From Thomson’s atom to Rutherford’s atom

An atom is a physical system with a tiny central nucleus and electrons around it. In Thomson’s earlier “plum-pudding” model, negative electrons sat inside a spread-out positive charge. The model was useful because it gave the atom internal structure, but it only predicted gentle deflections of fast positive particles.

A model is a simplified representation that explains selected observations while deliberately leaving out some detail. So older models aren’t just “stupid versions” of newer ones. A hard-sphere atom still helps in parts of gas theory; Thomson’s model was a genuine step forward once electrons had been discovered. Rutherford’s model is still the starting picture for nuclear structure.

The experiment

The Geiger–Marsden–Rutherford experiment fired alpha particles, positively charged helium nuclei, at a very thin gold foil in a vacuum. Around the foil, a fluorescent screen gave tiny flashes when alpha particles struck it. For this experiment, the required treatment is qualitative: know the set-up, the observations and the inference, not the full scattering equation.

Most alpha particles went straight through the foil, or changed direction only slightly. A very small fraction turned through large angles, and a few even came back towards the source. That rare back-scattering was the crucial surprise.

The conclusion

A spread-out positive charge could not repel a fast alpha particle strongly enough to reverse it. Rutherford inferred that an atom contains a nucleus, a very small, dense, positively charged central region containing nearly all the atom’s mass. The electrons take up the much larger region outside it, which leaves the atom mostly empty space.

This is a neat example of evidence forcing a model to change. The observation was not “one lucky flash”; it came from painstaking counting of many scintillations. Modern atomic and nuclear models have developed in the same spirit: observations first, model second, then better observations to test the model again.

What the symbols count

A proton is a positively charged nucleon in the nucleus. A neutron is an uncharged nucleon in the nucleus. A nucleon means any nuclear particle that is either a proton or a neutron. In a neutral atom, the number of electrons equals the number of protons, since electron and proton charges have equal magnitude and opposite sign.

The proton number is the number of protons in a nucleus. It is written as Z, and it identifies the element. The nucleon number is the total number of protons and neutrons in a nucleus, written as A. The neutron number is the number of neutrons in a nucleus, written as N.

A = Z + N, where A is the nucleon number (dimensionless count), Z is the proton number (dimensionless count), and N is the neutron number (dimensionless count).

Nuclear notation

Nuclear notation is written as ^A_ZX, where X is the chemical symbol of the element, not a physical quantity. You aren’t expected to recall chemical symbols from memory in this topic; in questions, the relevant symbol or proton number will be supplied or inferable.

For example, ^235_92U has 92 protons and 235 nucleons, so it has 143 neutrons. The same element can have different nucleon numbers because its neutron number can vary. That’s why ^16_8O and ^17_8O have the same proton number but different neutron numbers.

Continuous spectra and line spectra

A spectrum is an ordered display of electromagnetic radiation separated by wavelength or frequency. Hot dense matter, such as a solid filament or dense gas, usually gives a continuous spectrum: a smooth spread of colours or wavelengths.

A hot low-pressure gas produces an emission line spectrum, with bright lines at particular wavelengths on a dark background. Put a cooler low-pressure gas in front of a continuous source and it produces an absorption line spectrum, a continuous spectrum crossed by dark lines at particular wavelengths.

The key word is “particular”. A low-pressure gas does not emit or absorb every possible wavelength. It emits and absorbs only certain wavelengths, and the pattern depends on the atom. This is evidence that atoms can have only certain allowed internal energies.

Why line spectra imply discrete energy levels

An energy level is an allowed value of the total energy of an atom. A discrete quantity can take only separated values, rather than any value in a continuous range.

If atomic energies were continuous, an atom could change energy by any amount, so the spectrum would be continuous. Line spectra show the opposite: only selected energy changes occur. The dark lines in an absorption spectrum and the bright lines in an emission spectrum line up because both processes involve the same allowed energy gaps, just in opposite directions.

