The electrons that pay in fixed amounts

In 1914, in Berlin, James Franck and Gustav Hertz built a glass tube containing a heated cathode, a wire-mesh grid a centimetre or so away and a collecting plate just beyond the grid, filled it with mercury vapour by warming a drop of mercury, and measured the current reaching the plate as they raised the voltage accelerating electrons from the cathode towards the grid. A small reverse voltage between grid and plate meant that only electrons arriving at the grid with more than about a volt and a half of energy could reach the plate.

The current rose with voltage, as it should. At about five volts it fell sharply, almost to nothing. Then it rose again, and at about ten volts it fell again; and again near fifteen, and twenty, and twenty-five — a sawtooth, its dips spaced 4.9 volts apart, continuing as far as they cared to raise the voltage. It was the first experiment to show directly that an atom can take up energy only in fixed amounts. Its authors, for several years, thought it showed something else.

Bounce, or pay in full #

An electron crossing the tube collides with mercury atoms many times; the vapour is dense enough that the distance between collisions is much less than the gap. Most collisions are elastic: the electron bounces off a mercury atom four hundred thousand times its mass, and loses almost no energy. The kick a fast particle can give an electron found that a heavy body can give a light one at most twice its own speed; turned round, a light electron striking a heavy atom can give it at most a fraction $4m/M$ of its energy, which for mercury is about one part in a hundred thousand. An electron can bounce off thousands of atoms and still arrive at the grid with nearly all the energy the voltage gave it.

That changes once the electron has 4.9 electronvolts. Then a collision can be inelastic: the electron gives up exactly 4.886 electronvolts, lifting one of the atom’s outer electrons from its ground level to the first excited level, and carries on with whatever it had left over. It cannot give up 3 electronvolts, or 4, or 4.5: the atom has no level for that energy to go into, as the box that allows only some energies found for any wave confined to a finite space. The atom will accept 4.886 eV or nothing.

The simulation behind the figure contains exactly that rule and nothing else of the atom. Each simulated electron gains energy steadily as it crosses the gap; when it has at least 4.886 eV, it travels on a random distance — the free path for an inelastic collision — and then loses 4.886 eV. At the grid, it reaches the plate if it has more than 1.5 eV left. Below 4.9 volts every electron arrives with all its energy, and the current is full. Just above, every electron gets enough energy just before the grid, gives it away in a collision, and arrives with almost nothing — too little to climb the last 1.5 volts. The current collapses. Raise the voltage a little more and the electrons, after their one collision, regain enough before the grid; the current recovers. At twice 4.9 volts they can afford two collisions and arrive with nothing again.

Why the tube has to be hot #

The experiment needs the electrons to meet many atoms on their way across, but not so many that their energy is spread out by countless small losses before they can reach 4.9 electronvolts. That is a statement about the mean free path, the average distance between collisions, which how far a molecule gets found to depend on how many molecules there are per unit volume and how large a target each presents. For mercury at room temperature the vapour pressure is only a quarter of a pascal, the atoms are sparse, and an electron crosses a centimetre-wide tube with few collisions of any kind; the dips are faint. Heating the tube to about 180 °C raises the vapour pressure several thousand times, bringing the inelastic free path down to a fraction of the gap, which is the regime the simulation assumes.

Too much vapour would be as bad as too little. The elastic collisions, each taking only a hundred-thousandth of the electron’s energy, would at high enough density add up to a steady drag, and they also scatter the electrons sideways, lengthening their paths and blurring the energy at which they arrive. The working temperature is a compromise between enough inelastic collisions to make the dips deep and few enough elastic ones to keep them sharp. Franck and Hertz chose it by trial, as every student repeating the experiment still does.

An electron’s energy on its way across #

The sawtooth in the current is a sawtooth in the electrons’ energy, seen at the end of the tube.

With 15.5 volts across the tube an electron makes three inelastic collisions and arrives with a little under two electronvolts — enough to reach the plate. Each collision costs it exactly the same 4.886 eV; between collisions it climbs steadily under the field. The figure shows the price of the free path: an electron that travels on a little after reaching the threshold arrives at its collision with more than 4.886 eV and keeps the surplus afterwards, so its later collisions come a little sooner. Its total energy budget is unchanged — it still pays 4.886 eV three times — but where the payments happen shifts.

