mostly at the outside. The result is that the inner electrons are drawn in tighter. A uranium atom holds half its electrons within about 0.2 Å, closer than hydrogen holds its one. So a heavy atom is a very dense core, inside a thin haze. It is not much larger than a light atom because it is not filled in the same way, not because something stops it. The metal that is yellow because it is heavy depends on this concentration too: gold’s inner electrons move so close to the nucleus that they are significantly relativistic.

There is also a simple way to see the $-1/3$. For the kinetic energy of a Fermi gas of $Z$ electrons confined in a radius $R$ to balance the attractive energy with the nucleus of charge $Z$, the kinetic energy per electron goes as $Z^{2/3}/R^2$ and the potential as $Z/R$. Equality gives $R \propto Z^{-1/3}$. It is a one-line argument with the same structure that gave the size of hydrogen, just with a bigger number. That is why it has the same form as the one in the mass no cold matter can hold up, where a white dwarf’s size falls as its mass to the minus a third. The heavier one is smaller. Heavy atoms and white dwarfs are the same problem in very different ranges, of a charge or mass with its degenerate electrons, and they both contract under load.

A ceiling on the outside #

The outer region is more curious. If half of an atom’s electrons are drawn in as it gets heavier, what happens further out? The density at a fixed distance can be written in terms of $\phi$ and $Z$, and in the far tail, where $\phi$ follows its $144/x^3$, the powers of $Z$ all cancel. The density at a given distance tends to

$$n(r) \to \frac{95.5}{r^6}$$

in atomic units, independent of $Z$.

This is as close as the model gets to explaining why atoms are the same size. A heavier atom’s outer layers contain almost the same density of electrons, at the same distance, as a lighter atom’s do. So where a neighbour would feel its outer edge, set by when it has dropped to a low enough density, is nearly fixed. The chart makes the point visually and also shows that the real approach to the ceiling is slow.

In truth the model’s tail is wrong. A real atom’s density decays exponentially, as the outermost electron has a definite binding energy, and with a decay rate set by the ionisation energy. A power law like this never reaches zero, so the model atom has an infinite size. And the model has no shells, so it does not get the wiggles in the real sizes of atoms, in which a noble gas is tight and the following alkali metal is suddenly large. A caesium atom has a radius of about 2.6 Å, the largest of any stable atom, and neon’s is far smaller than its neighbour sodium’s. Those are the shell effects in the order the shells fill, and they control all of chemistry. The Thomas–Fermi atom is therefore no use for chemistry at all. It does not bind molecules: Teller proved in 1962 that in this model, two atoms never form a stable molecule, so that no molecule exists in it. That result is exact, and from a chemist’s point of view rather damning.

Where the model works is as the smooth trend behind the shells. The periodic table’s variation goes up and down around a mean, and the mean is what this curve describes.

A theorem about a rough model #

The model’s total energy is the place to test it. Add up the kinetic and potential energies and integrate the curve, and the binding energy comes out as

$$E = 0.7687\, Z^{7/3}$$

in hartrees. That is, the energy needed to strip every electron from an atom goes as the seven-thirds power of its charge. And for atoms of any size, accurate numbers for this quantity exist, from Hartree–Fock calculations and experiment.

The points are real atoms, from helium to radon, and they trend toward the model’s number. In the plot, heavy atoms sit at the left where $Z^{-1/3}$ is small. Extrapolate to $Z$ going to infinity, and the points land on the intercept. In 1952, J. M. C. Scott found a next term: the innermost electrons near the nucleus do not behave like a gas, since the distance over which the potential changes is small compared with their wavelength there. Treating them properly adds a correction that goes as $Z^2$, the same as the energy of a hydrogen-like atom with just those electrons. Julian Schwinger worked out the next term in 1980–81, and with it, the formula reproduces real atoms’ energies to within a part in a hundred or better. So a smooth equation from 1927 turns out to be the first term in a real asymptotic series.

And then a theorem. In 1977 Elliott Lieb and Barry Simon proved that as $Z$ goes to infinity, the ratio of the true many-body quantum energy to the Thomas–Fermi energy goes to one. Rigorously, not numerically. A statement about $10^{23}$ coupled equations with every correlation included is settled by a one-line differential equation. Later work by Ivrii and Sigal, and by Siedentop and Weikard, proved the Scott term also. In a field where exact results are rare, this one has become a model of what mathematical physics can do. The nearly comic part is that real atoms go up to a $Z$ of about a hundred and twenty, while the theorem needs infinity. Nature does not get far enough to use it.

How much of an atom is where #

Two more readings of the same curve round out the picture.

The curve shows that an atom is not a fuzzy ball of uniform density. Half of it sits inside $x = 1.88$, while the last tenth spreads to more than four times that. The tail is a long and thin one. X-ray scattering works largely through this distribution and the model does well there. Early atomic form factors, used to interpret X-ray diffraction before computers, came from this same function. The Thomas–Fermi screening was also how the first electron scattering and energy-loss tables were built, and it still appears in the theory of how fast ions lose energy in matter. Dimensional reasoning that started with an atom was adopted in the same way for metal screening, in a form discussed by the long-range force that does not reach.

