From Rutherford's nucleus to Bohr's atom

The gold-foil evidence

In 1909 Hans Geiger and the 20-year-old student Ernest Marsden worked in Ernest Rutherford's laboratory in Manchester. They shot alpha particles, fast and heavy positive particles from a radioactive source, at very thin metal foils. Each alpha particle that hit a zinc sulfide screen made a tiny flash of light. In a darkened room they counted the flashes through a microscope, for hours at a time.

Most alpha particles went through the foil and were turned aside by only a small angle. But a few bounced back: from a thin platinum foil, about one alpha particle in 8000 was turned through more than $90^\circ$. Nobody had expected this. Rutherford later said that it was "almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you".

The spectrum of hydrogen

In 1885 the Swiss teacher Johann Balmer found a simple formula that gives these four wavelengths, but nobody could say why it worked. Every element has its own pattern of lines, like a fingerprint. At the Fotoni lab visit you will look at the spectra of lamps and gas tubes yourself, at the station on spectra (unit Fotoni lab visit: modern physics experiments).

Key idea

Rutherford's atom has a tiny, heavy, positive nucleus. The nucleus of gold has a radius of about $7\ \text{fm}$ ($1\ \text{fm} = 10^{-15}\ \text{m}$), while the gold atom has a radius of about $144\ \text{pm} = 1.44 \cdot 10^{-10}\ \text{m}$, about $20\,000$ times more. The atom is almost empty space, and the light electrons move around the nucleus.

But this atom raised a new problem. An electron moving round the nucleus is always accelerating, and classical physics says that an accelerating charge sends out electromagnetic radiation. The electron should lose energy, spiral into the nucleus and send out a continuous spectrum on the way. Real atoms are stable and give sharp lines. Classical physics had failed again.

Bohr's quantum atom

In 1913 the Danish physicist Niels Bohr, who had worked with Rutherford in Manchester, combined the nuclear atom with Planck's and Einstein's quanta. His model of hydrogen rests on three ideas:

1. The electron can move only in certain allowed orbits, the stationary states. In them it does not radiate. Each state has its own energy level $E_n$, numbered by $n = 1, 2, 3, \ldots$
2. The electron can jump from a higher level to a lower one. The atom then sends out one photon that carries away the difference in energy.
3. A photon can be absorbed only if its energy is exactly the difference between two levels.

Bohr's model gives the hydrogen lines very accurately, but it fails for atoms with more than one electron, and its electrons in neat circular orbits are not the whole truth. In the 1920s quantum mechanics replaced the orbits with waves and probabilities: the topic of the next unit. The idea of energy levels survived.

Energy levels of hydrogen

The energy levels of the hydrogen atom are

$$E_n = -\frac{13.6\ \text{eV}}{n^2}, \qquad n = 1, 2, 3, \ldots$$

The energy is negative, because the electron is bound: $13.6\ \text{eV}$ is needed to free it from the lowest level $n = 1$, the ground state. In a jump from level $n_\text{u}$ (upper) to $n_\text{l}$ (lower), energy is conserved:

$$hf = \frac{hc}{\lambda} = E_{n_\text{u}} - E_{n_\text{l}}$$

A useful product for these calculations is $hc = 1240\ \text{eV} \cdot \text{nm}$.

Worked example

Common misconception

"The atom is a tiny solar system." Bohr's circular orbits look like planets around the Sun, but electrons do not have paths like planets. The model was a bridge: it got the energy levels of hydrogen right, but quantum mechanics describes the electron with a wave and probabilities, not with a track.

A second mistake is to think that the wavelength of a line comes from one level alone. A photon carries the difference between two levels, $hf = E_{n_\text{u}} - E_{n_\text{l}}$, so you always need two levels. A third is to think that the electric repulsion stops the alpha particles in Thomson's atom too: there the positive charge was spread out, so the force on an alpha particle was never strong enough to turn it back.

Summary

• The Geiger–Marsden experiment (1909) showed that a few alpha particles bounce back from a thin foil. Rutherford (1911) explained this with a tiny, positive, heavy nucleus; the atom is mostly empty.
• Classical physics could not explain why such an atom is stable or why gases give line spectra.
• Bohr's model (1913): the electron has only certain energy levels, and a jump between two levels sends out one photon, $hf = E_{n_\text{u}} - E_{n_\text{l}}$.
• For hydrogen, $E_n = -13.6\ \text{eV}/n^2$. The visible lines are jumps to $n = 2$: $656$, $486$, $434$ and $410\ \text{nm}$.
• The nucleus is tiny and positive, so two nuclei repel each other strongly: this is the first obstacle to fusion.