Probing the atom

Atoms were barely established as real before they turned out to have parts. Thomson found the electron in 1897, before Perrin's Brownian motion work had finished convincing the sceptics that atoms exist, and by 1911 the atom had a nucleus. This lesson follows the experiments that took the uncuttable particle apart.

Rays that would not behave

By the 1890s every well equipped laboratory owned an evacuated tube with an electrode at each end. Raise the voltage across a good vacuum and something streams from the cathode, casting sharp shadows and bending in a magnetic field. These cathode rays bent as a negative charge should, which suggested particles, but Heinrich Hertz had failed to deflect them electrically, which suggested waves in the ether.

Thomson broke the deadlock by realising that Hertz's null result came from residual gas: the rays ionised it, and the ions drifted to the plates and cancelled the field. In a harder vacuum the beam bent electrically. He then sent it through an electric field $E$ and a magnetic field $B$ at right angles, arranged so the deflections opposed, and tuned them until the spot returned to its undeflected position. At balance $q E = q v B$ , so the speed follows without knowing the charge:

v = \frac{E}{B}

Switching the electric field off leaves the magnetic field to bend the beam into an arc of radius $r$ , where $q v B = m v^{2} / r$ , giving $q / m = E / (r B^{2})$ . Thomson found roughly $10^{11}$ C kg⁻¹ (now $1.759 \times 10^{11}$ ) against $9.58 \times 10^{7}$ for the hydrogen ion, until then the largest charge-to-mass ratio known for anything. Nor did the value shift with the gas in the tube or the metal of the cathode. Every kind of matter emitted the same particle at nearly two thousand times the hydrogen ratio, so it belonged to atoms in general rather than to any one element.

Weighing the electron

A ratio is not a mass, and until $e$ was measured on its own the electron stayed half known. Cavendish cloud methods leaked accuracy, since water droplets evaporate while you watch them. Robert Millikan and Harvey Fletcher solved that in 1909 with a mist of low volatility watch oil sprayed between two horizontal brass plates.

The method is a pair of force balances. With no field applied a drop falls at terminal velocity, its weight matched by the Stokes drag $6 π η a v$ , and since the mass is $\frac{4}{3} π a^{3} ρ$ , timing the fall gives the radius and hence the weight. X-rays then ionise the air so the drop captures a few ions, and the field is raised until it hangs still, giving $q E = m g$ .

The jumps were the real result. Charge was never continuous: every drop carried an integer multiple of one basic amount, and every change was a multiple of it. Millikan's $1.592 \times 10^{- 19}$ C sits half a percent below the modern value, an error traced to the viscosity of air he assumed. With Thomson's ratio it gives $m_{e} = 9.11 \times 10^{- 31}$ kg, about $1 / 1836$ of a hydrogen atom.

The plum pudding

Atoms are neutral and contain very light negative particles, so they must also hold positive charge carrying nearly all the mass. Thomson's 1904 model, remembered as the plum pudding, spread that charge as a uniform sphere about $10^{- 10}$ m across, with the electrons embedded in it in rotating rings.

It is easy to sneer at this and quite wrong to do so. Nobody had isolated a light positive particle, so there was no candidate for a compact core. The model was calculable, it made electron rings stable only in particular numbers, and Thomson used that to attempt an account of chemical periodicity. It also matched the small deflections seen when beta and alpha particles crossed thin foils.

A model that is neutral, tractable and consistent with every measurement to hand is a good model. This one was tested and failed, which is the honourable fate of good models, and the test that killed it was set up to check something else entirely.

One alpha in eight thousand

In 1909 Ernest Rutherford handed Hans Geiger and the undergraduate Ernest Marsden a check nobody expected to yield anything. A radium source in a lead block fired alpha particles, helium nuclei of about $7.7$ MeV, at a gold foil some $4 \times 10^{- 7}$ m thick, around two thousand atomic layers. Observers with eyes dark-adapted for half an hour counted flashes on a movable zinc sulphide screen. Marsden looked for alphas coming back towards the source and found them: roughly one in $8000$ turned through more than $90^{\circ}$ .

To see why that is impossible in the Thomson atom, ask how hard a diffuse sphere can push. The field of a uniform ball of charge peaks at its surface, at $Z e / (4 π ε_{0} R^{2})$ with $R \approx 10^{- 10}$ m, and an alpha crossing the atom in $10^{- 19}$ s feels it only briefly. The sideways momentum it gains corresponds to a deflection of a hundredth of a degree, and the electrons, thousands of times lighter, barely move it. Deflections in successive atoms are independent, so they accumulate as a random walk growing with the square root of the layers, and the chance of thousands of tiny kicks conspiring into a reversal is vanishing.

Concentrate the same charge into a ball of radius $10^{- 14}$ m and the alpha can approach ten thousand times closer, where the inverse square law gives it $10^{8}$ times the field, so one encounter can turn it round. Rutherford worked this out in 1911 for a point charge and a pure Coulomb force, predicting that the number scattered into unit solid angle at $θ$ varies as

\frac{d σ}{d Ω} = {(\frac{Z z e^{2}}{16 π ε_{0} E})}^{2} \frac{1}{\sin^{4} (θ / 2)}

Geiger and Marsden spent 1913 testing that prediction. The $\sin^{- 4} (θ / 2)$ factor held from $5^{\circ}$ to $150^{\circ}$ across a count rate varying by $10^{5}$ , and the yield went as the foil thickness, as $1 / E^{2}$ , and as the square of the nuclear charge. That agreement turned a startling anecdote into a measurement.

