# Properties of the Nucleus

> Alpha, beta and gamma decay, half-life, mass defect and binding energy, and how fission and fusion release energy.

From HSC Physics on WhizzNotes, free study notes for NSW students: https://whizznotes.com/physics/the-nucleus/. Updated 2 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The clock inside a bone
In 1991, two hikers in the Alps found a body frozen in a glacier. Police first thought it was a lost climber. Scientists measured the carbon-14 in his bones and found he had died about 5300 years ago. They named him Ötzi.

Every living thing takes in carbon-14 while it is alive. After death, no more comes in, and the carbon-14 already there slowly decays: half of it every 5730 years. By measuring how much was left, scientists could read the time since Ötzi died like a clock. This chapter explains how unstable nuclei decay, and where nuclear energy comes from.

**In short:** The nucleus holds protons and neutrons. An unstable nucleus decays by giving out alpha, beta or gamma radiation, and its half-life is the time for half of it to decay. Fission (splitting heavy nuclei) and fusion (joining light nuclei) both release energy, from the mass defect through $E = mc^2$.

## The nucleus

A nucleus is written $^{A}_{Z}\text{X}$:

- $Z$ = **atomic number** = number of protons
- $A$ = **mass number** = protons + neutrons
- **Isotopes** of an element have the same $Z$ but different $A$ (different numbers of neutrons).

The **strong nuclear force** holds protons and neutrons together. It is very strong but only acts over very short distances, about the width of a nucleus. It overcomes the electric repulsion between protons.

## Radioactive decay

Unstable nuclei give off radiation to become more stable.

| Type | What it is | Charge | Stopped by | Change to nucleus |
|---|---|---|---|---|
| **Alpha** (α) | helium nucleus $^4_2\text{He}$ | +2 | paper | $A$ − 4, $Z$ − 2 |
| **Beta minus** (β⁻) | fast electron $^{\;\,0}_{-1}e$ | −1 | a few mm of aluminium | $A$ same, $Z$ + 1 |
| **Beta plus** (β⁺) | positron $^{0}_{+1}e$ | +1 | a few mm of aluminium | $A$ same, $Z$ − 1 |
| **Gamma** (γ) | high-energy photon | 0 | thick lead or concrete | no change in $A$ or $Z$ |

In **beta minus** decay, a neutron turns into a proton, giving out an electron and an **antineutrino**. In **beta plus** decay, a proton turns into a neutron, giving out a positron and a **neutrino**.

In every nuclear equation, **mass numbers balance** and **atomic numbers balance**.

### Worked example: Writing decay equations
Alpha decay of uranium-238:

$$^{238}_{92}\text{U} \rightarrow\, ^{234}_{90}\text{Th} + \,^4_2\text{He}$$

Beta minus decay of carbon-14:

$$^{14}_{6}\text{C} \rightarrow\, ^{14}_{7}\text{N} + \,^{\;\,0}_{-1}e + \bar{\nu}$$

Check: top numbers $238 = 234 + 4$ and $14 = 14 + 0$. Bottom numbers $92 = 90 + 2$ and $6 = 7 - 1$.

## Half-life

Radioactive decay is **random**: you cannot say when one nucleus will decay. But with huge numbers of nuclei, the pattern is predictable.

The **half-life** $t_{1/2}$ is the time for **half** of the radioactive nuclei in a sample to decay.

| Time | Fraction left |
|---|---|
| 0 | 1 |
| 1 half-life | ½ |
| 2 half-lives | ¼ |
| 3 half-lives | ⅛ |

The number left after time $t$ is:

$$N_t = N_0 e^{-\lambda t} \qquad \lambda = \frac{\ln 2}{t_{1/2}}$$

where $\lambda$ is the **decay constant**.

### Worked example: Iodine in medicine
Iodine-131 has a half-life of 8.0 days. A patient is given $4.0 \times 10^{12}$ nuclei. How many are left after 24 days?