The collisions take place at roughly evenly spaced positions across the tube, a third, two-thirds and the whole way across, wherever the electrons have accumulated another 4.886 eV of energy from the field. In mercury vapour nothing marks those positions; the light emitted is ultraviolet. In neon it can be seen.

Sheets of light in neon #

Neon’s first excitation energies lie near 18.7 electronvolts, four times mercury’s, and when its excited atoms return to the ground state, part of the energy comes out as visible orange-red light.

In a Franck–Hertz tube filled with neon, glowing sheets of orange light hang between the cathode and the grid, perpendicular to the electrons’ path. At about 20 volts there is one, close to the grid, where the electrons first have enough energy to excite neon. At 40 volts there are two: the electrons excite neon about halfway across, lose their energy, regain it, and excite neon again near the grid. Each further 18.7 volts adds another sheet, and the sheets move towards the cathode as the voltage rises, since the electrons reach the excitation energy sooner. The number of sheets is the voltage divided by the excitation energy, rounded down — quantisation that can be counted by eye.

How the free path stretches the spacing #

The simple story says the dips should fall at whole multiples of 4.886 volts. A careful measurement does not quite give that, and the reason is the free path.

If an electron collides the instant it has 4.886 eV, the dips fall at exactly 4.886, 9.772 and so on. If it travels on a mean distance λ before colliding, it reaches the grid with nothing left — the condition for the current to vanish — only when the field is a little stronger, so that the electron covers its whole budget of collisions plus the final delay within the gap. In this model the electron keeps the energy it gains during each delay, so only the last delay matters, and every dip moves out by the same proportion: the spacing becomes $E_1/(1 - \lambda)$. For a free path of five per cent of the gap, the dips are 5.06 volts apart instead of 4.886.

Real Franck–Hertz curves show something more: their dip spacings increase from one dip to the next, by a few tenths of a volt over the first several. The explanation, worked out in 2006 by Rapior, Sengstock and Baev, is that the chance of an inelastic collision is not all-or-nothing at threshold but rises steadily with the electron’s excess energy, so an electron that is barely above threshold may travel some way before exciting an atom, and the delay depends on the field. That rising probability is not in the simulation drawn here, which treats every electron above threshold the same. Both effects warn that the spacing of the dips is a measurement of the excitation energy only after the geometry of the tube is allowed for.

Real curves also start late. The first dip in a real mercury tube appears near seven volts, not five, because the cathode and the grid are made of different metals, and the difference in their work functions — the energies needed to free an electron from each, the quantity the electrons a field pulls from cold metal began from — acts as an extra voltage in the circuit. Only the spacings, not the positions, are free of it.

The light that comes back out #

An excited mercury atom does not keep its energy. It falls back to the ground state within about a hundred nanoseconds and emits the energy as a single photon, of energy 4.886 eV and therefore of wavelength

$$\lambda = \frac{hc}{E_1} = \frac{1239.84\ \text{eV nm}}{4.886\ \text{eV}} \approx 253.7\ \text{nm},$$

in the ultraviolet.

The number of atoms each electron excites is a staircase: none below 4.886 volts, one above, two above twice that. Each excitation produces one ultraviolet photon of the same wavelength, the strongest line in mercury’s spectrum, which had been measured spectroscopically decades earlier. Franck and Hertz checked this in 1914: the vapour emitted the 253.7 nm line only once the voltage exceeded 4.9 volts. The energy an electron loses in a collision reappears as a photon whose energy, by light arriving in lumps of $h\nu$, is exactly the energy lost. The spectrum is a subtraction found that every spectral line is the difference between two levels; here the same difference is measured twice, once with a voltmeter and once with a spectrometer, and the two agree.

What they thought they had found #

The irony is that Franck and Hertz did not at first accept this interpretation. Bohr’s model of the atom had been published in 1913, with its discrete energy levels, but Franck and Hertz were working in a tradition that thought in terms of ionisation — of electrons knocked out of atoms altogether — and they concluded that 4.9 volts was the ionisation energy of mercury. Bohr himself pointed out in 1915 that this could not be right: his model predicted a first excitation energy close to the 4.9 volts they had found, and an ionisation energy roughly twice as large. Measurements soon showed mercury’s ionisation energy to be 10.4 volts, and the dips at 4.9 to be excitations, as Bohr had said.

