You can see the shells in the data

Strip the electrons off an atom one at a time and the energy needed does not climb smoothly. It steps — hugely, and at exactly the places the periodic table says it should. This is the rare case where the structure is not inferred from the measurement but printed in it.

1869 – 196216 min

By the end you should be able to

Read the shell structure of an atom straight off a column of measured energies
Identify an unknown element from where its ionisation-energy jump falls
Explain why the energies climb gently inside a shell and leap between shells
Say what photoelectron spectroscopy adds that successive ionisation energies cannot give

The last lesson gave chemistry to the electrons and said nothing whatever about how they are arranged. By the 1920s that had become the embarrassing gap in the middle of the subject. It was not for want of models. Thomson had rings inside his positive sphere. Bohr had orbits at fixed radii. Lewis had a cube with an electron at each corner. Every one of them assigned the shells populations of 2, 8 and 18, and every one of them worked. And every one of them got those numbers from the periodic table. That is the difficulty. Bohr’s shell populations were chosen so that the elements came out in the right groups; Lewis’s cube was built to make eight a special number because eight was already known to be special. These are not predictions. They are restatements. You cannot explain the shape of the table with a model whose only evidence is the shape of the table. Something has to break the circle: a measurement of the atom itself, in energy units, which mentions no chemical property at all — and which shows structure anyway.

There is such a measurement, and it is almost crude in its directness. Take an atom. Remove one electron. Write down what it cost. That is the first ionisation energy, and it is a number in kilojoules per mole, obtained from a discharge tube or a spectrometer. Nothing about valency, nothing about oxides, nothing about who reacts with whom. Now remove the next one, from the ion you just made. That is the second ionisation energy. Then the third, and so on, until nothing is left but the bare nucleus. For sodium that is eleven numbers. For calcium, twenty. And the whole argument of this lesson is: look at the list. If the electrons are all much the same — a swarm around a nucleus, as Rutherford left them — the numbers should climb steadily, because each removal leaves a more positive ion and the next electron is held harder. A smooth rise is the null hypothesis, and it is a perfectly reasonable one.

The experiment

Measuring an ionisation energy

James Franck and Gustav Hertz by electron impact; spectroscopists by series limits; Robert Millikan and Ira Bowen for the stripped ions · 1914 for the electron-impact tube; series limits from the 1890s; Millikan and Bowen’s vacuum-spark work from 1924 · Berlin, and then the Norman Bridge Laboratory at Caltech

The question: How much energy does it take to remove one electron from one atom — measured, rather than inferred from how the atom behaves chemically?
The apparatus: Two independent routes. By collision: a tube of vapour at low pressure, a heated filament, a grid at a controlled accelerating voltage, and a collector. Raise the voltage until the tube abruptly starts to conduct, because ions are being made. By spectrum: photograph the emission lines of the element. In every series the lines crowd closer as they go, and converge on a limit. That limit is the energy needed to take the electron away altogether.

Theory predicted

Both routes should give the same number, because they are the same physical quantity approached from opposite ends — one supplies the energy mechanically, the other measures the energy of the photon that just fails to be emitted.

They measured

They do. Sodium’s principal series converges at 41,449 cm⁻¹, which is 5.139 eV, which is 495.8 kJ/mol. For the doubly, triply and further-stripped ions the collision route becomes impractical, and Millikan and Bowen’s hot vacuum spark supplied the spectra instead — stripping light atoms of several electrons at once and photographing what the remains emitted.

How sure could they be? Series limits are the reason the tables are quoted to a tenth of a kilojoule. A convergence limit is read off a photographic plate against a wavelength scale, and wavelengths were the most precisely measured quantities in all of physics — better than one part in a million. The collision method is far cruder, and it also produced the most instructive error in the field: Franck and Hertz’s celebrated 4.9 V threshold in mercury vapour was not an ionisation at all. It was an excitation, as Bohr pointed out; mercury’s actual first ionisation energy is 10.44 eV. They had measured something more important than the thing they were looking for.

Why it mattered

An ionisation energy is a property of an atom, in joules, with no chemistry in it. That is what makes the next step legitimate rather than circular.

Sodium, all eleven electrons, in the order they come off. Look at the first two bars and nothing else to begin with. Then step the element along to magnesium and aluminium and watch where the break goes.

Loading the ladder…

A logarithmic scale, because sodium’s first ionisation energy is 495.8 kJ/mol and its last is 159,076, and no linear axis holds both. The gold dashes mark where the next electron has to come out of a shell closer in; the faint dashes mark a change of sub-shell within the same shell. Nothing in this chart was put there by a theory of the atom. It is a plot of measured energies against a counting index.

