---
title: X-Ray Spectra and the Franck-Hertz Experiment
module: Early Atomic Models and the Old Quantum Theory
moduleNumber: 1
lessonNumber: 3
order: 103
summary: >
  Two 1913-14 experiments confirmed the Bohr-Rutherford atom independently of
  optical spectra. Moseley found that the square root of a characteristic X-ray
  frequency is linear in atomic number, fixing Z as nuclear charge and ordering
  the periodic table. Franck and Hertz measured discrete atomic energy levels
  directly by scattering electrons through a mercury vapor.
topics: [Early Atomic Models and the Old Quantum Theory]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 4 — The Nuclear Atom; §4-4 X-Ray Spectra, Auger Electrons"
  - book: Tipler & Llewellyn
    ref: "§4-5 The Franck-Hertz Experiment, A Critique of the Old Quantum Mechanics"
  - book: Tipler & Mosca
    ref: "Ch. 36 — Atoms; §36-3 Energy Quantization in Other Systems"
draft: false
---

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.[^tl-xray]

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.

$$
% caption: An incident electron ejects a K-shell (n=1) electron; an L-shell
% electron dropping into the vacancy emits a K-alpha photon, and an M-shell
% electron emits a K-beta photon. Filling an L-shell vacancy gives the L series.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\fill[acc] (0,0) circle (3pt);
\node[acc, font=\scriptsize, below] at (0,-0.18) {nucleus};
\draw[black!70] (0,0) circle (0.75);
\draw[black!70] (0,0) circle (1.7);
\draw[black!70] (0,0) circle (2.7);
\node[font=\scriptsize, anchor=south] at (0.0,0.75) {K};
\node[font=\scriptsize, anchor=south] at (0.0,1.7) {L};
\node[font=\scriptsize, anchor=south] at (0.0,2.7) {M};
% incident electron ejecting K
\draw[->, black, thick] (-3.4,1.2) -- (-0.6,0.45);
\node[black, font=\scriptsize, anchor=east] at (-3.4,1.25) {fast electron};
% vacancy marker in K
\draw[black, thick] (0.53,0.53) circle (2.4pt);
% K-alpha transition L->K
\draw[->, acc, very thick] (1.2,1.2) -- (0.62,0.62);
\node[acc, font=\scriptsize, anchor=west] at (1.25,1.2) {$K_\alpha$};
% K-beta transition M->K
\draw[->, black, very thick, dashed] (-1.9,1.9) -- (-0.6,0.55);
\node[black, font=\scriptsize, anchor=east] at (-1.95,1.9) {$K_\beta$};
\end{tikzpicture}
$$

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 fit[^tl-moseley]

> **Definition (Moseley's law).** The frequency of a characteristic X-ray line
> obeys
> $$
> \sqrt{f} = A_n\,(Z - b),
> $$
> where $A_n$ is a constant for the line and $b$ is a shielding constant: $b=1$
> for the K series and $b \approx 7.4$ for the L series.

$$
% caption: Moseley plot: the square root of characteristic X-ray frequency is
% linear in atomic number Z. The K series and the L series each fall on a
% straight line, with the L line offset by a larger shielding constant.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (6.6,0) node[right, font=\scriptsize] {atomic number $Z$};
\draw[->, black] (0,0) -- (0,4.6) node[above, font=\scriptsize] {$f^{\frac{1}{2}}$};
\draw[acc, very thick] (0.9,0.3) -- (6.0,4.2);
\node[acc, font=\scriptsize, anchor=west] at (4.6,3.4) {K series};
\draw[black, very thick, dashed] (2.6,0.3) -- (6.0,2.75);
\node[black, font=\scriptsize, anchor=west] at (5.0,2.05) {L series};
\foreach \x in {2,4,6} \draw[black] (\x,0) -- (\x,-0.12) node[below, font=\scriptsize, black] {};
\end{tikzpicture}
$$

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 $n_f = 1$,

$$
f = cR\,(Z-1)^2\left(1 - \frac{1}{n^2}\right),
$$

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 $b \approx 7.4$.

> **Example ($K_\alpha$ of molybdenum).** For molybdenum ($Z=42$), the
> $K_\alpha$ line has $n=2$, $n_f=1$, and $(Z-1)=41$:
> $$
> \frac{1}{\lambda} = R\,(Z-1)^2\left(1 - \frac{1}{4}\right)
> = (1.097 \times 10^7)(41)^2\left(\tfrac34\right)\,\text{m}^{-1},
> $$
> giving $\lambda = 7.23 \times 10^{-11}\,\text{m} = 0.0723\,\text{nm}$, within
> $0.3\%$ of Moseley's measured $0.0721\,\text{nm}$.

