Inside the Atomic Nucleus: Size, Density and the Strong Force

At the centre of every atom sits the nucleus: less than a hundred-thousandth of the atom’s width, yet holding more than 99.9% of its mass. It’s where the element’s identity is fixed, where radioactivity comes from, and where the energy of stars and nuclear power stations is released. This article takes you inside the nucleus: how it was discovered, how big and dense it is, what holds it together, and why some nuclei fall apart.

Discovery: Rutherford’s gold foil experiment

In 1909, Hans Geiger and Ernest Marsden, working with Ernest Rutherford in Manchester, fired alpha particles at a very thin sheet of gold foil. The expectation, based on J.J. Thomson’s “plum pudding” model (positive charge spread evenly through the atom), was that the alpha particles would pass through with only slight deflections.

Most did. But a tiny fraction, roughly 1 in 8,000, bounced back at large angles. Rutherford later said it was “as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you”.

In 1911, he explained the result: all the positive charge and nearly all the mass must be concentrated in a tiny central nucleus. Most alpha particles miss it completely; the rare ones that come close are strongly repelled. See the history of atomic models.

What’s in a nucleus

The nucleus contains nucleons:

• protons: positive, number = atomic number Z (see the proton)
• neutrons: neutral, number = A − Z (see the neutron)

The total number of nucleons is the mass number A. See mass number explained.

How big is a nucleus?

Nuclei are measured in femtometres (fm): 1 fm = 10⁻¹⁵ m. Experiments scattering high-energy electrons from nuclei show that the radius follows a simple rule:

r ≈ r₀ × A^(1/3), with r₀ ≈ 1.2 fm

Nucleus	A	Radius (approx.)
hydrogen-1	1	1.2 fm (the proton’s charge radius is about 0.84 fm)
carbon-12	12	2.7 fm
iron-56	56	4.6 fm
lead-208	208	7.1 fm
uranium-238	238	7.4 fm

Because radius depends on the cube root of A, the volume (proportional to r³) is proportional to A. In other words, every nucleon takes up roughly the same volume: nucleons pack together like marbles in a bag.

Compare with atoms: atomic radii are around 30,000–300,000 fm. An atom is typically 10,000 to 100,000 times wider than its nucleus. See how small is an atom?

Nuclear density

Since the volume of a nucleus is proportional to the number of nucleons, all nuclei have about the same density:

• Mass of one nucleon ≈ 1.67 × 10⁻²⁷ kg
• Volume per nucleon ≈ (4/3)π(1.2 × 10⁻¹⁵)³ ≈ 7.2 × 10⁻⁴⁵ m³
• Density ≈ 1.67 × 10⁻²⁷ ÷ 7.2 × 10⁻⁴⁵ ≈ 2.3 × 10¹⁷ kg/m³

That’s about 10¹⁴ times denser than water. A sugar-cube-sized piece of nuclear matter would have a mass of hundreds of millions of tonnes. Neutron stars are essentially giant nuclei with this density.

The strong nuclear force

Protons repel each other electrically, and in a nucleus they’re extremely close together. So what holds nuclei together?

The strong nuclear force (strictly, the residual strong force between nucleons) has these properties:

• Very strong: at around 1 fm, it’s much stronger than the electrical repulsion between protons.
• Very short range: it drops to almost nothing beyond about 2–3 fm, so each nucleon only attracts its nearest neighbours.
• Repulsive at very short distances (below about 0.5 fm), which stops the nucleus collapsing.
• Charge-independent: it acts equally between p–p, n–n and p–n pairs.

The electrical repulsion, by contrast, is long-range: every proton repels every other proton in the nucleus. In a large nucleus, the total repulsion grows faster than the total strong-force attraction, which is why very heavy nuclei become unstable.

Mass defect and binding energy

If you add up the masses of the separate protons and neutrons in a nucleus, you get more than the actual mass of the nucleus. The difference is the mass defect.

That missing mass has been converted into energy, released when the nucleus formed. The binding energy is the energy needed to pull the nucleus completely apart into separate nucleons:

E = Δm × c²

Worked example: helium-4

• Mass of 2 protons = 2 × 1.007276 u = 2.014552 u
• Mass of 2 neutrons = 2 × 1.008665 u = 2.017330 u
• Total = 4.031882 u
• Actual mass of helium-4 nucleus = 4.001506 u
• Mass defect = 4.031882 − 4.001506 = 0.030376 u

Using 1 u ≡ 931.5 MeV:

Binding energy = 0.030376 × 931.5 ≈ 28.3 MeV Binding energy per nucleon = 28.3 ÷ 4 ≈ 7.1 MeV

That’s millions of times more energy per atom than chemical bonds, which are measured in a few electronvolts.

The binding energy per nucleon curve

Plotting binding energy per nucleon against mass number gives one of the most important graphs in physics:

• It rises steeply from hydrogen through helium-4 (which is unusually stable).
• It peaks around iron-56 and nickel-62, at about 8.8 MeV per nucleon. These are the most tightly bound nuclei.
• It falls gradually for heavier nuclei, to about 7.6 MeV per nucleon for uranium.

This curve explains nuclear energy:

• Fusion: joining light nuclei (to the left of iron) moves up the curve, releasing energy. This powers the Sun.
• Fission: splitting heavy nuclei (to the right of iron) also moves up the curve, releasing energy. This powers nuclear reactors.

It also explains why stars can make elements by fusion only up to about iron; heavier elements form in supernovae and neutron-star collisions. See nuclear fission vs fusion.

Stability and radioactivity

Of the thousands of known nuclides, only about 250 are stable. Stability depends on the balance of protons and neutrons:

• Light nuclei are stable with N ≈ Z.
• Heavier nuclei need N > Z (lead-208 has N/Z ≈ 1.55).
• No nucleus with more than 82 protons is truly stable; all elements beyond lead (Z = 82) are radioactive. (Bismuth-209, Z = 83, has such an enormously long half-life that it was long thought stable.)

Unstable nuclei decay towards stability:

• Alpha decay: very heavy nuclei shed 2 protons and 2 neutrons.
• Beta-minus decay: neutron-rich nuclei convert a neutron into a proton.
• Beta-plus decay / electron capture: proton-rich nuclei convert a proton into a neutron.
• Gamma emission: excited nuclei release energy as a photon.

See alpha, beta and gamma radiation, half-life explained and radioactive elements.

Magic numbers and shell structure

Nuclei with certain numbers of protons or neutrons, 2, 8, 20, 28, 50, 82 and 126, are especially stable. These magic numbers suggest that nucleons, like electrons, occupy shells. Nuclei that are “doubly magic” (magic numbers of both protons and neutrons), such as helium-4, oxygen-16, calcium-40 and lead-208, are exceptionally stable.

This shell model also underlies the prediction of an island of stability for some superheavy elements.

Key takeaways

• Rutherford’s gold foil experiment revealed a tiny, dense, positive nucleus in 1911.
• Nuclear radius ≈ 1.2 fm × A^(1/3), so all nuclei have roughly the same density, about 2 × 10¹⁷ kg/m³.
• The short-range strong force binds nucleons against the long-range electrical repulsion between protons.
• The mass defect, converted by E = mc², gives the binding energy; binding energy per nucleon peaks near iron.
• Unbalanced neutron–proton ratios make nuclei radioactive; magic numbers mark especially stable nuclei.