The nuclear atom

Thomson's model

Once Thomson had shown that atoms contain tiny negative electrons, and atoms are neutral overall, there had to be positive charge somewhere. In 1904 he proposed that the atom is a sphere of evenly spread positive charge, about $10^{-10}$ m across, with electrons embedded in it like plums in a pudding. Most of the atom's mass was assumed to be in the positive sphere. The model was simple and fitted the evidence available at the time.

The Geiger–Marsden experiment

In 1909, working under Rutherford in Manchester, Hans Geiger and an undergraduate, Ernest Marsden, fired alpha particles from a radium source at a very thin sheet of gold foil, in a vacuum. Alpha particles are helium nuclei: positively charged (+2e), about 7300 times as massive as an electron, and moving at about 5% of the speed of light. They detected where the alphas went by watching tiny flashes of light (scintillations) on a zinc sulfide screen through a microscope, which could be moved around the foil to any angle.

The results:

• Almost all the alpha particles went straight through, or were deflected by less than a degree or so.
• A small number were deflected through large angles.
• About 1 in 8000 was deflected by more than 90°, bouncing back towards the source.

In Thomson's model, the positive charge is spread out, so the electric field inside the atom is weak. An alpha particle should be deflected only very slightly by each atom, and the chance of a large deflection, even from many small ones adding up, is vanishingly small. Seeing 1 in 8000 bounce back meant the model was wrong.

Rutherford's nuclear model

In 1911 Rutherford explained the results. The positive charge and almost all the mass of an atom must be concentrated in a tiny central nucleus. Then:

• Most alphas pass through the empty space far from any nucleus and are hardly deflected.
• The few that pass close to a nucleus feel a very strong electrostatic repulsion (the field is intense because the charge is concentrated) and are deflected through large angles.
• Only those heading almost directly at a nucleus bounce back. Because the target area is so small, this is rare.
• Electrons, being so light, have negligible effect on the massive alpha particle.

Rutherford derived how the number of alphas scattered should depend on the angle, the alpha energy and the foil thickness, assuming only a Coulomb force from a point charge. Geiger and Marsden tested the predictions over the next two years and they matched, strong evidence for the model. In it, electrons orbit the nucleus like planets around the Sun, held by electrostatic attraction.

How small is the nucleus?

An alpha particle aimed straight at a nucleus slows down as it approaches, stops momentarily, and is pushed back. At the turning point all its kinetic energy has become electrical potential energy. The distance of closest approach, $d$, therefore gives an upper limit to the size of the nucleus:

$$K = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{d} \quad\Rightarrow\quad d = \frac{1}{4\pi\varepsilon_0}\frac{(2e)(Ze)}{K}$$

(The expression for electrical potential energy goes beyond the HSC data sheet, but the method uses only energy conservation.)

Limitations of Rutherford's model

The nuclear model explained scattering beautifully, but it had serious problems:

• Atoms should collapse. An orbiting electron is constantly accelerating (centripetal acceleration). Maxwell's theory says an accelerating charge radiates electromagnetic waves, losing energy. The electron should spiral into the nucleus in about $10^{-11}$ s. Atoms are clearly stable.
• Line spectra are unexplained. A spiralling electron would give out a continuous range of frequencies. Atoms actually emit and absorb only particular wavelengths.
• The nucleus's mass doesn't add up. A helium nucleus has twice the charge of a hydrogen nucleus but four times the mass. The model said nothing about what made up the extra mass, or what held the protons together against their electrostatic repulsion.
• No arrangement of electrons. It gave no reason for the chemical properties of elements or the periodic table.

The first two were tackled by Bohr in 1913 (page 8H). The third was solved by the discovery of the neutron.

Chadwick and the neutron

Rutherford proposed in 1920 that the nucleus also contained a neutral particle about the mass of a proton. It was hard to find: with no charge, it doesn't ionise gas or get deflected by fields, so ordinary detectors can't see it.

In 1930, Bothe and Becker found that alpha particles striking beryllium produced a very penetrating, uncharged radiation. They assumed it was high-energy gamma rays. In 1932 Irène Joliot-Curie and Frédéric Joliot showed this radiation knocked protons out of paraffin wax (which is rich in hydrogen) with energies up to about 5.7 MeV. For a gamma ray to give a proton that much energy, it would need about 50 MeV, far more than the reaction could supply.

James Chadwick, at the Cavendish Laboratory in Cambridge, repeated the experiment and also aimed the radiation at other targets, such as nitrogen gas, measuring the recoil speeds. He argued:

• The radiation is not deflected by electric or magnetic fields, so it is neutral.
• If it were photons, the recoil energies of protons and nitrogen nuclei would not fit with conservation of energy and momentum for any single photon energy.
• If it were a neutral particle of mass $m$, colliding elastically, both recoils fit, and the mass came out at about the mass of a proton.

He concluded that the radiation was neutrons, produced by $^{4}_{2}\text{He} + {}^{9}_{4}\text{Be} \rightarrow {}^{12}_{\ 6}\text{C} + {}^{1}_{0}n$, and won the 1935 Nobel Prize. The nucleus was now understood as protons and neutrons (nucleons), with the mass number $A$ counting both.