Read sodium’s ladder slowly, because everything is in it. The first electron costs 495.8 kJ/mol. That is an ordinary number — about the energy of a chemical bond, which is exactly why sodium gives this electron away to almost anything. The second costs 4,562. A factor of 9.2, for the very next electron off the very same atom. Nothing gradual has happened; a door has shut. Then eight electrons in a row that behave alike. From the second to the ninth the price climbs steadily — 4,562, 6,910, 9,543, and on up to 28,932 — each around a quarter dearer than the one before. A gentle, monotonous slope, over eight electrons. And then it leaps again, by a factor of 4.9, to 141,362. Two more electrons at that level and the atom is a bare nucleus. One. Then eight. Then two.

The rows of the periodic table, read off a set of measurements

One, eight, two — counting outward from the nucleus, two, eight, one. That is sodium’s electron configuration, and it was obtained here without using a single chemical fact. Sodium is in group 1 because it has exactly one electron that is cheap to remove. The ladder says one, and it says it with a factor of nine. The neon core underneath holds eight. The ladder says eight, by giving eight consecutive energies that differ from each other far less than they differ from their neighbours on either side. The innermost pair is helium-like, and there are two. The ladder says two, and then the atom is finished. This is worth pausing over because it is genuinely unusual. Most of what we believe about the structure of matter is inferred: you build a model, you compute what it predicts, you compare. Here the structure is present in the measurement. A person who had never heard of a shell, handed this column of numbers, would draw the lines in the same places. The theory does not organise the data. The data is already organised.

And now the part that makes it a law rather than a coincidence about sodium: step along the row and watch the break move. Magnesium, one place further on. Its first two electrons cost 737.7 and 1,451 kJ/mol — both ordinary — and the third costs 7,733. The jump has moved to after the second electron. Magnesium is in group 2. Aluminium, one further still. 577.5, 1,817, 2,745 — and then 11,577. After the third. Group 3. The position of the break walks one place along for each element you walk along the table, and lands exactly on the group number every time. An instrument that has never been told what a group is, telling you the group. That is not a check on the periodic table. It is an independent derivation of it.

Problem

Name the element

An unlabelled sample. Its successive ionisation energies, in kJ/mol, begin: 578, 1,817, 2,745, 11,577, 14,842, 18,379, … Other evidence places it in period 3. What is its atomic number?

Where the energies leap: read it off the list
Row of the table: period 3
Rule: electrons removed before the leap = electrons in the outer shell

The part the story usually rounds off

You will often see a ladder like this described as “the energies of the electrons in the atom”. It is not, and the difference is worth having straight. Every number belongs to a different species. The first is measured on a neutral sodium atom. The second on Na⁺. The third on Na²⁺. By the ninth you are working on an ion with eight units of positive charge, which is not sodium in any sense a chemist would recognise. And each removal tightens what is left. The nuclear charge has not changed, but there is one fewer electron to share the shielding and one fewer to repel the others, so every remaining electron is held harder than it was a moment ago. That, rather than any structural difference, is most of why the eight electrons of the neon core rise steadily from 4,562 to 28,932 instead of costing the same. So: the slope is bookkeeping. The steps are shells. The gentle rise within a group of eight is the ion getting more positive. The factor-of-nine jump between groups is a different shell, and no amount of increasing charge produces a discontinuity like that. The honest way to put it is that successive ionisation energies show you the sizes of the shells cleanly and their energies only roughly. Getting the energies of the electrons in a real, neutral atom needs a different experiment — which was eventually built.

The experiment

Photoelectron spectroscopy

Kai Siegbahn in Uppsala for the X-ray version; David Turner at Imperial College for the ultraviolet one · From the mid-1950s; Turner’s ultraviolet instrument in 1962 · Uppsala and London

The question: Successive ionisation energies come from a chain of different ions. Can all the electrons of one neutral atom be surveyed at once?
The apparatus: Photons of a single known energy — the helium resonance line at 21.22 eV for the ultraviolet instrument, aluminium Kα X-rays at 1,486.6 eV for the deeper electrons — fired at a gas. Every electron ejected has its kinetic energy measured. The binding energy is the difference: photon energy in, kinetic energy out, remainder is what held it. This is Einstein’s photoelectric equation from the physics path, turned into a measuring instrument.

Theory predicted

If a shell were a single uniform level, each shell would give one peak: neon should show two, for its n = 1 and n = 2 electrons, in a height ratio of 2 to 8.

They measured

Neon gives three. Peaks at 2.08, 4.68 and 84 MJ/mol, with relative heights 6 : 2 : 2. The eight electrons of the second shell are not all alike — six sit together and two sit apart, at more than twice the binding energy.

