Nuclear Decay

Q: Do I need to integrate for decay questions?

No, the integrated exponential decay formula is provided on the formula sheet. You only need to use the given formula and derive log-linear forms if explicitly asked, no calculus is required.

Q: How do I convert atomic mass units (u) to energy?

First convert u to kg using kg, then use to get energy in joules. If the question provides the conversion factor, you can also use MeV directly.

Edexcel International A-Level Physics· Spec 5.4, statements 133–142· 60 min read

1. Mass Deficit and Nuclear Binding Energy★★☆☆☆⏱ 10 min

📘 Definition

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 $1.66 \times 1 0^{- 27}$ 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, $c = 3.00 \times 1 0^{8}$ m/s, $1 u = 1.66 \times 1 0^{- 27}$ kg.

1
Calculate total mass of 2 protons and 2 neutrons: $2 (1.00728 + 1.00867) = 4.0319$ u
2
Calculate mass deficit: $Δ m = 4.0319 - 4.00153 = 0.03037$ u
3
Convert mass deficit to kg: $0.03037 \times 1.66 \times 1 0^{- 27} = 5.04 \times 1 0^{- 29}$ kg
4
Calculate total binding energy: $Δ E = (3.00 \times 1 0^{8})^{2} \times 5.04 \times 1 0^{- 29} = 4.54 \times 1 0^{- 12}$ J
5
Divide by number of nucleons (4) to get binding energy per nucleon: $\frac{4.54 \times 1 0 ^{- 12}}{4} = 1.14 \times 1 0^{- 12}$ 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 $Δ m$ .

2. Fusion, Fission and Binding Energy Curves★★★☆☆⏱ 10 min

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

📘 Definition

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 ( $_{1}^{2}$ H) nuclei fuse to form helium-3 ( $_{2}^{3}$ 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 $Δ E = c^{2} Δ m$ , 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).

3. Ionising Radiation Properties and Nuclear Equations★★☆☆☆⏱ 12 min

Radiation type	Nature	Penetration	Ionising ability	Range in air
$α$	Helium nucleus ( $_{2}^{4}$ He)	Stopped by paper/skin	Very high	~5 cm
$β^{-}$	High-speed electron ( $_{- 1}^{0}$ e)	Stopped by 3 mm aluminium	Medium	~1 m
$γ$	High-energy EM photon ( $_{0}^{0} γ$ )	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 ( $_{6}^{14}$ C) to nitrogen.

1
Beta minus decay emits a $_{- 1}^{0}$ 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: $_{6}^{14}$ C → $_{7}^{14}$ N + $_{- 1}^{0}$ 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.

4. Exponential Decay, Half-Life and Log-Linear Plots★★★★☆⏱ 15 min

📘 Definition

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 $N = N_{0} e^{- λ t}$ 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{d N}{d t} = - 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: $40 = 640 e^{- λ \times 7200} ⟹ ln (\frac{1}{16}) = - 7200 λ ⟹ λ = \frac{4 l n 2}{7200} ⟹ t_{1/2} = \frac{l n 2}{λ} = 1800$ s, confirming the result.

Exam tip:

Always subtract background radiation from measured activity values before performing decay calculations.

5. Core Practical 15: Absorption of Gamma Radiation by Lead★★★☆☆⏱ 10 min

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: $I = I_{0} e^{- μx}$ , 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.

6. Common Pitfalls

Wrong move:

Using atomic mass instead of nuclear mass for mass deficit calculations

Why:

Atomic mass includes electron mass, adding extra mass not part of the nucleus, leading to incorrect $Δ m$ values

Correct move:

Use nuclear mass if given, or subtract mass of orbiting electrons from atomic mass before calculations

Wrong move:

Forgetting to subtract background radiation from measured count rates

Why:

Background radiation adds extra counts, leading to overestimation of sample activity and wrong decay calculations

Correct move:

Measure background count first, subtract it from all sample measurements before analysis

Wrong move:

Using the negative gradient of a $ ln N$ vs $t$ plot directly as $λ$

Why:

The linear decay equation $ ln N = ln N_{0} - λ t$ has a negative gradient, so $λ$ is the magnitude of the gradient

Correct move:

Calculate the gradient, take its absolute value to get the decay constant $λ$

Wrong move:

Balancing only nucleon number in nuclear equations, ignoring proton number

Why:

This leads to incorrect identification of decay products, e.g. wrong element after beta decay

Correct move:

Check that both top (nucleon) and bottom (proton) numbers sum to equal values on both sides of the equation

Wrong move:

Using half-life in minutes/hours when calculating $λ$ in s⁻¹

Why:

Decay constant units are s⁻¹, so inconsistent time units lead to incorrect activity values

Correct move:

Always convert $t_{1/2}$ to seconds before calculating $λ$ , or use consistent time units for all values in the formula

7. Quick Reference Cheatsheet

Concept	Key Formula / Rule	Units
Binding energy	$Δ E = c^{2} Δ m$	$Δ E$ in J, $Δ m$ in kg, $c = 3 \times 1 0^{8}$ m/s
u conversion	$1 u = 1.66 \times 1 0^{- 27}$ kg ≈ 931.5 MeV
Half-life & Decay constant	$λ = \frac{l n 2}{t _{1/2}}$	$λ$ in s⁻¹, $t_{1/2}$ in s
Exponential decay	$N = N_{0} e^{- λ t}, A = A_{0} e^{- λ t}$	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

8. Frequently Asked

Do I need to integrate $\frac{d N}{d t} = - λ N$ for decay questions?

No, the integrated exponential decay formula $N = N_{0} e^{- λ t}$ is provided on the formula sheet. You only need to use the given formula and derive log-linear forms if explicitly asked, no calculus is required.

How do I convert atomic mass units (u) to energy?

First convert u to kg using $1 u = 1.66 \times 1 0^{- 27}$ kg, then use $Δ E = c^{2} Δ m$ to get energy in joules. If the question provides the conversion factor, you can also use $1 u \approx 931.5$ MeV directly.

Why do I have to subtract background radiation from my measurements?

Background radiation adds extra counts to your measured activity, leading to overestimation of the sample's activity and incorrect decay calculations. Always measure background count first and subtract it from all sample readings.

Going deeper

official_specEdexcel IAL Physics 2018 SpecificationRefer to section 5.4 for nuclear decay content
practical_guideCore Practical 15: Gamma Absorption by Lead

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.