Franck and Hertz accepted the correction, and in 1925 they shared the Nobel Prize for the experiment. It is often cited as a confirmation of Bohr’s model, which in a sense it was; but it is more fundamental than that. It shows that the internal energy of an atom is quantised by a method that never uses light, so that the quantisation cannot be blamed on anything peculiar to the way atoms emit or absorb radiation. The levels are a property of the atom, visible to any probe that can deliver energy to it.

What the collisions can do that light cannot #

There is one way in which the electrons see more than light does. Mercury’s 4.886 eV level is, strictly, a state of different total spin from the ground state — the two outer electrons have parallel spins instead of opposite ones — and a transition between states of different spin is forbidden for a photon, which cannot flip an electron’s spin in the dipole approximation. The spin a photon has to carry away found the rules that decide which transitions light can drive. In mercury the rule is broken weakly, because the heavy nucleus mixes the spin states, and the 253.7 nm line is strong anyway; in lighter atoms it holds well, and such levels are almost invisible to light.

An electron is under no such restriction. The incoming electron can exchange places with one of the atom’s electrons, leaving the atom with a flipped spin, so electron collisions excite these “forbidden” levels freely. This is why electron-impact spectroscopy — measuring the energy electrons lose in passing through a gas, the modern descendant of the Franck–Hertz tube — finds levels that optical spectroscopy misses, and why gas discharges, in which electrons do the exciting, populate levels that a lamp shining on the same gas would never reach.

The same method, a century on #

Measuring what a probe loses in passing through matter is now one of the standard ways of finding a material’s energy levels. In an electron microscope, electron energy-loss spectroscopy sorts the transmitted electrons by the energy they have given up, and the losses — sharp edges where inner-shell electrons are knocked out, broad peaks where the electrons of the whole solid oscillate together — identify the elements in a region a few atoms across and reveal their chemical bonding. Neutrons scattered from a crystal lose or gain energy in the amounts that create or absorb its quantised vibrations, the phonons that make up the gas of sound that carries a diamond’s heat, and the pattern of losses against scattering angle maps out how those vibrations’ energies depend on their wavelength.

In each case the logic is Franck and Hertz’s: a probe of known energy passes through, and the energies it can lose are only those the target can accept. The probe can be an electron, a neutron, a photon or an atom; the levels it reveals can belong to an atom, a molecule, a crystal or a nucleus. The tube with its sawtooth current was the first instrument of a whole family.

A one-dimensional tube with one level and no space charge #

The simulation is one-dimensional, with a uniform field across the gap and no space charge — no slowing of the electrons by the cloud of other electrons in the tube — and with all elastic collisions ignored; in a real tube, elastic scattering lengthens the electrons’ paths and spreads their arrival energies. The electrons start from rest, where real ones leave a hot cathode with a spread of a few tenths of an electronvolt, which rounds every corner of the current curve. Every collision above threshold is treated alike, as noted above, and only the first excited level is included: at higher voltages, higher levels and eventually ionisation also become available, and the curves of real tubes grow more complicated beyond about 15 volts.

The neon simulation uses a single excitation energy of 18.7 eV, where neon has a cluster of levels between about 16.6 and 19 eV, and it shows where atoms are excited rather than where light is emitted, which is the same place to within a fraction of a millimetre. The free-path figure uses the falling edge of each dip rather than its lowest point, because the simulated dips have flat floors; real dips are rounded and their minima are what experimenters measure.

Still open: what a single collision does #

The Franck–Hertz tube averages over millions of electrons and billions of collisions. Modern experiments reach the single collision: an electron of known energy and direction is fired at an atom, and the scattered electron, the excited atom and the photon it later emits are all detected together, so that the orientation and alignment of the excited atom’s electron cloud can be reconstructed — the shape, not just the energy, of what the collision produced. Such measurements test calculations of electron–atom scattering that matter for plasmas, lighting, the upper atmosphere and the interstellar medium, and for the heavier atoms they still disagree with theory in detail.

What the 1914 tube shows does not depend on those details. An atom can accept energy from a collision only in the amounts that separate its levels; an electron that has less bounces off almost unchanged, and one that has enough pays exactly the gap and keeps the rest. The current through a tube of mercury vapour falls every 4.9 volts because that is the price of the cheapest thing a mercury atom can do — and the light that comes back out has exactly that energy.