Experiment: Franck–Hertz
Everything in Atomic Models and Spectra that speaks of atomic energy levels speaks of them indirectly. Spectroscopy measures frequencies, Ritz's principle [Ritz:1908] shows the frequencies are differences of terms, and Bohr's postulates [Bohr:1913a] interpret the terms as energies — but no optical measurement can reach a level except as one end of a difference, and a sceptic in 1913 could hold that the terms were a bookkeeping device. Franck and Hertz's experiment removes the intermediary. It delivers energy to a mercury atom mechanically, by bombarding it with electrons of controlled kinetic energy, and asks at what energy the atom begins to accept any. The answer is: at nothing below \(4.9\,\mathrm{eV}\), and then in units of \(4.9\,\mathrm{eV}\). Discreteness is measured with a voltmeter and an ammeter, in a tube containing no spectroscope at all [Franck:1914a].
The chapter is placed at the end of Part VIII — The Transition to Quantum Physics
because it is the point at which the quantized atom stops being a
model that fits spectra and becomes a mechanical fact. It is also a
lesson in reading one's own data: Franck and Hertz took the
\(4.9\,\mathrm{V}\) threshold to be the ionization potential of
mercury and said so in print, a reading Bohr corrected within a year
[Bohr:1915] and which the experimenters themselves abandoned only
after further work [Davis:1917] [Franck:1920]. That history is set
out here in full rather than tidied away, because the treatise's
standard is that a reader must be able to reconstruct what was
actually claimed and when. When written out, the chapter will use the
structured experiment environment of the house style, with
apparatus, procedure, observations in SI with uncertainties,
interpretation and primary references as separate fields, and the
digitized current–voltage curve deposited in the evidence store as
docs/library/data/Franck\_1914a.csv.
Experiment: Franck–Hertz: all derivations and data analysis of this chapter are pending.
Historical context and the prediction under test
What the Bohr postulates predict for a collision
[Reserved: state the prediction sharply before any apparatus is described. If an atom possesses stationary states [Bohr:1913a], an electron of kinetic energy below the first excitation energy \(E_{1}-E_{0}\) can only scatter elastically, and because the electron is some \(4\times 10^{5}\) times lighter than a mercury atom it then loses a fraction of order \(10^{-5}\) of its energy per collision — that is, nothing measurable. Above that threshold an inelastic channel opens and the electron can lose the whole quantum at once. A classical atom with a continuum of internal energies predicts a gradual onset instead. The two predictions differ qualitatively, which is what makes the experiment decisive.]
What was known in 1913
[Reserved: Franck and Hertz's earlier collision work on noble gases, undertaken to measure ionization potentials and mean free paths, not to test Bohr; the state of the art in electron-impact technique; and the fact — important for the honest history of Section 72.5.2 — that the pair were working within an ionization-potential programme and were not looking for excitation. Bohr's papers [Bohr:1913a] [Bohr:1913b] had appeared months earlier and are not cited in [Franck:1914a].]
Apparatus
The tube
[Reserved: an evacuated glass vessel containing a drop of mercury, held in an oven at about \(450\,\mathrm{K}\) so that the saturated vapour pressure gives an electron mean free path much shorter than the cathode–grid distance — the condition that guarantees many collisions and hence a periodic structure rather than a single step. A heated platinum-wire cathode; a wire-gauze grid at a positive accelerating potential; a collecting anode a short distance beyond it. Geometry, materials, dimensions and the oven's temperature stability to be given as measured, together with the vapour-pressure curve that converts oven temperature to number density.]
Electrical arrangement and the systematics it controls
[Reserved: the accelerating voltage between cathode and grid; the small retarding bias of a few tenths of a volt between grid and anode, which is the essential element of the design — an electron that has just lost its energy in an inelastic collision cannot climb it and is not collected, so the anode current falls. Measurement of the current with a galvanometer. Systematics to be enumerated and bounded: the contact-potential difference between the electrodes, which offsets the whole curve without changing the spacing; space charge; thermionic energy spread of the cathode; and the possibility of double inelastic collisions.]