## 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.[^tl-periodic]

| Pair | Order by weight | $Z$ from Moseley | Chemistry demands |
| --- | --- | --- | --- |
| Ar, K | K (39.10) before Ar (39.95) | Ar $= 18$, K $= 19$ | Ar inert, K reactive |
| Co, Ni | Co (58.93) before Ni (58.69)? | Co $= 27$, Ni $= 28$ | fixed by $Z$, not weight |

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.[^tl-auger]

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.[^tl-fh]

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 $V_0$.

$$
% caption: The Franck-Hertz tube. Electrons from the cathode accelerate through
% V0 to the grid, then must overcome a small retarding potential to reach the
% plate; the plate current is read against V0 as the tube's vapor is excited.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% cathode
\draw[black, very thick] (0,-1.1) -- (0,1.1);
\node[font=\scriptsize, below] at (0,-1.1) {cathode};
% grid (dashed vertical)
\draw[black, thick, dash pattern=on 2pt off 2pt] (4.0,-1.1) -- (4.0,1.1);
\node[font=\scriptsize, below] at (4.0,-1.1) {grid};
% plate
\draw[black, very thick] (5.4,-1.1) -- (5.4,1.1);
\node[font=\scriptsize, below] at (5.4,-1.1) {plate};
% electron drift
\draw[->, acc, thick] (0.2,0.55) -- (3.8,0.55);
\fill[acc] (1.4,0.55) circle (1.8pt);
\fill[acc] (2.6,0.55) circle (1.8pt);
\node[acc, font=\scriptsize, above] at (2.0,0.6) {electrons};
% labels for potentials
\node[font=\scriptsize, anchor=north] at (2.0,-0.3) {accelerate through $V_0$};
\node[font=\scriptsize, anchor=north] at (4.7,-0.3) {retard};
% ammeter
\draw[black] (5.4,0) -- (6.3,0);
\draw[black] (6.3,0) circle (0.28);
\node[font=\scriptsize] at (6.3,0) {$I$};
\end{tikzpicture}
$$

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
$eV_0$ 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.

$$
% caption: Franck-Hertz plate current versus accelerating voltage. Current rises,
% then drops each time electrons gain just enough energy to excite the atoms;
% for mercury the dips repeat every 4.9 volts.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (7.6,0) node[right, font=\scriptsize] {accelerating voltage};
\draw[->, black] (0,0) -- (0,3.7) node[above, font=\scriptsize] {plate current};
\draw[acc, very thick]
  (0.2,0.1) .. controls (1.0,1.6) and (1.4,2.1) .. (1.8,2.1)
  .. controls (2.0,2.1) and (2.05,0.9) .. (2.3,0.9)
  .. controls (2.9,1.0) and (3.3,2.6) .. (3.7,2.6)
  .. controls (3.9,2.6) and (3.95,1.35) .. (4.2,1.35)
  .. controls (4.8,1.45) and (5.2,3.1) .. (5.6,3.1)
  .. controls (5.8,3.1) and (5.85,1.8) .. (6.1,1.8)
  .. controls (6.6,1.9) and (7.0,3.0) .. (7.3,3.4);
\foreach \x in {2.0,3.85,5.75} \draw[black, dashed] (\x,0) -- (\x,0.15);
\node[black, font=\scriptsize, anchor=north] at (2.0,-0.05) {4.9};
\node[black, font=\scriptsize, anchor=north] at (3.85,-0.05) {9.8};
\node[black, font=\scriptsize, anchor=north] at (5.75,-0.05) {14.7};
\node[font=\scriptsize, anchor=south, black] at (6.9,-0.55) {volts};
\end{tikzpicture}
$$

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:

$$
\lambda = \frac{hc}{eV_0} = \frac{hc}{4.9\,\text{eV}} = 253\,\text{nm}.
$$

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](/atomic-physics/early-models-and-old-quantum-theory/bohr-model-hydrogen)
had inferred from optical spectra.

| Feature | Optical spectra | Franck-Hertz |
| --- | --- | --- |
| Probe | emitted photons | scattered electrons |
| Instrument | spectroscope | voltmeter and ammeter |
| Reads | photon energy $E_i - E_f$ | excitation energy $E_2 - E_1$ |
| Evidence for | discrete transitions | discrete levels |

## 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 $L = n\hbar$. 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](/quantum-mechanics/matter-waves/de-broglie-waves-and-electron-diffraction) and
the [Schrödinger equation](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension).

[^tl-xray]: **Tipler & Llewellyn**, _Modern Physics_, §4-4 — characteristic X-ray production, inner-shell vacancies, and the K/L series notation.
[^tl-moseley]: **Tipler & Llewellyn**, §4-4, Eqs. 4-34 to 4-37 — Moseley's law, the shielding constants, and its derivation from the Bohr theory; Example 4-8.
[^tl-periodic]: **Tipler & Llewellyn**, §4-4 — the reordering of the periodic table by $Z$, the argon-potassium inversion, and the predicted gaps at $Z=43, 61, 75$.
[^tl-auger]: **Tipler & Llewellyn**, §4-4, "Auger Electrons" — the radiationless Auger process and its use in surface analysis.
[^tl-fh]: **Tipler & Llewellyn**, §4-5 — the Franck-Hertz apparatus, the interpretation of the current dips, the $4.9\,\text{V}$ mercury threshold, and the corresponding $253\,\text{nm}$ line.