Photons and transitions

A photon is a quantum of electromagnetic radiation with a definite amount of energy. A quantum is a discrete packet of a physical quantity, not an amount that can be divided up arbitrarily.

An atomic transition happens when an atom changes from one allowed energy level to another. If the atom drops from a higher energy level to a lower one, it emits a photon. If it goes from a lower energy level to a higher one, it absorbs a photon.

A better physics description is that the atom changes energy, rather than just saying “the electron changes energy”. In hydrogen, the atom is a proton–electron system; the energy level belongs to the whole bound system.

Absorption followed by re-emission

For absorption to occur, the incoming photon must have exactly the right energy to match an allowed energy gap. The atom is then left in a higher-energy state. Later, it may return to a lower level and emit a photon. That emitted photon does not have to travel in the original direction, which is why absorption lines look dark when viewed along the original beam.

Energy carried by one photon

A photon’s energy is proportional to its frequency:

E = h**f, where E is the energy of one photon (J), h is the Planck constant (J s), and f is the photon frequency (Hz). The Planck constant is about 6.63 × 10⁻³⁴ J s.

For an atomic transition, ΔE = E = h**f, where ΔE is the magnitude of the energy difference between the two atomic energy levels (J). A larger energy gap produces a higher-frequency photon; a smaller energy gap produces a lower-frequency photon.

Using wavelength

For electromagnetic radiation in a vacuum, c = fλ, where c is the speed of electromagnetic waves in a vacuum (m s⁻¹) and λ is the wavelength of the radiation (m). Combine this with the photon-energy equation to get:

E = hc / λ.

An electronvolt is a unit of energy equal to the energy transferred when an electron moves through a potential difference of one volt. Numerically, 1 eV = 1.60 × 10⁻¹⁹ J. Atomic transitions are often given in eV because joules are awkwardly small for these energies.

A useful data-booklet value is hc = 1.99 × 10⁻²⁵ J m = 1.24 × 10⁻⁶ eV m. If an energy gap is given in eV, use hc in eV m and the wavelength comes out directly in metres. Keep the units consistent; that’s where most mistakes in these calculations begin.

Spectra as fingerprints

Each element has its own set of allowed energy levels, which gives it a particular pattern of emission and absorption lines. A line spectrum acts like a fingerprint: not one line for each element, but a distinctive arrangement of lines taken together.

A flame test is the simple version of this idea. For more precision, a spectrometer is used: light from a gas or flame is split by a diffraction grating or prism, then the line wavelengths are measured. Comparing those wavelengths with known patterns identifies the chemical composition of the sample.

Stars and astronomical information

The same approach works for stars. Their light contains absorption and emission features from atoms in the outer layers and in the gas around them, so astronomers can infer chemical composition without collecting a sample — handy, since stars are inconveniently far away.

Spectra also help astronomers deduce other stellar properties. The overall continuous spectrum is linked to surface temperature. Line positions can be checked against laboratory wavelengths: shifts towards longer or shorter wavelengths show motion along the line of sight. In cosmology and stellar astronomy, those shifts can be combined with other relationships, such as luminosity and redshift methods, to estimate distances and velocities of celestial bodies.

Radius depends on nucleon number

Experiments suggest that nuclei behave roughly as though each nucleon takes up about the same volume. So, the nuclear volume is proportional to the nucleon number.

For a spherical nucleus, V = (4/3)πR³, where V is the nuclear volume (m³) and R is the nuclear radius (m). Since volume is proportional to A, the radius is proportional to the cube root of A:

R = R₀A^(1/3), where R₀ is the nuclear radius constant (m). A typical value is R₀ ≈ 1.2 × 10⁻¹⁵ m.

Doubling the radius, then, does not double the nucleon number. It makes the volume eight times larger, so it matches roughly eight times as many nucleons.

Nuclear density

Nuclear density is enormous because almost all of an atom’s mass sits inside the tiny nucleus. The density can be estimated as:

ρₙ = (A**u) / V = 3u / (4πR₀³), where ρₙ is nuclear density (kg m⁻³) and u is the unified atomic mass unit (kg), approximately 1.66 × 10⁻²⁷ kg.

