Size of the nucleus

How big is a nucleus?

Narrated video. Transcript:

Recall Rutherford’s experiment. Alpha particles fired at a thin gold foil. Most go straight through; a few bounce back from the tiny, positive gold nucleus.

An alpha particle of 5.5 MeV, heading straight at a gold nucleus, slows down, stops and turns back about 4.0 × 10⁻¹⁴ metre from its centre. The Coulomb force alone explains this.

Since the alpha never touched the nucleus, the nucleus must be smaller than 4.0 × 10⁻¹⁴ metre. That is only an upper limit.

Fire faster alpha particles, and they get closer. At some point the scattering stops matching Rutherford’s Coulomb-only calculation: the short-range nuclear force has started to act. That distance tells us the size of the nucleus.

Even better, fire fast electrons at the nuclei. Such scattering experiments have measured the sizes of the nuclei of many elements accurately.

The result is simple. A nucleus of mass number A has radius R equals R₀ times A to the power one-third, where R₀ is 1.2 × 10⁻¹⁵ metre.

10⁻¹⁵ metre is called a femtometre, or a fermi, written fm. So R₀ is 1.2 fm.

Aluminium-27: the cube root of 27 is 3, so R is 1.2 × 3, which is 3.6 fm. Gold-197: the cube root of 197 is about 5.82, so R is about 7.0 fm.

Gold has more than seven times as many nucleons as aluminium, yet its nucleus is not even twice as wide. The radius grows only as the cube root of A.

Compare with the atom: atoms are about 10⁻¹⁰ metre across, so a nucleus is about 10⁴ times smaller than its atom.

The same density everywhere

Narrated video. Transcript:

Treat the nucleus as a sphere. Its volume is four-thirds π R cubed. With R equal to R₀ A to the one-third, R cubed is R₀ cubed times A.

So the volume is four-thirds π R₀ cubed, times A. The volume is proportional to A: each nucleon brings the same amount of space with it.

The mass is also proportional to A: about A atomic mass units. Mass over volume, the A cancels. The density of a nucleus does not depend on A at all.

Every nucleus, from carbon to uranium, is like a drop of the same liquid. A bigger nucleus is just a bigger drop.

Let us work it out for iron. Its nucleus has a mass of 55.85 u, which is 9.27 × 10⁻²⁶ kilogram, and A equals 56.

The volume is four-thirds π, times 1.2 × 10⁻¹⁵ cubed, times 56. That is about 4.05 × 10⁻⁴³ cubic metre.

Divide: 9.27 × 10⁻²⁶ over 4.05 × 10⁻⁴³ gives 2.29 × 10¹⁷ kilogram per cubic metre. That is the density of all nuclear matter, about 2.3 × 10¹⁷.

Water is 10³ kilogram per cubic metre. Nuclear matter is about 10¹⁴ times denser. One cubic centimetre of it would have a mass of about 2.3 × 10¹¹ kilogram: as much as a crore of trucks, each carrying 20 tonnes.

Why is ordinary matter so much lighter? Because atoms are mostly empty space. All their mass is squeezed into a nucleus 10⁴ times smaller than the atom.

Neutron stars have densities like this. Their matter is crushed so hard that the whole star is like one giant nucleus, about 20 kilometres across.

Try it: nuclei to scale

Interactive simulation. Things to try:

• Both nuclei are drawn to the same scale, R = 1.2 A^(1/3) fm. The dots show the front layer of nucleons.
• Set A₁ = 27 and A₂ = 216: A is 8 times bigger, but R only doubles.
• Watch the two density readouts as you move the sliders. They never change: about 2.3 × 10¹⁷ kg/m³.
• Compare gold (197) and silver (107): the ratio of radii is (197/107)^(1/3) ≈ 1.23.

Eight times the nucleons

The radius of the ²⁷Al nucleus is 3.6 fm. What is the radius of a nucleus with A = 216?

1. 28.8 fm
2. 7.2 fm
3. 14.4 fm
4. 4.5 fm

Uranium against helium

What is the ratio of the nuclear density of ²³⁸U to that of ⁴He?

1. 59.5
2. 3.9
3. 1
4. 1/59.5

Summary

• Nuclear radius: R = R₀ A^(1/3), R₀ = 1.2 fm = 1.2 × 10⁻¹⁵ m
• Volume: V = (4/3)π R₀³ A ∝ A
• Nuclear density: ρ ≈ 2.3 × 10¹⁷ kg m⁻³ (same for all A)

• Alpha scattering gave an upper limit on nuclear size (4 × 10⁻¹⁴ m for gold); fast electron scattering measures it accurately.
• The radius grows as the cube root of A: 8 times the nucleons, only twice the radius.
• Volume and mass are both proportional to A, so all nuclei have the same density, like drops of one liquid.
• Nuclear density is about 10¹⁴ times that of water, because ordinary matter is mostly empty space. Neutron stars are about as dense as a nucleus.