Nuclear Radius and Fission Reactors

The closest approach of an α particle gives an upper limit on nuclear size; electron diffraction gives better values. Nuclear radius follows $R=R_0A^{1/3}$ with $R_0\approx1.2$ fm, so all nuclei have about the same density, $\sim10^{17}$ kg m⁻³. In a thermal reactor, slow neutrons sustain a chain reaction in uranium-235, controlled by a moderator and control rods.

AQA 7408 Paper 2≈ 40 minRevision 1 · updated 2026-10-27

Section 1 of 14

Learning objectives

Estimate a nuclear radius from α scattering and electron diffraction.
Use $R=R_0A^{1/3}$ and show nuclear density is constant.
Explain the roles of fuel, moderator, control rods, coolant and shielding.

Pause and think

If a nucleus were the size of a marble, how big would the atom be?

Section 2 of 14

Core idea

Start hereHow big is a nucleus?

You can't see a nucleus or measure it with a ruler. Instead, physicists fire particles at it and see how they bounce or spread. The answer is astonishing: all nuclei, light or heavy, are packed to the same incredible density.

The big idea

$R=R_0A^{1/3}$ , so volume is proportional to nucleon number and density is the same for all nuclei. In a reactor, U-235 fission is kept steady by slowing neutrons and absorbing the extras.

1. Measuring nuclei

Worked exampleClosest approach

A $5.0$ MeV α particle heads straight at gold ( $Z=79$ ).

Goal: Closest distance.
Strategy: All KE becomes electric potential energy.

$E_k=\frac{(2e)(79e)}{4\pi\varepsilon_0r}$
Energy conservation at the turning point.
$r=\frac{1.44\times158}{5.0}\text{ fm}$
Use $\frac{e^2}{4\pi\varepsilon_0}=1.44$ MeV fm.
$r=45\text{ fm}$
Calculate.

Answer

45

fm: an upper limit on gold's radius.

✓ Check

R_0A^{1/3}=1.2\times197^{1/3}=7

fm, much smaller ✓.

Constant nuclear density

$m\approx Au$
Each nucleon has mass about $u$ .
$V=\tfrac43\pi R^3=\tfrac43\pi R_0^3A$
Cube $R=R_0A^{1/3}$ .
$\rho=\frac mV=\frac{3u}{4\pi R_0^3}$
$A$ cancels.
$\approx2.3\times10^{17}\text{ kg m}^{-3}$
$R_0=1.2$ fm.

Nucleons pack like marbles in a bag: more marbles, bigger bag, same density.

2. Thermal reactors

Inside a thermal reactor

Drawing diagram…

Each part has one job.

Key takeawayCritical mass Below it, too many neutrons escape to sustain the chain reaction.

Common misconceptionModerator vs control rods Moderators slow neutrons (light nuclei, elastic collisions); control rods absorb them.

Pause and think

Why is a light-nucleus moderator best?

Summary

Closest approach: upper limit.
$R=R_0A^{1/3}$ .
Density constant.
Moderator, control rods, coolant, shielding.

Moderator: Material that slows neutrons.
Critical: One further fission per fission on average.
Closest approach: Where the α particle momentarily stops.

HistoryChicago Pile-1. On 2 December 1942, under a Chicago football stadium, Enrico Fermi's team ran the first controlled chain reaction, using graphite blocks as moderator.

Fun factA teaspoon of neutron star. Neutron stars have roughly nuclear density: a teaspoon would weigh around a billion tonnes.

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