Energy levels are real

Every clue about atomic energy levels had come from light. Two physicists in Berlin fired electrons through mercury vapour instead, found the same numbers, and then misunderstood their own result for a year.

1914 – 192516 min

By the end you should be able to

Explain why an independent probe matters more than a more precise version of the same one
Say what the periodic dips in the current mean, and why elastic collisions cost so little
Describe how the same energy was confirmed by two completely different measurements

Everything in this act so far has one source: light. Fraunhofer's dark lines, Kirchhoff's fingerprints, Balmer's formula, Bohr's ladder — all of it is about photons coming out of atoms. Even the photoelectric effect is light going in. And that leaves a genuine worry. What if the discreteness is a fact about how atoms emit light, rather than a fact about atoms? Emission and absorption might be quantised for some reason peculiar to the interaction with the electromagnetic field, while the atom underneath is a perfectly ordinary continuous thing. To rule that out, you need to find the same energies using a probe with nothing to do with light.

A Franck–Hertz tube filled with neon, glowing in several distinct orange bands separated by dark gaps. — A Franck–Hertz tube running on neon, and you can simply see the answer. Each glowing band is a place where electrons have picked up exactly enough energy to excite an atom and have lost it all again. Turn the voltage up and another band appears. The energy is not going in smoothly — it is going in **in equal lumps**, and here the lumps are spaced out along the tube.

James Franck and Gustav Hertz did exactly that in 1914. (Gustav was a nephew of the Heinrich Hertz who has now appeared three times in this course — proving Maxwell right in Act 1, and accidentally discovering the photoelectric effect two lessons ago.) Instead of shining light at atoms, they threw electrons at them. The apparatus is simple to state. A sealed tube of mercury vapour. At one end a hot filament boiling off electrons. Partway along, a grid held at an adjustable positive voltage, which accelerates them. Beyond the grid, a collector plate held slightly negative — so an electron only reaches it if it still has some energy left after crossing. That last detail is what turns the tube into a measuring instrument. The collector counts electrons that survived the journey with energy to spare.

Compare the measured curve with the classical expectation, then switch to neon and watch the spacing change.

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The dip spacing is the physics and is exact — minima fall at multiples of the excitation energy. The overall shape of the curve depends on tube geometry, vapour pressure and mean free path, and is reproduced here illustratively rather than derived. Note also that the measurement is the spacing between dips, not the position of the first: contact potentials between the electrodes shift the whole curve sideways.

And then the pattern repeats — which is what makes it conclusive rather than suggestive. Past 9.8 V, an electron has enough to do the trick twice: excite one atom, get re-accelerated over the remaining distance, and excite another before reaching the grid. Past 14.7 V, three times. The dips continue for as far as the voltage can be pushed, evenly spaced. Evenly spaced is the whole point. Each dip is another identical unit of the same fixed energy being handed over. If atoms absorbed energy in a range, or in amounts that varied, or in decreasing instalments, the dips would drift, blur, or fade. They do none of these. Atoms accept energy in exact, identical, indivisible amounts.

The experiment

Firing electrons through mercury vapour

James Franck and Gustav Hertz · 1914 · Physikalisches Institut, University of Berlin

The question: Do atomic energy levels exist independently of light? If atoms really have discrete internal states, an electron beam should reveal the same energies that spectroscopy does — using no photons at all.
The apparatus: A sealed tube containing mercury vapour, heated to control the pressure. A hot filament supplied electrons; a wire grid at adjustable positive potential accelerated them; a collector plate beyond the grid was held about 0.5 V negative, so only electrons retaining some kinetic energy could reach it. The collector current was measured against the accelerating voltage.

Theory predicted

Classically, a current rising smoothly with voltage. Elastic collisions can remove at most 1 part in 91,000 of an electron’s energy against an atom 400,000 times heavier, so scattering should not produce structure.

They measured

The current rose, then dropped sharply at 4.9 V, rose again and dropped at 9.8 V, and again at 14.7 V — evenly spaced minima continuing as far as the voltage was raised. Below 4.9 V the vapour was dark; above it, the tube emitted ultraviolet light at 253.6 nm and nothing else.

How sure could they be? The spacing was reproducible to better than 0.1 V. Absolute positions are shifted by contact potentials between the electrodes, which is why the measurement is the interval between minima rather than where the first one falls — a point that also matters for the modern teaching version of the experiment.

Why it mattered

The first demonstration of atomic energy levels by a method with no light in it. And the two numbers agree: 4.89 eV from a voltmeter and an electron beam, 4.88 eV from hc/λ of the emitted line. Two entirely unrelated measurements landing on the same quantity is what settled that the levels belong to the atom rather than to its dealings with the electromagnetic field. Franck and Hertz received the 1925 Nobel Prize.

You might think

Franck and Hertz set out to confirm Bohr’s model, and did.

Actually

They did neither. They were investigating the ionisation of gases by electron impact, and interpreted their 4.9 V figure as the ionisation energy of mercury. It is not: mercury ionises at 10.44 eV, more than twice as much. What they had measured was the first excitation energy — a different quantity, and the one that matters. Bohr pointed this out in 1915, observing that 4.9 eV corresponds exactly to the 253.6 nm line their own tube was emitting, and that this was precisely what his theory required. Franck and Hertz resisted the correction for several years. The experiment is one of the cleanest confirmations of quantised atomic energy levels ever performed, and its authors did not know that is what they had done.

A pattern worth noticing

That is now the fourth time in this act: Hertz discovered the photoelectric effect in 1887 and reported it as an unexplained nuisance. Planck derived energy quantisation in 1900 and spent a decade trying to remove it. Millikan confirmed Einstein's equation to 0.5% and called the theory "wholly untenable" in the same paper. Franck and Hertz demonstrated atomic energy levels and thought they had measured an ionisation potential. Experiments are frequently more intelligent than the people performing them. The meaning of a result is often established by somebody who was not in the room, sometimes years later — and the person who did the work may go on disagreeing. This is not a failing of those individuals, all of whom were excellent. It is what genuinely new physics looks like from the inside: you get the right answer before you have any framework in which it makes sense, and the framework is what takes the decade.

1914Franck and Hertz find evenly spaced current dips at 4.9 V intervals in mercury vapour.
1914They interpret 4.9 V as mercury’s ionisation energy. It is not.
1915Bohr identifies it as the first excitation energy, matching the 253.6 nm line.
1919Improved tubes resolve additional, smaller steps corresponding to other levels.
1925Franck and Hertz receive the Nobel Prize — the same year quantum mechanics replaces the model their result supported.

Where this act leaves us

The quantum is now established beyond argument. Energy comes in discrete amounts. Atoms have discrete internal levels, confirmed by two independent methods. Light arrives in packets carrying energy and momentum. Every piece of it measured repeatedly, by sympathetic and hostile hands alike. And none of it is explained. There is no theory here. There is a collection of rules that work, several of which contradict the physics they are bolted onto. Light is a wave and light is a particle. Electrons orbit but decline to radiate, for no stated reason. Planck's constant appears everywhere and nobody can say what it is. What comes next is not another patch. Between 1924 and 1927 the whole structure is rebuilt from the foundations — twice, in two mathematically different forms, by people who disliked each other's approach intensely and were eventually shown to be saying the same thing. And it starts with a doctoral thesis so strange that the examiners, unwilling to pass it and unwilling to fail it, sent a copy to Einstein and asked whether it was nonsense.