How sure could they be? The heights are the check that it means what it appears to mean. Peak area is proportional to the number of electrons in that sub-shell, so an electron configuration can be read straight off the chart. Sodium adds a fourth peak at 0.496 MJ/mol, which is 496 kJ/mol — sodium’s first ionisation energy, arrived at by a completely different instrument on a completely different principle.

Why it mattered

Sub-shells are real, measurable, and they come in fixed sizes: 2 and 6 inside the second shell, and later 10 and 14. Those numbers are the blocks of the periodic table — the s-block two wide, the p-block six wide, the d-block ten. Which raises the question this act still has not answered: where do 2, 6, 10 and 14 come from?

What the peaks are, and what the heights are

A photoelectron spectrum carries two independent pieces of information, and it is worth separating them. Where a peak sits is an energy. How tightly that group of electrons is held, in megajoules per mole, measured on the neutral atom rather than on a chain of ions. How large a peak is, is a count. Twice as many electrons, twice the area. Neon’s three peaks come in the ratio 6 : 2 : 2 — 2p⁶, 2s², 1s². That is not an interpretation of the spectrum. It is the y-axis. And the two together are the configuration. 1s² 2s² 2p⁶ for neon, read off a chart, with no model of the atom involved at any point. Successive ionisation energies gave the shells: 2, 8, 8, 18. Photoelectron spectra split the eights: 2 and 6. Between them, the entire block structure of the periodic table has been established by measurement — the s-block two columns wide, the p-block six, the d-block ten. What neither can do is say why. Nothing in a column of energies explains why a sub-shell stops at six.

You might think

First ionisation energy increases steadily across a period.

Actually

It increases overall, and there are two dips in every period which are not exceptions to be excused but the most informative points on the chart. Across period 2: beryllium is 899.5 kJ/mol and boron is 800.6 — a drop of 99 going the wrong way. And nitrogen is 1402.3 while oxygen is 1313.9 — another drop. The same two dips appear in period 3, in the same places: magnesium 737.7 down to aluminium 577.5, and phosphorus 1011.8 down to sulphur 999.6. A pattern that repeats in the same two positions every row is structure, not scatter. The first dip is where the outermost electron moves from an s sub-shell into a p, which is higher in energy and better screened. The second is where a p sub-shell is forced to double up — four electrons into three orbitals — and the pair that shares an orbital repel each other, so one of them is easier to remove. Both dips are sub-shell effects, visible in the very first number of the ladder, and a rule that says the energy always rises has quietly deleted them.

There can never be two or more equivalent electrons in an atom for which, in strong fields, the values of all the quantum numbers are the same.
Wolfgang PauliZeitschrift für Physik 31, 765 (1925), in the standard English rendering. This is the sentence that says why a shell fills up and stops — why the answer is 2, 8, 8, 18 and not "as many as will fit". It is also, as Pauli himself said, a rule he could state but not derive; the derivation waited for relativistic quantum field theory and his own spin-statistics theorem of 1940.

1869Mendeleev’s table: periodicity established as a fact about chemical properties and atomic weights.
1904Thomson computes ring populations inside his positive sphere, and suggests they explain periodicity.
1913Bohr postulates shells with populations chosen to reproduce the groups. It works, and it is circular.
1914Franck and Hertz measure a 4.9 V threshold in mercury and take it for an ionisation. Bohr shows it is an excitation.
1916Lewis’s cubical atom makes eight the special number, because eight was already known to be special.
1924Millikan and Bowen’s hot vacuum spark reaches multiply-stripped light atoms; successive ionisation energies become measurable.
1925Pauli’s exclusion principle: a stated rule for why a shell closes, with no derivation behind it.
1926Schrödinger’s equation. The sub-shell sizes 2, 6, 10, 14 finally have a source.
1962Turner’s ultraviolet photoelectron spectrometer: sub-shells appear as separate peaks, with heights equal to their occupancies.

Where this leaves us

The shells are no longer a proposal. They are a measurement, and they hold 2, 8, 8, 18 — read off a column of energies that never mentions a chemical property. The photoelectron spectra then split those eights into a 2 and a 6, and hand back the block structure of the table: s two wide, p six, d ten, f fourteen. That is as far as measurement alone will take it. The numbers are established beyond argument and completely unexplained. Why does a shell close at 2 and then at 8? Pauli could state the rule in 1925 — no two electrons alike — but he could not say where the number of available states comes from, and said so. And why 2, 6, 10, 14? Look at the differences: 4, 4, 4. Something is counting in fours, plus one, doubled. That is not an accident and it is not chemistry. The answer is not in any table of energies. It is in the shapes a wave is permitted to take when it is trapped around a point — and there are exactly one of the first kind, three of the second, five of the third, seven of the fourth. Double each for the two ways an electron can spin, and you have 2, 6, 10, 14. The electron is not orbiting anything.