The X-ray line that counts the protons

In the autumn of 1913 a twenty-six-year-old physicist named Henry Moseley, working in Oxford after two years in Rutherford’s laboratory in Manchester, built a long glass tube in which a little trolley could carry one metal target after another into the path of a beam of electrons. Each target, struck by the electrons, gave off X-rays, and Moseley measured their wavelengths by reflecting them from a crystal and photographing where they landed. Every metal gave a continuous background and, on top of it, a few sharp lines whose wavelengths depended on the metal. He photographed the lines of calcium, titanium, vanadium, chromium, manganese, iron, cobalt, nickel, copper and zinc, and found that a single simple formula fitted all of them.

The formula said that the frequency of the strongest line rose as the square of a number that went up by exactly one from each element to the next. Moseley identified that number with the element’s position in the periodic table, and its position with the electric charge on its nucleus. Eighteen months later he was dead, shot at Gallipoli at the age of twenty-seven. In that time he had shown what the periodic table was counting.

Lines that come from deep inside #

The spectrum is a subtraction found that every spectral line is the difference between two energy levels of an atom, and that the lines of hydrogen’s visible spectrum come from its single electron jumping between outer levels a few electronvolts apart. X-ray lines come from much deeper. When a fast electron knocks out one of an atom’s innermost electrons, from the shell physicists call K, it leaves a vacancy close to the nucleus, where the binding is thousands of electronvolts. An electron from the next shell out, L, drops into the vacancy and emits the difference as an X-ray photon: the line called Kα. An electron dropping from the shell beyond, M, emits Kβ.

These energies are thousands of times larger than the visible lines because the electrons involved are deep inside the atom, close to a nucleus of large charge. Why an atom is the size it is found that a single electron around a charge $Z$ has energy levels $Z^2$ times hydrogen’s and an orbit $Z$ times smaller; an innermost electron of copper, $Z = 29$, sits close enough to the nucleus that most of the other electrons are outside it and hardly matter. It behaves almost like the lone electron of a hydrogen atom with the nuclear charge turned up.

Moseley’s insight was to treat the falling electron that way, with one correction: the vacancy in the K shell still has one electron left in it, which partly shields the nucleus. So the falling electron sees a charge of $Z - 1$ rather than $Z$, and its energy levels are those of hydrogen scaled by $(Z-1)^2$. The jump from $n = 2$ to $n = 1$ in hydrogen releases three-quarters of the Rydberg energy, $13.6$ electronvolts, so

$$E_{K\alpha} = \tfrac{3}{4}\,R_y\,(Z-1)^2.$$

For copper that is $\tfrac34 \times 13.6 \times 28^2$ electronvolts, 8.00 kiloelectronvolts, against a measured 8.05.

The figure is honest about what the model gets wrong. Copper’s real K-shell binding is not 10.7 keV but 9.0, and its L shell is bound by under one keV, not 2.7, because every electron in the atom screens the nucleus from every other and Moseley’s single correction does not capture that. The model works for Kα because the extra screening shifts the K and L levels by nearly the same amount, so their difference survives. For Kβ it does not: the M electron is screened by the whole of the L shell as well, and the simple formula overestimates its energy by six per cent.

The background the lines stand on #

The sharp lines are not all an X-ray tube produces. Most of its output is a continuous spectrum, from the electrons of the beam braking in the target: a charge that decelerates radiates, as a charge that turns must glow found for one that changes direction, and a fast electron stopped in metal gives up its energy as X-rays of every energy up to its own. That continuum has a sharp upper edge. An electron accelerated through 30 kilovolts carries 30 kiloelectronvolts, and no single photon it makes can carry more, so the continuum stops at a wavelength fixed by the tube’s voltage alone — the Duane–Hunt limit of 1915, one of the cleanest early demonstrations that light arrives in lumps, since a wave picture gives no reason for an edge.

The characteristic lines sit on top of the continuum, and they appear only when the tube’s voltage is high enough to knock out a K electron at all. Below the K-shell binding of the target — 9 kilovolts for copper — the K lines are simply absent, however long the tube runs; above it, they switch on and grow. Moseley’s targets were run well above their thresholds, and the lines stood out clearly against the brake radiation.

