# Nuclear Decay

> Edexcel International A-Level Physics · IAL Physics Unit 5 (WPH15) 2018 Spec
> Source: https://www.owlsprep.com/study/edexcel-ial-physics-u5-nuclear-decay/

This guide covers all Edexcel IAL Physics Unit 5 nuclear decay content, including binding energy calculations, radiation properties, exponential decay, half-life graphs, and Core Practical 15 for gamma absorption by lead.

**Prerequisites:** [Atomic structure (Edexcel IAL Physics Unit 1)](https://www.owlsprep.com/study/edexcel-ial-physics-u1-atomic-structure/); [Exponentials and logarithms (IAL Maths P1)](https://www.owlsprep.com/study/edexcel-ial-maths-p1-exponentials-logarithms/)

## Learning objectives

- Calculate mass deficit and nuclear binding energy using $ΔE=c^2Δm$
- Convert between atomic mass units (u) and SI units for nuclear mass/energy
- Interpret binding energy per nucleon curves to distinguish fission and fusion reactions
- Compare properties of $α$, $β^-$, $β^+$ and $γ$ radiation and write balanced nuclear equations
- Calculate activity, decay constant and half-life using exponential decay formulae and log-linear plots
- Account for background radiation in practical measurements and decay calculations
- Describe the random and spontaneous nature of nuclear decay
- Complete Core Practical 15 (absorption of gamma radiation by lead) calculations and analysis

## Mass Deficit and Nuclear Binding Energy

**Mass Deficit** — The difference between the total mass of individual separate nucleons (protons and neutrons) in a nucleus and the actual mass of the intact nucleus.

Nuclear binding energy is the energy required to split a nucleus into its separate nucleons, or the energy released when nucleons bind together to form a nucleus. It is calculated using the mass-energy equivalence relation:

$$ΔE = c^2Δm$$

Nuclear masses are measured in atomic mass units (u), where $1 u = \frac{1}{12}$ the mass of a neutral carbon-12 atom, equivalent to $1.66 \times 10^{-27}$ kg.

**Worked example:** Calculate the binding energy per nucleon for a helium-4 nucleus, given: mass of proton = 1.00728 u, mass of neutron = 1.00867 u, mass of He-4 nucleus = 4.00153 u, $c = 3.00 \times 10^8$ m/s, $1 u = 1.66 \times 10^{-27}$ kg.

1. Calculate total mass of 2 protons and 2 neutrons: $2(1.00728 + 1.00867) = 4.0319$ u
2. Calculate mass deficit: $Δm = 4.0319 - 4.00153 = 0.03037$ u
3. Convert mass deficit to kg: $0.03037 \times 1.66 \times 10^{-27} = 5.04 \times 10^{-29}$ kg
4. Calculate total binding energy: $ΔE = (3.00 \times 10^8)^2 \times 5.04 \times 10^{-29} = 4.54 \times 10^{-12}$ J
5. Divide by number of nucleons (4) to get binding energy per nucleon: $\frac{4.54 \times 10^{-12}}{4} = 1.14 \times 10^{-12}$ J per nucleon

> **Exam tip:** Always use nuclear mass (not atomic mass) for mass deficit calculations to avoid including electron mass in your value of $Δm$.

## Fusion, Fission and Binding Energy Curves

A plot of binding energy per nucleon against nucleon number peaks at ~56 nucleons (iron-56), the most stable nucleus. Nuclei with lower binding energy per nucleon can undergo reactions to increase stability, releasing energy:

- **Fusion**: Two light nuclei combine to form a heavier nucleus, increasing binding energy per nucleon and releasing energy
- **Fission**: A heavy unstable nucleus splits into two smaller nuclei, increasing binding energy per nucleon and releasing energy

**Nuclear Fusion** — Reaction where two light nuclei combine to form a heavier nucleus, releasing energy. Requires very high temperature (to overcome electrostatic repulsion between positive nuclei) and high density (to increase collision frequency).

**Worked example:** Explain why energy is released when two deuterium ($^2_1$H) nuclei fuse to form helium-3 ($^3_2$He) and a neutron, with reference to binding energy per nucleon.

1. The binding energy per nucleon of helium-3 is higher than the binding energy per nucleon of deuterium.
2. The total mass of the products (He-3 + neutron) is less than the total mass of the two deuterium reactants.
3. The missing mass is converted to energy via $ΔE = c^2Δm$, equal to the difference between the total binding energy of the products and reactants.

> **Exam tip:** You may be asked to sketch the binding energy per nucleon curve: label the peak at ~56 nucleons, mark the fusion region (A < 56) and fission region (A > 56).

## Ionising Radiation Properties and Nuclear Equations

| Radiation type | Nature | Penetration | Ionising ability | Range in air |
| --- | --- | --- | --- | --- |
| $α$ | Helium nucleus ($^4_2$He) | Stopped by paper/skin | Very high | ~5 cm |
| $β^-$ | High-speed electron ($^0_{-1}$e) | Stopped by 3 mm aluminium | Medium | ~1 m |
| $γ$ | High-energy EM photon ($^0_0γ$) | Attenuated by cm of lead | Low | Unlimited (reduced by matter) |

Nuclear equations for decay reactions must balance both total nucleon number (top value) and total proton number (bottom value) on both sides of the equation.

**Worked example:** Write the balanced nuclear equation for the beta minus decay of carbon-14 ($^{14}_6$C) to nitrogen.

1. Beta minus decay emits a $^0_{-1}$e particle, so proton number of the product increases by 1, nucleon number stays the same.
2. Product proton number = $6 + 1 = 7$, which corresponds to nitrogen (N).
3. Balanced equation: $^{14}_6$C → $^{14}_7$N + $^0_{-1}$e + $ar{
u}_e$ (an electron antineutrino)

> **Exam tip:** Always check that both top (nucleon) and bottom (proton) numbers sum equally on both sides of the nuclear equation to avoid errors.