The same physics bounds the size of the nucleus. Head-on, an alpha stops where all its kinetic energy has become potential energy, at a distance given by $E = 2 Z e^{2} / (4 π ε_{0} d)$ . Taking $e^{2} / 4 π ε_{0} = 1.44$ MeV fm, a $7.7$ MeV alpha on gold stops at $2 (79) (1.44) / 7.7 \approx 30$ fm, an upper limit since it never touched anything. For light elements the scattering later broke away from the Coulomb prediction, showing the alphas had reached something near $10^{- 15}$ m.

The emptiness of matter

A nucleus of $10^{- 15}$ m inside an atom of $10^{- 10}$ m is a ratio of $10^{5}$ in radius and $10^{15}$ in volume. Put a grain of rice on the centre spot of a football stadium and the nearest electrons are up in the tiers.

That grain carries more than $99.9 %$ of the mass, giving nuclear matter a density near $2 \times 10^{17}$ kg m⁻³. A cubic millimetre would weigh two hundred thousand tonnes, and a neutron star is essentially a nucleus the size of a city.

The obvious objection is that matter does not behave as though it were empty, since a table stops your hand. Solidity, though, is electromagnetic rather than geometric. What stops your hand is repulsion between the outer electrons of your skin and those of the table, which fills space even when particles do not. An alpha particle, small and fast and positive, notices only the rare close approach to a nucleus.

The proton and the missing mass

If nuclear charge is $Z e$ , the natural guess is that it is built from hydrogen nuclei, the lightest positive particles known. Rutherford supplied evidence in 1919 by bombarding nitrogen with alphas and detecting hydrogen nuclei emerging, the first artificial transmutation, ¹⁴N + α → ¹⁷O + p. He named the particle the proton the following year.

That created a bookkeeping problem. Helium carries charge $2 e$ but four times the mass of hydrogen, and nitrogen carries $7 e$ with mass $14$ , so nuclei weigh about twice what their charge implies. The stopgap was to add $A - Z$ electrons to $A$ protons, but confining an electron to $10^{- 14}$ m demands tens of MeV, far above anything seen in beta decay, and the measured spin of ¹⁴N came out wrong.

James Chadwick settled it in 1932. Bothe and Becker had found that beryllium struck by polonium alphas emits a penetrating uncharged radiation, and the Joliot-Curies showed it knocks protons out of paraffin wax, calling it gamma radiation. Chadwick objected that a photon would need some $50$ MeV to eject protons of the observed energy, and measured recoils from hydrogen and nitrogen instead. Elastic collision kinematics gave a projectile mass close to the proton's: the neutron, uncharged, of mass $1.0087$ u.

Isotopes and the weighted average

The neutron also explained something chemists had puzzled over. In 1913 Thomson deflected positive rays of neon and found two traces, at masses $20$ and $22$ , from a chemically single gas. Frederick Soddy had already named such variants isotopes, meaning same place in the periodic table, and Francis Aston built a mass spectrograph that eventually catalogued over two hundred. Isotopes share $Z$ , so they are chemically identical, and differ in neutron number, so they differ in mass.

A mass spectrometer refines Thomson's apparatus. Atoms are ionised and accelerated through a potential difference $V$ , so that $\frac{1}{2} m v^{2} = q V$ , then bent by a magnetic field into an arc of radius $r = m v / (q B)$ , which combine to give $m / q = B^{2} r^{2} / 2 V$ . Where an ion lands identifies its mass, and the current it deposits measures how much of it there is.

This is why relative atomic masses are rarely whole numbers. A relative atomic mass is an abundance-weighted average over the isotopes in a natural sample. Chlorine is the standard case: ³⁵Cl has mass $34.969$ u and abundance $75.76 %$ , ³⁷Cl has mass $36.966$ u and abundance $24.24 %$ , and $0.7576 \times 34.969 + 0.2424 \times 36.966 = 35.45$ , the figure on the periodic table. No chlorine atom weighs $35.45$ u: three in four weigh $35$ and one in four weighs $37$ .

Why the nucleus holds

Packing positive charges into $10^{- 15}$ m ought to be impossible. Two protons $2$ fm apart repel with a force of about $58$ N, trivial for a person and colossal for a particle of mass $1.7 \times 10^{- 27}$ kg. Gravity between them is weaker than that repulsion by a factor of $10^{36}$ , so nothing in the forces known in 1932 could hold a nucleus together.

The answer is a further force, the strong nuclear force, attractive between nucleons, indifferent to whether they are protons or neutrons, and about a hundred times stronger than electrostatic repulsion at close quarters. Its defining property is that it dies away almost entirely beyond $2$ fm. Yukawa explained that range in 1935 by proposing an exchange particle whose mass sets the scale, predicting some $150$ MeV; the pion, found in 1947, weighs $140$ MeV.

The short range has consequences you will meet again. Each nucleon binds only to its neighbours, so binding energy grows in step with nucleon number and settles near $8$ MeV each, while Coulomb repulsion is long ranged and every proton pushes every other. Heavy nuclei therefore need surplus neutrons to dilute the repulsion, and the neutron to proton ratio climbs from $1$ in light elements to about $1.5$ in lead. Past bismuth, $Z = 83$ , nothing is stable.

By 1932 the picture looked complete: a dense nucleus of protons and neutrons held by the strong force, with electrons somewhere outside. The trouble is that this atom cannot exist. An orbiting electron is accelerating, classical electromagnetism says an accelerating charge radiates, and the orbit should collapse in about $10^{- 11}$ s. Classical mechanics also allows any orbital radius at all, so it cannot say why every hydrogen atom is the same size, nor why atoms emit sharp spectral lines. Rutherford's nucleus is correct and mechanically impossible at once, and that contradiction is where the next lesson begins.