Not every vacancy produces an X-ray. The energy released when an L electron falls into the K shell can instead be handed to another electron, which is ejected from the atom — the Auger effect, after Pierre Auger, who saw the electrons’ tracks in a cloud chamber in 1923. For light elements this wins almost every time: in carbon fewer than one vacancy in a hundred fills by emitting an X-ray. The chance of X-ray emission rises steeply with atomic number, roughly as its fourth power at first, and passes a half near zinc. That is why X-ray fluorescence analysis struggles with the lightest elements, and why Moseley’s line, built from elements calcium and above, was so clean. The shell structure that decides which electrons are available for the Auger process is the one the order the shells fill worked out.

One step per element #

The test of the law is to plot the square root of each line’s energy against the number that is supposed to be $Z$. If the law holds, the points lie on a straight line rising by $\sqrt{3/4}$ for each step.

The points march up the line in equal steps. Each element’s line is one step higher than the one before, and Moseley could not have produced that regularity from atomic weights, which go up irregularly — by one unit here, by four there. Whatever was being counted went up by exactly one from each element to the next, and the obvious candidate was the number of positive charges on the nucleus. Antonius van den Broek, a Dutch lawyer and amateur physicist, had suggested in 1911 that an element’s position in the periodic table equals its nuclear charge, after Rutherford’s experiments showed that the nucleus existed and that its charge was roughly half the atomic weight. Moseley’s line turned that suggestion into a measurement.

It also showed where elements were missing. A gap in the sequence of steps, with no known element to fill it, meant an element not yet discovered, and its X-ray energy could be predicted in advance. Moseley’s data put gaps at atomic numbers 43, 61 and 75, and his law located 72 among the rare earths. Hafnium, element 72, was found in 1923 in Copenhagen by Dirk Coster and George de Hevesy, who looked for its predicted X-ray lines in zirconium ores. Rhenium, 75, followed in 1925; technetium, 43, was made artificially in 1937, the first element produced before it was found in nature; promethium, 61, was isolated from fission products in 1945. Moseley’s line also settled how many rare earths there could be — exactly fifteen between lanthanum and hafnium — a question chemists had found impossible to answer by chemistry alone.

Ordered by weight, the elements step backwards #

Mendeleev’s periodic table of 1869 ordered the elements by atomic weight, and mostly that gave an order in which chemically similar elements fell into columns. In a few places it did not, and Mendeleev had swapped pairs of elements against the order of their weights to keep the columns right.

Argon is heavier than potassium, cobalt heavier than nickel, tellurium heavier than iodine. Ordered by weight, potassium would come before argon and fall in the column of the noble gases, which is chemically absurd: potassium is a violently reactive metal and argon an inert gas. Chemists had put these pairs in the “wrong” order of weight on the evidence of their chemistry. Moseley’s X-rays showed why that was right. In each pair the X-ray energy rises from the first element to the second, as it does everywhere else, so the nuclear charge rises too: argon has 18 protons and potassium 19, cobalt 27 and nickel 28. The weights are out of step because the heavier element of each pair happens to carry more neutrons, which add weight without adding charge. Atomic number, not atomic weight, is what the periodic table is ordered by, and what decides an element’s chemistry.

Where the law bends #

The figure of the line hides a small, systematic departure that turns out to be informative. Divide each measured energy by Moseley’s prediction and the ratio is not quite one.

For light elements the law is good to a per cent. For heavy elements the measured energy runs ahead of it, by eleven per cent for gold and twelve for lead. The reason is relativity. An innermost electron in a heavy atom moves very fast — in Bohr’s picture its speed is about $Z/137$ of the speed of light, so in lead nearly 60 per cent of it — and at such speeds its energy is no longer given by Bohr’s non-relativistic formula. Relativity deepens the binding, by an amount that grows as the square of $Z/137$ — the fine-structure constant $\alpha$ times $Z$ — and the grey guide in the figure has that shape. The metal that is yellow because it is heavy traced gold’s colour to the same relativistic contraction of its inner orbitals; here it appears in the X-ray energies directly.

Relativity also splits the line. The L shell has two sublevels, separated by the coupling of the electron’s spin to its orbit, which the line that is really two found splitting sodium’s yellow light into a doublet. The same splitting appears in X-rays, and scales up steeply with nuclear charge: Kα is really two lines, Kα₁ and Kα₂, separated by twenty electronvolts in copper and by over four thousand in lead. The figures here use Kα₁, the stronger.