Procedure
[Reserved: bring the oven to temperature and verify stability; set the retarding bias; sweep the accelerating voltage from zero upwards in steps and record the anode current at each step, repeating in both directions to bound hysteresis; repeat the sweep at several oven temperatures to show that the spacing of the features is independent of vapour pressure while their sharpness is not; repeat with the retarding bias varied to show the features are not an artefact of it. The analysis procedure is to be fixed in advance: locate each current maximum, fit the maxima against their ordinal number, and take the slope as the excitation energy, so that the contact potential appears only in the intercept and drops out of the result.]
Observations
The current–voltage curve
[Reserved: the anode current rises with accelerating voltage, drops abruptly near \(4.9\,\mathrm{V}\), rises again, drops near \(9.8\,\mathrm{V}\), and so on, giving a series of maxima equally spaced by \(4.9\,\mathrm{V}\) [Franck:1914a]; the original work followed the sequence far enough to see several periods. To be tabulated: the accelerating voltage at each maximum with its uncertainty, the fitted spacing, the intercept and its interpretation as a contact potential, and the oven temperature and derived vapour density for each run — all in SI, with the uncertainty budget assembled in the manner of [Taylor:1997] [JCGM:2008].]
In a tube containing mercury vapour, with electrons accelerated from a hot cathode towards a grid and collected beyond it against a small retarding bias, the anode current rises with the accelerating voltage, drops abruptly near \(4.9\,\mathrm{V}\), rises again, drops again near \(9.8\,\mathrm{V}\), and continues so for as many periods as the tube will sustain: the current maxima are equally spaced, and the spacing is \(4.9\,\mathrm{V}\) [Franck:1914a]. The spacing does not depend on the vapour pressure, on the geometry of the tube or on the size of the retarding bias — these govern only how sharp the features are and where the first one falls — so it is a property of the mercury atom and not of the apparatus.
Derivation. Suppose with [Bohr:1913a] that the atom possesses discrete internal energies and that the lowest excitation costs \(\varepsilon=E_{1}-E_{0}\). An electron of kinetic energy below \(\varepsilon\) cannot excite the atom and can only scatter elastically. Because the electron is lighter than a mercury atom by a factor of about \(4\times 10^{5}\), an elastic collision transfers at most the fraction \(4m_{\text{e}}/M\) of its energy, some parts in \(10^{5}\), which is nothing measurable; the electron therefore crosses the tube carrying essentially the whole energy \(eV\) it gained from the field, climbs the retarding bias and is collected. The current rises with \(V\).
When \(eV\) first reaches \(\varepsilon\) — which happens just in front of the grid, where the accelerating potential has been fully traversed — the inelastic channel opens. An electron that takes it is left with almost no kinetic energy, cannot climb the retarding bias between grid and anode, and is not collected: the current falls. Raising \(V\) further, the electron regains energy after its collision and is collected again, so the current recovers; but as soon as it can acquire a further \(\varepsilon\) before reaching the grid it suffers a second inelastic collision and is again stopped. The condition for the \(k\)-th drop is therefore
with \(V_{\text{c}}\) the contact-potential difference between the electrodes, so that successive maxima are equally spaced by
Note what Equation (72.1) says about the systematics: the contact potential enters the intercept and cancels from the spacing, which is why the analysis fits the maxima against their ordinal number and takes the slope rather than reading a single threshold. A short mean free path — that is, a high vapour pressure — is what makes the collisions numerous enough for the structure to be periodic at all, but it does not enter Equation (72.2). A classical atom, whose internal energy could be changed by any amount however small, predicts a gradual onset of energy loss and no periodic structure whatever.
∎The emission line at 253.7 nanometres
[Reserved: the companion measurement, and the one that closes the argument [Franck:1914b]. When and only when the electron energy exceeds the threshold, the vapour emits ultraviolet light at a single wavelength of \(253.7\,\mathrm{nm}\). The photon energy \(hc/\lambda=4.89\,\mathrm{eV}\) agrees with the electrical threshold, so the energy the electron loses is the energy the atom radiates — the mechanical and the optical measurements of the same level interval, made in the same tube. This is the point at which the experiment tests [Bohr:1913a] and not merely the existence of a threshold.]