24 days is 3 half-lives, so $N = 4.0 \times 10^{12} \times \frac{1}{8} = 5.0 \times 10^{11}$.

## Mass defect and binding energy

Weigh a nucleus and it has **less** mass than the protons and neutrons that make it up, weighed separately. The missing mass is the **mass defect** $\Delta m$.

That mass was released as energy when the nucleus formed. To pull the nucleus apart, you would need to put the same energy back. This is the **binding energy**:

$$E = \Delta m c^2$$

In atomic mass units, use $1 \text{ u} = 931.5$ MeV.

### Worked example: Helium-4
Two protons ($2 \times 1.007276$ u) and two neutrons ($2 \times 1.008665$ u) have a total mass of 4.031882 u. A helium-4 nucleus has a mass of 4.001506 u.

$\Delta m = 4.031882 - 4.001506 = 0.030376$ u

Binding energy $= 0.030376 \times 931.5 = 28.3$ MeV.

## Binding energy per nucleon

Dividing the binding energy by the number of nucleons shows how **tightly bound** each nucleon is. Plotting this against mass number gives a curve that:

- rises steeply for light nuclei,
- peaks near **iron-56** (the most stable nucleus),
- falls slowly for heavy nuclei.

Energy is released by any reaction that moves nuclei **towards the peak**:

- **Fusion**: joining light nuclei (like hydrogen into helium).
- **Fission**: splitting heavy nuclei (like uranium) into middle-sized ones.

## Fission

A slow neutron hits a uranium-235 nucleus, which splits into two smaller nuclei and gives out **2 or 3 neutrons** and a lot of energy:

$$^{235}_{92}\text{U} + \,^1_0n \rightarrow\, ^{141}_{56}\text{Ba} + \,^{92}_{36}\text{Kr} + 3\,^1_0n + \text{energy}$$

The new neutrons can split more uranium nuclei: a **chain reaction**.

- **Uncontrolled**: the number of fissions grows very fast. This is a nuclear bomb.
- **Controlled**: in a reactor, **control rods** absorb spare neutrons so that, on average, exactly one neutron from each fission causes another. A **moderator** (water or graphite) slows the neutrons down so uranium-235 captures them more easily.

Enrico Fermi built the first controlled nuclear reactor in 1942.

**Mind map**

```text
The Nucleus
  Notation
    Z protons, A nucleons
    Isotopes
    Strong nuclear force
  Decay
    Alpha: A − 4, Z − 2
    Beta minus: Z + 1
    Beta plus: Z − 1
    Gamma: photon
  Half-life
    Random but predictable
    N = N₀ e^(−λt)
    λ = ln 2 / t½
  Binding energy
    Mass defect
    E = Δmc²
    1 u = 931.5 MeV
    Peak at iron
  Energy
    Fusion: light nuclei
    Fission: heavy nuclei
    Chain reaction
    Control rods, moderator
```

### Year 11: Which radiation is most dangerous?
**Outside** the body, gamma is the biggest risk because it passes through skin and deep into tissue. Alpha cannot get through skin.

**Inside** the body (breathed in or swallowed), alpha is the most dangerous. It gives up all its energy in a very short distance, badly damaging nearby cells. Alpha particles **ionise** strongly, gamma rays only weakly.

### Year 12: Energy released in a reaction
For any nuclear reaction: energy released $= (\text{mass before} - \text{mass after}) \times c^2$.

**Alpha decay of polonium-210:**

$^{210}\text{Po}$ (209.982874 u) $\rightarrow$ $^{206}\text{Pb}$ (205.974465 u) $+$ $^4\text{He}$ (4.002603 u)

$\Delta m = 209.982874 - 209.977068 = 0.005806$ u, so $E = 0.005806 \times 931.5 = 5.41$ MeV.

The products have **less mass** but a **greater total binding energy**: they are more tightly bound. Most of the released energy goes to the alpha particle as kinetic energy.