The model has also spread outside atoms entirely. Density functional theory, now the dominant way to compute the electronic structure of molecules and solids, descends from the idea that an atom’s energy can be written as a functional of its density. Thomas and Fermi’s kinetic term is the first attempt at the key part of that functional. Modern versions replace it with Kohn and Sham’s orbitals, because it was the kinetic energy that the simple form got most wrong, and they keep the idea that the density alone determines what is needed. Kohn’s Nobel Prize came in 1998. Its ancestor is this very crude model.

What pressing on an atom shows #

A model with no shells has one further use that is easy to overlook. Squeeze matter hard enough and the shells stop mattering. In a solid at ordinary pressure, each atom’s outer electrons are bound to it in a way that depends on all of chemistry. At the pressures inside a giant planet or a white dwarf, the outer orbitals overlap and lose their identity, in the process that the pressure that ionises described, and the atom’s electrons become exactly what Thomas and Fermi assumed from the beginning: a locally uniform gas of fermions, with a varying density around the nucleus.

So the model gets better the harder matter is squeezed. Richard Feynman, Nicholas Metropolis and Edward Teller wrote this down in 1949, confining a Thomas–Fermi atom in a sphere whose radius is set by the density of the material, and they obtained an equation of state usable at very high density. It comes from the same equation, with the outer boundary moved in. Their calculation came out of work on hot dense matter in the years after the war, and it has gone on being used for stars. It joins smoothly onto the other limit, in which the electrons are entirely free and the pressure is that of the pressure that is not a temperature, the pressure that holds up the cloud that stops shrinking.

The compressed atom has a simple message about the sizes above. A free atom’s outer radius has no fixed meaning because its tail goes on for ever, in this model and in reality. What gives an atom a size in a solid is its neighbours. Two atoms brought together overlap in their tails, and the exclusion principle starts charging for it. A hard wall seems to appear at about the place where the tails have the density at which the charge exceeds the attraction. So the figure’s ceiling on density is the reason atoms of very different mass can pack at nearly the same spacing. Where the density at a given distance no longer depends on what is in the middle, neighbours arriving from outside meet the same thing in each case.

This also accounts for the density of a metal, which is mostly its nuclear mass over a nearly constant volume. Osmium is about forty times as dense as lithium. Its atoms are about twenty-seven times as massive, and that accounts for most of it, with the packing supplying only the remainder. Anyone who has picked up a small lump of tungsten has felt the size of the $Z^{-1/3}$ effect directly: a nucleus nearly two hundred times the proton’s mass, in an atom no larger than iron’s.

Where the model has nothing to say #

Its largest omission is the shell structure, and with it all the things that make elements different. An atom in this model has no valence and no chemistry. It predicts no periodic table. It gets the total energy right because that is dominated by the many inner electrons, where a gas approximation is good, and it gets the outermost electrons wrong because there are few of them and they live in a low-density tail, where the local gas idea is invalid. These last few are where nearly all the phenomena that interest a chemist reside.

Also missing is exchange. The Pauli principle gives an energy correction to a real gas of electrons, and Dirac added a term for that in 1930. Relativity is also absent, and at $Z \approx 90$ the innermost electrons have a Lorentz factor substantially above one; the model’s $Z^{7/3}$ eventually overestimates or underestimates accordingly. The nucleus is a point, and its size matters at high $Z$. Beyond around 137, the Dirac treatment of the innermost electron needs revising anyway. So the limit proved rigorous by Lieb and Simon applies to a nonrelativistic theory, which is not what the universe’s actual heavy atoms obey. The result holds as a theorem, for that theory. It is not known whether a fully relativistic atom would obey an analogue of the formula.

The radii drawn in the figures compare a calculated median with a measured van der Waals radius, and those are different quantities. They are placed on the same chart to make one point about trend. They should not be taken to agree or disagree. No single radius exists for an atom, and various definitions differ by up to 50 per cent.

Still open: what a very heavy atom looks like #

The heaviest atoms made so far have been around $Z = 118$, in a few at a time for fractions of a second. Calculations for such an atom indicate that shells begin to dissolve: relativistic effects spread energy levels so far that electrons cease to have the clean shell structure of lighter atoms, and the outermost levels get close together. Calculations suggest that oganesson’s electrons may be spread so smoothly that it looks more like the Thomas–Fermi gas than like a noble gas, and that this smoothness could be a real property of the element at the end of the table. Whether this will be testable is a matter for experiments that measure a single atom’s properties when only a handful are ever available. It is a curious prospect: the crude model might describe best the atoms at the very end of the table, those that nature makes least of.

Treating an atom’s electrons as a locally Fermi gas reduces every element to one equation, with one solution whose length scale shrinks as Z to the minus a third: the middle of a heavy atom is drawn in, from 0.41 Å at Z = 10 to 0.20 Å at Z = 90, the outside rises at most 2.2 times for a ninefold increase in charge, and the binding energy tends to a constant 0.7687 $Z^{7/3}$ proved exact at infinite Z. It has no shells, no chemistry and no molecules, and it is the exact first term for what atoms do. The habit worth carrying is to ask which question a crude model is right for, because being wrong about the part everyone looks at does not stop it from being exactly right about the rest.