Here, A cancels. In this simple model, all nuclei have roughly the same density. Using the usual value of R₀ gives a density of order 10¹⁷ kg m⁻³. By everyday standards, that is absurdly large, but it follows naturally from packing nucleons into a femtometre-scale volume.

One practical way to model the relationship is to press equal-sized balls of modelling clay into larger spheres, measure their diameters with callipers, and graph the number of balls against radius cubed. If the model has the same scaling, the graph is approximately linear. It’s a crude model, but the scaling idea is the point.

When Rutherford scattering works

Rutherford scattering treats the alpha particle and the nucleus as repelling each other only through the electric force, with the nucleus modelled as a small charged centre. If the incident kinetic energy is low enough, the alpha particle never gets close enough for other nuclear forces to have an effect. That is the assumption used in scattering calculations for this topic.

What changes at high energy

With higher incident kinetic energies, alpha particles can get much closer to the nucleus. At some point, the observed scattering stops matching the Rutherford prediction. At these very small separations, the alpha particle feels the strong nuclear interaction as well as electric repulsion.

This failure of the Rutherford prediction is actually useful. The energy where the deviations first appear gives information about the effective size of the nucleus: it shows roughly when the alpha particle has come close enough for nuclear forces to matter.

Energy conservation in a head-on approach

The distance of closest approach is the smallest separation between an incoming alpha particle and a target nucleus in a head-on scattering event. In the ideal head-on case, the alpha particle briefly stops, then is repelled back along the same path.

At the start, the alpha particle has kinetic energy. At the closest point, all of that kinetic energy has become electric potential energy:

Eα = ½mαvα² = kqαQ / r𝚌, where Eα is the initial kinetic energy of the alpha particle (J), mα is the alpha-particle mass (kg), vα is the initial alpha-particle speed (m s⁻¹), k is the Coulomb constant (N m² C⁻²), qα is the alpha-particle charge (C), Q is the target nuclear charge (C), and r𝚌 is the distance of closest approach (m).

For an alpha particle, qα = 2e, where e is the elementary charge (C). For a target nucleus, Q = Ze. Substituting these gives:

r𝚌 = 2kZee / Eα = 2kZe² / Eα.

What the estimate means

The value of r𝚌 gives an upper estimate of the nuclear radius or diameter scale; it is not a direct photograph of the nucleus. This simple calculation assumes the target nucleus does not recoil and that the particles interact only through electric repulsion. In this syllabus, the scattering energies are chosen so this electric-repulsion model is the only force you need to include.

Why Bohr was needed

Rutherford’s nuclear atom had a classical problem. If an electron orbited a proton like a tiny planet, it would be accelerating, and an accelerating charge should radiate energy continuously. The electron would then spiral into the nucleus. Stable hydrogen atoms clearly don’t do that.

Bohr kept Rutherford’s nucleus, but added a new idea: hydrogen atoms can exist only in certain stationary states, and radiation is emitted or absorbed only when the atom moves between these states.

Hydrogen energy levels

For hydrogen in the Bohr model:

Eₙ = −13.6 / n² eV, where Eₙ is the total energy of the hydrogen atom in level n (eV) and n is the principal quantum number (dimensionless positive integer).

The ground state is the lowest allowed energy state of an atom. For hydrogen, this is n = 1. An excited state is an allowed energy state above the ground state. The zero-energy level represents ionization: the electron and proton are no longer bound together.

A bound state is a state of a system in which energy must be supplied to separate the parts to infinity. On this scale, bound atomic states have negative total energy. The negative sign is fine; it says the atom has less energy than the separated proton-and-electron state.

Spectral series now fit the model. Transitions down to n = 1 produce ultraviolet lines; transitions down to n = 2 produce the visible Balmer lines; transitions down to n = 3 produce infrared lines. The old empirical patterns in hydrogen spectra were clues to these allowed energies.