**Using the decay constant.** Starting with $N_0$ nuclei, the number left after a time that is not a whole number of half-lives needs the exponential. For a half-life of 2.6 hours, $\lambda = \dfrac{0.693}{2.6} = 0.267$ per hour. After 6.0 hours, $\dfrac{N}{N_0} = e^{-0.267 \times 6.0} = 0.20$, so 20% is left.

**Activity** (decays per second, in becquerels) also halves every half-life. A detector also picks up **background radiation**, which must be subtracted before using the half-life.

**Spotting the decay.** If $Z$ drops by 1 and $A$ stays the same (phosphorus-30 to silicon-30), a positron and a neutrino were given off (beta plus).

**Background and percentages.** Subtract the background first: 340 Bq read with 20 Bq background is 320 Bq from the source; after 4 half-lives it is 20 Bq, so the detector reads 40 Bq. After 6 half-lives, $\left(\frac{1}{2}\right)^6 = 1.6\%$ remains, so 98.4% has decayed.

**From a graph.** $\ln\dfrac{N}{N_0} = -\lambda t$, so a graph of $\ln\dfrac{N}{N_0}$ against $t$ is a straight line with gradient $-\lambda$. Then $t_{1/2} = \dfrac{\ln 2}{\lambda}$: a gradient of $-0.0502$ s⁻¹ gives 13.8 s.

**Dating.** Solve $\dfrac{N}{N_0} = e^{-\lambda t}$ for $t$: wood with 30% of its carbon-14 is $t = \dfrac{\ln(1/0.30)}{\ln 2} \times 5730 \approx 9950$ years old. Falling to $\frac{1}{8}$ always takes exactly 3 half-lives.

**Sharing the energy.** A nucleus at rest splits with equal and opposite momentum, so the lighter particle gets most of the kinetic energy: $K_\alpha = E\dfrac{M}{m + M}$. For polonium-210, $5.41 \times \dfrac{206}{210} = 5.31$ MeV.

**Decay chains.** From U-238 to Pb-206: $A$ drops by 32, and only alpha decays change $A$, so there are $\frac{32}{4} = 8$ alphas. These alone would lower $Z$ by 16, but it only drops by 10, so 6 beta minus decays raised it back.

**Fuel use.** A 1000 MW reactor for a day releases $8.64 \times 10^{13}$ J. Each fission gives $173 \times 1.602 \times 10^{-13} = 2.77 \times 10^{-11}$ J, so about $3.1 \times 10^{24}$ fissions, which is about 1.2 kg of U-235.

## Quick summary

- $Z$ = protons, $A$ = protons + neutrons. Isotopes differ in neutron number.
- Alpha: $A - 4$, $Z - 2$. Beta minus: $Z + 1$. Beta plus: $Z - 1$. Gamma: no change.
- Half-life is the time for half the nuclei to decay. $N = N_0 e^{-\lambda t}$ and $\lambda = \dfrac{\ln 2}{t_{1/2}}$.
- Mass defect $\Delta m$ gives the binding energy $E = \Delta m c^2$ (1 u = 931.5 MeV).
- Binding energy per nucleon peaks at iron. Fusion of light nuclei and fission of heavy nuclei both release energy.
- Fission neutrons can cause a chain reaction, controlled in reactors by control rods and a moderator.

## Quiz

Try these questions from the chapter quiz.

1. The number of protons in a nucleus is its:
   - A. Atomic number
   - B. Mass number
   - C. Half-life
   - D. Binding energy
   - Answer: A. Atomic number. Z is the atomic number.

2. Isotopes of an element have:
   - A. Different numbers of protons
   - B. The same number of protons but different numbers of neutrons
   - C. Different numbers of electrons only
   - D. The same mass number
   - Answer: B. The same number of protons but different numbers of neutrons. Same element means same Z. Different isotopes have different A.

3. An alpha particle is:
   - A. An electron
   - B. A helium nucleus
   - C. A photon
   - D. A neutron
   - Answer: B. A helium nucleus. Alpha is ⁴₂He: 2 protons and 2 neutrons.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/the-nucleus/#quiz