Inside the Atom: Rutherford's Nucleus

What an atom had to contain

By 1900 two facts about atoms were known. An atom is electrically neutral, and it contains electrons, which are very light and negatively charged. So an atom must also hold an equal positive charge, and almost all of its mass must be somewhere other than in its electrons. The question was where.

J. J. Thomson, who had discovered the electron, offered an answer in 1904. The positive charge fills the whole atom, a sphere about \(10^{-10}\,\text{m}\) across, like the red flesh of a watermelon. The electrons sit inside it like the seeds. In this picture the charge and the mass are spread thinly through the whole atom.

Probing the atom with alpha particles

To test such a picture, you need a probe small and fast enough to enter an atom. Ernest Rutherford chose the alpha particle, which some radioactive atoms emit. It is the nucleus of a helium atom, with charge \(+2e\) and about 7300 times the mass of an electron. It leaves its source with a kinetic energy of a few MeV, at about \(2 \times 10^7\,\text{m/s}\).

In the experiment a narrow beam of alpha particles struck a gold foil only a few hundred atoms thick. A screen coated with zinc sulphide surrounded the foil. Each alpha particle that hit the screen made a tiny flash. An observer in a dark room counted the flashes through a microscope, angle by angle.

The counts showed three things:

1. Almost all the alpha particles passed straight through the foil, or turned by a degree or two.
2. A small fraction turned through large angles. About 1 in 8000 turned through more than \(90°\).
3. A very few came almost straight back towards the source.

Only a nucleus can turn an alpha particle back

Think about what could turn a fast, heavy alpha particle back. The electrons cannot: an alpha particle meeting an electron is like a cricket ball meeting a mosquito. The positive charge of Thomson's atom cannot either. Spread over a whole atom, its push is weak at every point. It could turn an alpha particle by only a small fraction of a degree.

A strong push needs the alpha particle to come very close to a large charge. So Rutherford concluded in 1911 that all the positive charge, and nearly all the mass, sits in a tiny nucleus at the centre. An alpha particle that happens to pass close to a nucleus feels an enormous repulsion and turns sharply. Everything else in the atom is nearly empty space, which is why most alpha particles pass straight through.

The line along which an alpha particle would travel if nothing pushed it passes the nucleus at some distance \(b\), the impact parameter. In the figure above, an alpha particle with \(b = 60\,\text{fm}\) is turned through only \(28°\). One with \(b = 15\,\text{fm}\) is turned through about \(90°\). One aimed straight at the nucleus, \(b = 0\), stops and comes straight back.

Now picture the whole foil. A gold atom is about \(10^{5}\,\text{fm}\) across, so the paths drawn in the figure fill a tiny patch at its centre. An alpha particle aimed at random almost always has a large \(b\) and hardly turns. Only the rare one aimed within a few tens of femtometres of a nucleus turns through a large angle. That is exactly the pattern of the counts.

A head-on approach

Now aim an alpha particle straight at a gold nucleus. The two charges repel, so the alpha particle slows as it comes in. Predict first what happens next.

Two charges \(q_1\) and \(q_2\) a distance \(r\) apart have electric potential energy \(U = kq_1q_2/r\), with \(k = 9 \times 10^9\,\text{N m}^2/\text{C}^2\) and \(U = 0\) far apart. The chapter on electric potential energy derives this; here we only use it. For an alpha particle, \(q_1 = 2e\). For a nucleus of atomic number \(Z\), \(q_2 = Ze\).

In the figure above, the kinetic energy \(K\) falls as the potential energy \(U\) rises, and their sum stays fixed. The alpha particle stops where all of \(K\) has become \(U\). That point is the distance of closest approach, \(d\). Mechanical energy is conserved, as in the chapter on Potential Energy and Conservation of Energy, so one equation gives \(d\).

The product \(ke^2\) turns up in every atomic calculation, so it pays to know it in handy units. In SI units it is \(9 \times 10^9 \times (1.6 \times 10^{-19})^2 = 2.3 \times 10^{-28}\,\text{J m}\). In MeV and femtometres this is exactly \(1.44\,\text{MeV fm}\), which is the same as \(1.44\,\text{eV nm}\).

A distance of closest approach is an upper limit on the size of a nucleus: the alpha particle turned back without touching it. Later measurements put the radius of the gold nucleus at about \(7\,\text{fm}\), about fourteen thousand times smaller than the atom. If the nucleus were a marble \(1\,\text{cm}\) across, the atom would be as wide as a cricket ground.

What keeps the electrons out?

The nucleus attracts the electrons. An electron at rest would simply fall in. So Rutherford pictured the electrons circling the nucleus, as planets circle the Sun, with the electric pull supplying the centripetal force. This is the planetary model of the atom.

Here the planetary picture meets a problem. An electron on a circle has a centripetal acceleration, as in the chapter on Dynamics of Circular Motion. The laws of electricity and magnetism say that an accelerating charge sends out electromagnetic waves. The chapter on electromagnetic waves shows why; here we take it as known. The waves carry away energy.

Follow the chain of Fig 1.1. The electron loses energy, so it falls to a smaller orbit. On a smaller orbit it goes round faster, so it radiates at a higher frequency. The calculation says it reaches the nucleus in about \(10^{-11}\,\text{s}\). So the planetary atom fails twice. It could not last, and on its way in it would give light of every wavelength, a continuous band.

Real atoms do neither. A hydrogen atom lasts for ever if left alone. Hot hydrogen gives light at only a few sharp wavelengths, the red, blue-green and two violet lines of the lower strip. Rutherford's nucleus was right. What was wrong was the classical rule for the electrons around it. The next lesson gives Bohr's repair.