When, and only when, the electron energy exceeds the threshold of Phenomenon 72.1, the mercury vapour in the tube emits ultraviolet light — and at one wavelength only, \(253.7\,\mathrm{nm}\) [Franck:1914b]. The energy of a single quantum of that light equals, within the precision of the 1914 measurement, the energy the electron is observed to lose. The mechanical and the optical determinations of the same atomic level interval are thus made on the same atoms in the same apparatus, and they agree.
Derivation. By the Planck–Einstein relation the energy of one quantum of light of wavelength \(\lambda\) is \(E=hc/\lambda\). With \(h\) and \(c\) at their present SI values [BIPM:2019] the product \(hc\) is \(1239.84\,\mathrm{eV}\,\mathrm{nm}\), so a wavelength of \(253.7\,\mathrm{nm}\) corresponds to \(4.89\,\mathrm{eV}\). The electrical threshold of Phenomenon 72.1 is \(4.9\,\mathrm{V}\), so by Equation (72.2) the two numbers are the same quantity measured two ways.
The identification is Bohr's postulate [Bohr:1913a] that a transition between stationary states emits a single quantum whose energy is the difference of the two level energies: the electron delivers \(\varepsilon\) to the atom by collision, and the atom returns \(\varepsilon\) to the field as one quantum on decaying. The agreement is what makes this a test of that postulate rather than of discreteness alone — a threshold by itself shows only that the atom accepts energy in a definite amount, whereas the coincidence of the threshold with \(hc/\lambda\) shows that the amount accepted is the energy of a level whose existence the spectrum independently records.
∎Uncertainties and the dataset
[Reserved: statistical uncertainty from the current measurement and from locating a maximum on a rounded peak; systematic contributions from the voltage calibration, the electron energy spread, and residual gas. Present-day teaching-laboratory realizations reach a few parts in a thousand on the spacing, which is enough to resolve the discrepancy discussed in Section 72.6.2. The digitized curve and the fitted maxima are the chapter's dataset; the SI traceability of the voltage scale is that of Measurement, SI Units, and the Theory of Errors and rests on the defining constants of [BIPM:2019].]
Interpretation
Discrete energy transfer
[Reserved: below threshold the electron loses essentially nothing per collision, so it arrives at the grid with the full energy \(eV\); at threshold it can transfer exactly one quantum and is then stopped by the retarding bias; beyond that it re-accelerates and can transfer a second quantum after a further \(4.9\,\mathrm{V}\), which is why the structure is periodic rather than a single edge. The spacing is therefore an atomic energy interval measured in volts, and its independence of vapour pressure, of tube geometry and of the retarding bias is what identifies it as a property of the mercury atom. This is the phenomenon environment the chapter will carry, with its derivation inline.]
What the authors concluded, and why it was wrong
[Reserved: Franck and Hertz reported \(4.9\,\mathrm{V}\) as the ionization potential of mercury [Franck:1914a], on the reasoning that a sudden onset of energy loss meant an electron had been torn off, and they explicitly denied that their result supported Bohr. The reading is wrong: the true first ionization potential of mercury is \(10.44\,\mathrm{V}\), and \(4.9\,\mathrm{eV}\) is the excitation energy of the lowest triplet state that radiates at \(253.7\,\mathrm{nm}\). The section states the error plainly and identifies what would have settled it at the time — the resonance line of Section 72.4.2 was in their own data.]
Bohr's correction and its experimental confirmation
[Reserved: Bohr's reinterpretation [Bohr:1915], which identified the threshold as a resonance potential — an excitation, not an ionization — and noted that the emitted line is exactly what his postulates require of a return to the ground state. Independent confirmation by Davis and Goucher, who separated excitation from ionization in the same vapour by collecting positive ions [Davis:1917], and by Franck and Einsporn's refined measurement [Franck:1920], which resolved further excitation potentials and in which Franck accepted the correction. The episode is a case study for Epistemology and the Scientific Method: a correct measurement, a wrong inference, and the further measurement that decided between them.]
