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 small deflections when fast positive particles passed through.

A model is a simplified representation that explains selected observations while deliberately leaving out some detail. Older models, then, aren’t just “stupid versions” of newer ones. A hard-sphere atom still works well in parts of gas theory; Thomson’s model was a real advance 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, which are 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 IB, the 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 were deflected 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 turn it back. Rutherford inferred that an atom contains a nucleus, which is a very small, dense, positively charged central region containing nearly all the atom’s mass. The electrons occupy the much larger region outside it, so the atom is mostly empty space.

This is a neat example of evidence changing a model. The key 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, then a model, then better observations that test the model again.

What the symbols count

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

The proton number counts the protons in a nucleus. It is written as $Z$ and identifies the element. The nucleon number counts all the protons and neutrons in a nucleus. It is written as $A$. The neutron number counts the neutrons in a nucleus. It is written as $N$.

Nuclear notation

Nuclear notation is written as ${}^{A}_{Z}X$, where $X$ is the chemical symbol of the element (not a physical quantity). You don't need to recall chemical symbols from memory in this topic; in questions, the relevant symbol or proton number will be supplied or inferable.

For example, ${}^{235}_{92}\text{U}$ 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 is why ${}^{16}_{8}\text{O}$ and ${}^{17}_{8}\text{O}$ 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 range of colours or wavelengths.

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

The key word here is “particular”. A low-pressure gas does not emit or absorb every possible wavelength. It only emits and absorbs certain wavelengths, and the pattern depends on the atom. That gives 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, not 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 something different: only selected energy changes happen. 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 one quantum of electromagnetic radiation, carrying a definite amount of energy. A quantum is a discrete packet of a physical quantity, not an amount that can be divided arbitrarily.

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

Physically, it is more accurate to say the atom changes energy, rather than just “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

During absorption, the incoming photon must have exactly the right energy to match an allowed energy gap. The atom is then in a higher-energy state. Later, it may drop back 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 carries energy in proportion to its frequency:

The Planck constant has a value of about $6.63 \times 10^{-34}\ \text{J s}$.

For an atomic transition,

A bigger energy gap produces a higher-frequency photon; a smaller gap produces a lower-frequency photon.

Using wavelength

For electromagnetic radiation in a vacuum,

Combine this with the photon-energy equation to get:

$E = \frac{hc}{\lambda}$.

An electronvolt is a unit of energy equal to the energy transferred when an electron moves through a potential difference of one volt. Numerically, $1\ \text{eV} = 1.60 \times 10^{-19}\ \text{J}$. Atomic transitions are often given in eV because joules are awkwardly small for these values.

A useful data-booklet value is $hc = 1.99 \times 10^{-25}\ \text{J m} = 1.24 \times 10^{-6}\ \text{eV m}$. 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 start.

Spectra as fingerprints

Every element has its own allowed energy levels. Because of that, every element produces its own pattern of emission and absorption lines. A line spectrum works a bit like a fingerprint: not because one line identifies one element, but because the full arrangement of lines is distinctive.

A flame test shows the basic idea. For a more precise method, a spectrometer separates light from a gas or flame using a diffraction grating or prism, then the line wavelengths are measured. By matching those wavelengths with known patterns, we can identify the chemical composition of the sample.

Stars and astronomical information

The same approach works for stars, since their light contains absorption and emission features from atoms in their outer layers and surrounding gas. Spectra let astronomers infer chemical composition without collecting a sample — useful, given that stars are inconveniently far away.

Emission and absorption spectra can also be used to deduce other stellar properties. The overall continuous spectrum links to surface temperature. Line positions are compared with laboratory wavelengths: shifts towards longer or shorter wavelengths show motion along the line of sight. In cosmology and stellar astronomy, these 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 act roughly as though each nucleon takes up about the same volume. So nuclear volume is proportional to nucleon number.

For a spherical nucleus,

Since volume is proportional to $A$, the radius is proportional to the cube root of $A$:

A typical value is $R_0 \approx 1.2 \times 10^{-15}\,\text{m}$.

Doubling the radius, then, doesn’t 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 huge because almost all of an atom’s mass is packed into its tiny nucleus. The estimate is:

Here, $A$ cancels. In this simple model, all nuclei have about the same density. Using the usual value of $R_0$ gives a density of order $10^{17}$ kg m$^{-3}$. By everyday standards that is absurdly large, but it follows naturally when nucleons are packed into a femtometre-scale volume.

One practical model is to press equal-sized balls of modelling clay into larger spheres, measure their diameters with callipers, and plot 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 assumes the alpha particle and the nucleus repel one another only through the electric force, and that the nucleus behaves like a small charged centre. If the incident kinetic energy is low enough, the alpha particle never gets close enough for other nuclear forces to matter. That is the assumption used in the scattering calculations for this topic.

What changes at high energy

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

The breakdown of the Rutherford prediction is useful, rather than just a problem. The energy where deviations first appear tells us about the effective size of the nucleus: roughly, it shows when the alpha particle has got close enough for nuclear forces to become important.

Energy conservation in a head-on approach

The distance of closest approach is the smallest separation reached 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 first, the alpha particle has kinetic energy. At the closest point, all of that kinetic energy has become electric potential energy:

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

What the estimate means

The value of $r_c$ gives an upper estimate of the nuclear radius or diameter scale. It is not a direct photograph of the nucleus. The calculation assumes the target nucleus does not recoil and that the only interaction is electric repulsion. In this syllabus, scattering energies are chosen so that this electric-repulsion model is the only force you need to include.

Why Bohr was needed

Rutherford’s nuclear atom created a problem for classical physics. 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 one new rule: hydrogen atoms can exist only in certain stationary states. Radiation is emitted or absorbed only when the atom moves between these states.

Hydrogen energy levels

For hydrogen in the Bohr model:

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 isn’t a problem; it shows that the atom is energetically below the separated proton-and-electron state.

Spectral series now fit the picture. 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.

Angular momentum quantization

Angular momentum is a vector quantity that measures rotational motion. For a particle moving in a circular orbit, its magnitude is:

Bohr’s key quantization condition was:

$mvr = \frac{nh}{2\pi}.$

So the angular momentum isn’t free to have just any value. It has to be an integer multiple of $\frac{h}{2\pi}$. With angular momentum restricted like this, only certain orbital radii and energies can exist. That is how quantized angular momentum leads to quantized orbits and quantized energy.

Standing-wave picture

The de Broglie interpretation gives a neat way to picture the same idea. In an allowed orbit, the electron behaves like a wave that fits exactly around the orbit’s circumference.

The standing-wave condition is $2\pi r = n\lambda_\mathrm{d}$. Combining these gives Bohr’s angular momentum condition again.

If the wave fails to join smoothly after one full circuit, it cancels itself instead of forming a stable stationary state. It’s a hand-waving picture, yes, but it gets the main point across: allowed states are chosen by a wave condition, not by classical planetary motion.

Where the Bohr model fails

The Bohr model works very well for hydrogen and other one-electron ions, but it is not a general model of atoms. Multi-electron atoms cause problems because electron–electron interactions change the energy structure. The model also cannot explain why some allowed transitions are more probable than others.

More precise spectroscopy revealed fine structure too: lines that had seemed single were actually closely spaced. Electron spin and spin–orbit interaction, for example, sit outside Bohr’s assumptions. This is a useful nature-of-science lesson: a model can be powerful, predictive and historically essential, yet still be replaced when sharper evidence exposes its limits.