The steep growth of the line energies with $Z$ is what makes them useful at the hospital as well as the laboratory. A medical X-ray tube has a tungsten anode partly because tungsten survives the heat, and partly because its K lines, near 59 kiloelectronvolts, are hard enough to cross a body; a mammography tube uses molybdenum or rhodium, whose lines near 17 and 20 kiloelectronvolts are soft enough to show the small differences between soft tissues. The choice of target is a choice of where on Moseley’s line to sit.

Reading an alloy by its lines #

The law that ordered the periodic table is also one of the most widely used methods of chemical analysis. Shine X-rays on any material and its atoms are ionised in their inner shells; the vacancies fill and emit their characteristic lines, a process called X-ray fluorescence. Measure the energies of the lines and the elements present can be read off, one by one, from their positions on Moseley’s line.

The figure is not a measurement but a picture of what one looks like. Each element in the sample contributes a pair of peaks, Kα and the weaker Kβ, at energies set by its atomic number alone, so a peak at 8.05 keV means copper whatever the copper is combined with. Neighbouring elements’ lines are a few hundred electronvolts apart, and a good detector separates them: in the figure, nickel’s Kβ is a shoulder on copper’s Kα. Handheld analysers built on this identify the grade of steel in a scrap yard in seconds, check jewellery for lead and toys for cadmium, and read the composition of rocks on Mars from the end of a rover’s arm. The X-rays are cheap, the method does not destroy the sample, and its foundation is a formula written down in 1913.

The crystal that measured the X-rays #

Moseley’s measurement was possible only because of a discovery made the year before. In 1912 Max von Laue’s group showed that X-rays passing through a crystal are diffracted by its regularly spaced rows of atoms, as light is by a grating, and William Lawrence Bragg showed how to turn this into a spectrometer: X-rays reflect strongly from the planes of a crystal only at angles where the waves reflected from successive planes add, $2d\sin\theta = n\lambda$, so the angle of strong reflection measures the wavelength. What a thousand slits buy found how a grating’s many slits sharpen its lines; a crystal is a grating with millions of slits stacked in three dimensions, and its spacing — a fraction of a nanometre — is the right size for X-rays.

Moseley used crystals of potassium ferrocyanide and rock salt, measured the angles at which each target’s lines reflected, and so obtained their wavelengths to a fraction of a per cent. The precision mattered: equal steps in the square root of the energy are visible only if each energy is measured well enough to place it between its neighbours. The modern detectors in the figure above do not use crystals; they measure each X-ray photon’s energy directly, from the charge it frees in a cooled semiconductor, which is faster and less precise. The physics of the lines is unchanged.

One electron, one screening charge and tabulated lines #

The level diagram is a single-electron model with one screening electron; real atoms have many electrons screening one another in a way that requires a full calculation of the atom’s electronic structure, and the K and L binding energies it implies are not the real ones. The guide curve on the residual figure is a rule of the right form — a correction growing as the square of $\alpha Z$ — scaled to pass near the points, not a derived relativistic correction; the full calculation of a heavy atom’s inner levels, with relativity and screening together, reproduces the measured energies to better than a part in a thousand. The line energies are typed in from standard tables, and the atomic weights are the standard values.

The fluorescence spectrum is a picture, not data: its peak heights are invented, its continuous background is a flat floor, and it leaves out escape peaks, sum peaks and the scattering of the exciting X-rays, all of which a real spectrum shows. And all the figures concern the K lines only; the L and M lines of heavy elements, at lower energies, follow similar laws with different screening and are used in the same way.

Still open: where the periodic table ends #

Moseley’s law counts protons for every element that has been measured, up to the heaviest. The question it raises is where the counting stops. Relativity, which pulls the measured energies above the law, grows stronger with every added proton; Dirac’s equation for a single electron round a point nucleus breaks down when $Z$ exceeds 137, where the innermost level would bind the electron by more than its rest energy, and for a nucleus of finite size the limit moves out to about 170. Beyond it, the vacuum near the nucleus is expected to become unstable and produce electron–positron pairs spontaneously. Elements up to 118, oganesson, have been made; none of them lives long enough for an X-ray spectrum, and whether nuclei anywhere near 170 can be made at all, or whether the table ends long before chemistry and relativity reach their limit, is not known.

What Moseley found is the general rule. An electron deep inside an atom sees a nucleus barely screened, so it behaves like hydrogen’s electron with the charge turned up; the energy of the X-ray it emits therefore grows as the square of the nuclear charge, and the square root of that energy counts the protons one by one. The periodic table had been built by chemists from the behaviour of outer electrons; the inner ones, counted by X-rays, showed what it had been counting all along.