Modern repetitions and precision
The neon variant
[Reserved: neon replaces mercury in most teaching laboratories — no oven is needed and the excitation region is visible as a sequence of orange glow layers that migrate towards the cathode as the voltage is raised, so the periodic structure can be seen as well as plotted. The spacing corresponds to excitation of the neon levels near \(18.7\,\mathrm{eV}\), and because several close-lying levels contribute, the measured spacing depends on the current and on the geometry in a way that the mercury curve does not — a systematic studied and quantified by Rapior, Sengstock and Baev [Rapior:2006].]
With neon in place of mercury the current–voltage curve shows the same periodic structure, with a spacing near \(18.7\,\mathrm{V}\), and the excitation regions are directly visible: the tube glows in a sequence of orange layers, one more appearing for each further period, each layer migrating towards the cathode as the accelerating voltage is raised. The periodicity can therefore be seen as well as plotted. Because several close-lying neon levels are accessible at once, the measured spacing depends on the discharge current and on the electrode geometry in a way that the mercury spacing does not [Rapior:2006].
Why the glow layers sit where they do: an electron starting from rest at the cathode reaches the excitation energy after falling through a fixed potential difference, so the luminous shells mark equal potential intervals, and their migration towards the cathode as the voltage is raised follows from the field distribution in the tube. The dependence of the effective spacing on current and geometry requires in addition the branching among the accessible neon levels and the collision cross-section of each.
Fine structure of the mercury curve
[Reserved: the canonical figure hides real structure. Mercury has a metastable level at \(4.67\,\mathrm{eV}\) as well as the radiating one at \(4.89\,\mathrm{eV}\), and Franck and Einsporn resolved both [Franck:1920]; higher-resolution modern curves show additional steps and a spacing that drifts with the number of the maximum [Rapior:2006]. The honest statement is that “the” Franck–Hertz spacing is an effective quantity averaging over accessible levels, and that agreement to three figures with a single level energy is a coincidence of the usual apparatus, not a property of the method.]
The canonical mercury curve hides real structure. Mercury has a metastable level at \(4.67\,\mathrm{eV}\) as well as the radiating one at \(4.89\,\mathrm{eV}\), and electrons excite both; Franck and Einsporn resolved the two, together with further excitation potentials above them [Franck:1920]. Higher-resolution modern curves show additional steps, and a spacing that drifts systematically with the ordinal number of the maximum rather than staying constant [Rapior:2006]. What the experiment measures is therefore an effective interval averaging over the levels the electrons can reach in the particular tube; its agreement to three figures with a single level energy [Kramida:2023] is a property of the usual apparatus, not of the method.
The effective spacing: the excitation cross-sections of the competing mercury levels as functions of electron energy, and the weighted mean interval they produce in a tube of given vapour density and geometry; and the mechanism of the drift with maximum number, in which electrons surviving several accelerating stages sample a different mixture of levels from those excited near the first drop.
Present-day level energies
[Reserved: the mercury level energies as now known from precision spectroscopy [Kramida:2023], quoted to the digits the experiment can be compared against, together with the current values of \(h\) and \(e\) [Mohr:2025] [BIPM:2019]; the electrical measurement of an atomic energy interval is today a demonstration rather than a determination, since the spectroscopic route (Experiment: Precision Spectroscopy and Atomic Clocks) is better by many orders of magnitude. A short table comparing the 1914 value, the 1920 refinement and the modern value belongs here.]
What the experiment settles
[Reserved: closing assessment. The experiment establishes, by a purely mechanical measurement, that an atom accepts energy only in discrete amounts, and it ties those amounts to the emitted spectrum; it does not establish anything about orbits, and the Bohr model's mechanical picture is not tested by it — which is why the result survives the model's abandonment intact and constrains The Postulates of Quantum Mechanics and Atoms and Molecules directly. Franck and Hertz received the Nobel Prize in Physics for 1925, awarded in 1926, and Franck's Nobel lecture [Franck:1926b] is also the clearest statement of how he came to accept Bohr's reading. Cross-references to the other experiments of this part (Experiment: Stern–Gerlach and Experiment: Wave Optics) and to the inelastic-collision formalism of Scattering Theory.]