X-Ray Spectra and the Franck-Hertz Experiment

The Bohr model rests on optical spectra, but two experiments carried out just as Bohr published gave the nuclear atom independent support and pushed past what the model could reach. Moseley measured the characteristic X-ray lines of forty elements and found a regularity so clean it redefined the atomic number. Franck and Hertz drove electrons through mercury vapor and read the atom's discrete energy levels off a voltmeter and an ammeter, with no spectroscope at all. Both confirm the picture of a positively charged core surrounded by electrons in quantized states, and both expose the point where the old quantum theory runs out.

Characteristic X-rays and inner shells

When a target is bombarded with fast electrons in an X-ray tube, the emitted spectrum has two parts: a continuous background (bremsstrahlung, from decelerating electrons) and, superimposed on it, sharp characteristic lines whose wavelengths depend only on the target element. The characteristic lines come from transitions of the atom's innermost electrons.¹

A fast electron can knock an inner electron completely out of the atom, leaving a vacancy in a low-$n$ shell. An electron from a higher shell drops into the vacancy and emits a photon whose energy is the difference between the two levels. For a heavy element that difference is large enough to fall in the X-ray range. The shells are labeled by their principal quantum number: $n=1$ is the K shell, $n=2$ the L shell, $n=3$ the M shell.

• $K_{\alpha }$: an $n=2\to n=1$ transition filling a K-shell vacancy, the lowest-energy K line.
• $K_{\beta }$: an $n=3\to n=1$ transition, higher energy than $K_{\alpha }$.
• L series: transitions filling an $n=2$ vacancy, lower energy than the K series.

Inner electrons are well shielded from the outer electrons and from interatomic forces, so their energies depend almost entirely on the nuclear charge, not on the complicated outer-electron structure that makes optical spectra irregular. This is why the X-ray lines vary smoothly from element to element while optical spectra do not.

Moseley's law

Moseley plotted the square root of the frequency of a given characteristic line against atomic number $Z$ and found a straight line for each line in each series. The data fit²

The law follows from the Bohr theory. A $K_{\alpha }$ photon is a one-electron transition to the $n=1$ shell, but the transitioning electron sees the nuclear charge partly screened by the one electron remaining in the K shell, so $Z$ is replaced by $(Z-1)$. Using the Bohr frequency formula with $n_f=1$,

so $\sqrt{f}\propto (Z-1)$, which is Moseley's law with $b=1$. The L series involves transitions to $n=2$, whose electrons sit farther out and see more inner electrons screening the nucleus, giving the larger $b\approx 7.4$.

Ordering the periodic table

Before Moseley, the atomic number was merely an element's position in a table ordered by atomic weight, and that ordering had known errors. Geiger and Marsden's scattering showed the nuclear charge was about $A/2$, and Barkla's X-ray scattering showed the electron count was also about $A/2$, consistent with neutrality, but neither fixed $Z$ exactly. Moseley's law did: an element's $Z$ is whatever integer places it on the $\sqrt{f}$-versus-$Z$ line.³

Ordering by $Z$ instead of weight put argon and potassium in their correct chemical columns. The Moseley plot also exposed gaps at $Z=43,61,75$, predicting undiscovered elements; all three were later found. The atomic number was thereby established as a physical quantity, the nuclear charge, not a bookkeeping index.

The Auger effect

Ejecting an inner electron leaves an ionized atom, and filling the vacancy need not produce a photon. In the Auger effect, the energy released when an outer electron fills the vacancy is transferred to a third electron, which is ejected instead of a photon being emitted. This radiationless path leaves the atom doubly ionized.⁴

If the $K$-vacancy energy $\Delta E=E_2-E_1$ is handed to an $n=3$ electron of binding energy $|E_3|$, that electron leaves with kinetic energy $\Delta E-|E_3|$, a value fixed by the atom's own level structure. Each element therefore has a characteristic Auger electron spectrum, which makes Auger spectroscopy a sensitive surface-analysis tool: it identifies impurities on clean surfaces and detects the small level shifts caused by chemical bonding.

The Franck-Hertz experiment

Moseley confirmed quantized inner-shell energies through the light atoms emit. Franck and Hertz confirmed quantized energy levels without any light at all, by measuring how electrons lose energy in collisions with atoms.⁵

Electrons boil off a heated cathode and accelerate through a potential $V_0$ toward a grid. Past the grid they must climb a small retarding potential $\Delta V$ to reach the plate and register as current $I$. The tube holds a low-pressure vapor, mercury in the original. The measurement is the plate current as a function of the accelerating voltage $V_0$.

An electron colliding with an atom cannot transfer energy unless it carries at least the excitation energy $\Delta E=E_2-E_1$ of the atom's first excited state, because the atom has no level in between. Below that threshold the collisions are elastic: the electron keeps its kinetic energy, overcomes the retarding potential, and contributes to the current, which rises with $V_0$. Once $e{V}_0$ reaches $\Delta E$, an electron can excite an atom, losing $\Delta E$ in a single inelastic collision. Drained of energy near the grid, it can no longer climb the retarding potential, and the current drops sharply.

For mercury the first dip is at $4.9\text{V}$, so the first excited state lies $4.9\text{eV}$ above the ground state. Raising $V_0$ further produces more dips at regular $4.9\text{V}$ intervals: an electron reaccelerated after one inelastic collision reaches $4.9\text{eV}$ again near the grid and loses it once more.

The excited mercury atoms fall back to the ground state and emit a photon of the corresponding energy:

Mercury has exactly this ultraviolet line, and it appears only when $V_0$ exceeds $4.9\text{V}$, closing the argument: the current dips and the emitted line report the same discrete level. Franck and Hertz detected quantized atomic energies with nothing but voltmeters and ammeters, an independent confirmation of the discrete levels the Bohr model had inferred from optical spectra.

Where the old quantum theory ends

The Bohr-Rutherford picture, confirmed from three independent directions — optical spectra, X-ray spectra, and electron collisions — is nonetheless a patchwork of classical orbits with quantum rules bolted on. It works only for one-electron systems, cannot predict line intensities, gives no account of the fine structure Moseley's L lines already showed, and offers no principle behind the quantization $L=n\hslash$. These are not gaps to be filled by more careful bookkeeping; they mark the limit of the old quantum theory. Resolving them requires treating the electron as a wave, the subject of the matter-waves module and the Schrödinger equation.