## Exponential Decay, Half-Life and Log-Linear Plots

**Half-life ($t_{1/2}$)** — The average time taken for half the number of undecayed nuclei in a sample to decay, or for the activity of a sample to reduce to half its initial value.

Key decay formulae (provided on the formula sheet) are:

$$A = λ N, λ = \frac{\ln2}{t_{1/2}}, N = N_0e^{-λ t}, A = A_0e^{-λ t}$$

Taking the natural logarithm of $N = N_0e^{-λ t}$ gives the linear form: $​\ln N = \ln N_0 - λ t$. A plot of $​\ln N$ against time $t$ is a straight line with gradient $-λ$, so you can calculate the decay constant from the magnitude of the gradient. No integration of $​\frac{dN}{dt} = -λ N$ is required.

**Worked example:** A radioactive sample has an initial activity of 640 Bq. After 2 hours, the activity is 40 Bq. Calculate the half-life of the sample in seconds.

1. Count the number of half-lives passed: $640 \to 320 \to 160 \to 80 \to 40$, so 4 half-lives = 2 hours = 7200 s
2. Half-life $t_{1/2} = \frac{7200}{4} = 1800$ s
3. To verify with the decay formula: $40 = 640e^{-λ \times 7200} \implies \ln(\frac{1}{16}) = -7200λ \implies λ = \frac{4\ln2}{7200} \implies t_{1/2} = \frac{\ln2}{λ} = 1800$ s, confirming the result.

> **Exam tip:** Always subtract background radiation from measured activity values before performing decay calculations.

## Core Practical 15: Absorption of Gamma Radiation by Lead

This practical investigates how the intensity of gamma radiation decreases with increasing thickness of lead absorber. Key steps:

- Measure background count rate for 5 minutes, calculate average background count per second
- Place a gamma source at a fixed distance from a Geiger-Muller (GM) tube
- Insert lead sheets of increasing measured thickness between the source and GM tube, record count rate for each thickness
- Subtract background count rate from all measured values to get corrected count rate (proportional to gamma intensity)

Intensity decreases exponentially with lead thickness: $I = I_0e^{-μ x}$, where $μ$ is the linear attenuation coefficient of lead, and $x$ is absorber thickness.

**Worked example:** A student measures a background count rate of 20 counts per minute (cpm). With a 1 cm thick lead sheet, the measured count rate from a gamma source is 220 cpm. Calculate the corrected count rate.

1. Corrected count rate = measured count rate - background count rate
2. Corrected count rate = $220 - 20 = 200$ cpm

> **Exam tip:** Safety precautions for this practical: use tongs to handle gamma sources, keep sources at arm's length, store in a lead container when not in use.

## Common pitfalls

- **Wrong:** Using atomic mass instead of nuclear mass for mass deficit calculations
  - Why it fails: Atomic mass includes electron mass, adding extra mass not part of the nucleus, leading to incorrect $Δm$ values
  - Correct: Use nuclear mass if given, or subtract mass of orbiting electrons from atomic mass before calculations
- **Wrong:** Forgetting to subtract background radiation from measured count rates
  - Why it fails: Background radiation adds extra counts, leading to overestimation of sample activity and wrong decay calculations
  - Correct: Measure background count first, subtract it from all sample measurements before analysis
- **Wrong:** Using the negative gradient of a $​\ln N$ vs $t$ plot directly as $λ$
  - Why it fails: The linear decay equation $​\ln N = \ln N_0 - λ t$ has a negative gradient, so $λ$ is the magnitude of the gradient
  - Correct: Calculate the gradient, take its absolute value to get the decay constant $λ$
- **Wrong:** Balancing only nucleon number in nuclear equations, ignoring proton number
  - Why it fails: This leads to incorrect identification of decay products, e.g. wrong element after beta decay
  - Correct: Check that both top (nucleon) and bottom (proton) numbers sum to equal values on both sides of the equation
- **Wrong:** Using half-life in minutes/hours when calculating $λ$ in s⁻¹
  - Why it fails: Decay constant units are s⁻¹, so inconsistent time units lead to incorrect activity values
  - Correct: Always convert $t_{1/2}$ to seconds before calculating $λ$, or use consistent time units for all values in the formula

## Cheatsheet

| Concept | Key Formula / Rule | Units |
| --- | --- | --- |
| Binding energy | $ΔE = c^2Δm$ | $ΔE$ in J, $Δm$ in kg, $c=3\times10^8$ m/s |
| u conversion | $1 u = 1.66\times10^{-27}$ kg ≈ 931.5 MeV |  |
| Half-life & Decay constant | $λ = \frac{\ln2}{t_{1/2}}$ | $λ$ in s⁻¹, $t_{1/2}$ in s |
| Exponential decay | $N=N_0e^{-λ t}, A=A_0e^{-λ t}$ | A in Bq (1 count per second) |
| Nuclear equations | Balance nucleon and proton numbers on both sides |  |
| Core Practical 15 | Corrected count rate = measured rate - background rate | Count rate in counts per second/minute |

## What's next

Now that you have mastered nuclear decay for Edexcel IAL Physics Unit 5, you can move on to other Unit 5 topics including oscillations, thermodynamics and cosmology. This content is frequently tested alongside practical skills questions in Paper 15, so practice past paper questions to familiarize yourself with common exam phrasing and calculation formats. Make sure you can interpret log-linear plots of decay data and apply the exponential decay formulae to both numerical and written questions. Remember to always account for background radiation in any practical decay questions, and double-check nuclear equation balances for both nucleon and proton numbers.

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