What is radioactive decay in the context of Nuclear Physics?

Radioactive decay is a process by which an unstable atomic nucleus loses energy by emitting radiation. This process is fundamental in Nuclear Physics and involves the transformation of an unstable isotope into a more stable one, often resulting in the emission of alpha particles, beta particles, or gamma rays. Understanding radioactive decay is crucial for Cambridge International A Levels students as it explains the behavior of radioactive materials and their applications.

How do half-life and radioactive decay relate to each other?

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. It is a key concept in understanding radioactive decay, as it provides a measure of the rate at which a radioactive isotope transforms. In the Cambridge International A Levels curriculum, students learn to calculate half-lives to predict the behavior of radioactive materials over time, which is essential for applications in fields like medicine and archaeology.

What are the different types of radioactive decay?

There are three primary types of radioactive decay: alpha decay, beta decay, and gamma decay. Alpha decay involves the emission of an alpha particle, which consists of two protons and two neutrons. Beta decay occurs when a neutron in the nucleus is transformed into a proton, emitting a beta particle (electron or positron). Gamma decay involves the emission of gamma rays, which are high-energy photons. Each type of decay results in the transformation of the original nucleus into a different element or isotope, a concept explored in depth in the Cambridge International A Levels Physics syllabus.

Why is understanding radioactive decay important in Physics?

Understanding radioactive decay is crucial in Physics because it explains the stability of elements and isotopes, the energy released in nuclear reactions, and the principles behind nuclear power and medical imaging technologies. For Cambridge International A Levels students, mastering this topic is essential for comprehending broader concepts in Nuclear Physics, such as nuclear fission and fusion, and their practical applications in energy production and healthcare.

How is radioactive decay measured and detected?

Radioactive decay is measured and detected using instruments like Geiger-Müller counters, scintillation counters, and semiconductor detectors. These devices measure the radiation emitted by decaying nuclei, allowing scientists to determine the type and intensity of radiation. In the Cambridge International A Levels curriculum, students learn about these detection methods to understand how radioactive materials are analyzed and monitored in various scientific and industrial applications.

Why is "Mixing $\ln 2$ and $\log 2$" a common mistake in Radioactive decay?

Why it happens: Calculator confusion. How to avoid it: Use 2 = 0.693 (natural log). t_1/2 = 0.693. Source: 9702 Examiner Reports 2022-2024

Why is "Forgetting random AND spontaneous keywords" a common mistake in Radioactive decay?

Why it happens: Only stating one. How to avoid it: Mark scheme rewards both 'random' AND 'spontaneous'.

Radioactive decay – Study Notes & Past Paper Style Questions | Cambridge International A Level Physics 9702

Short Notes - Radioactive decay

Detailed Study Notes

Detailed notes on Nuclear Physics for Cambridge International A Levels Physics, covering key concepts, explanations, examples, and exam-focused revision points.

Radioactive Decay Study Notes — Cambridge International A Level Physics 9702 (2025-2027 syllabus)

Decay law, activity, half-life. The exponential framework for nuclear decay.

At a glance

Decay is spontaneous + random.
$A = \lambda N$ .
$N = N_0 e^{-\lambda t}$ .
$t_{1/2} = \ln 2 / \lambda$ .
Units: Bq, s $^{-1}$ .

What you’ll learn

Mapped to the Cambridge International A Level 9702 syllabus (2025-2027).

23.2.1 — Describe radioactive decay as spontaneous and random.
23.2.2 — Use $A = \lambda N$ .
23.2.3 — Solve exponential decay problems.

Spontaneous and random decay

Two key keywords.

Spontaneous. Decay is unaffected by external physical conditions (pressure, temperature, chemical state).

Random. Cannot predict WHEN a specific nucleus will decay — only probability per unit time.

Decay constant $\lambda$ = probability per unit time that a nucleus decays. Units: s $^{-1}$ .

Figure 1 — Random decay: unpredictable one by one, reliable in bulk

Random means you cannot say which nucleus goes next, only the probability per unit time. Spontaneous means nothing external alters that probability.

Cambridge tip. Always use BOTH words in definition questions.

Spontaneous: unaffected by external conditions.
Random: probabilistic.
$\lambda$ in s $^{-1}$ .

Decay law and half-life

Exponential.

Activity: $A = \lambda N$ . Rate of decay. Bq = decays/s.

Decay equation: $N = N_0 e^{-\lambda t}$ . Also applies to A and count rate R.

Half-life: time for N (or A) to halve. $t_{1/2} = \frac{\ln 2}{\lambda}$

Figure 2 — Exponential decay: every half-life removes half of what is left

Decay is exponential: N halves every t½. The equal spacing of t½, 2t½ and 3t½ along the axis is the signature of an exponential.

Useful shortcut. After $n$ half-lives: $N = N_0 / 2^n$ — great for whole-number half-life problems.

The 'n half-lives' shortcut.

Half-lives elapsed, $n$	Fraction left, $N/N_0$	Percentage left
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%
4	1/16	6.25%
$n$	$1/2^{\,n}$	—

Use this only when the elapsed time is a whole number of half-lives; otherwise go back to $N = N_0 e^{-\lambda t}$ . Activity $A$ and count rate $R$ follow exactly the same pattern.

Background. Always subtract background count rate from measured rate before fitting.

Figure 3 — Subtracting the background count rate

Only the corrected curve is a true exponential. The uncorrected one flattens onto the background and gives a half-life that is far too long.

Cambridge tip. Linearise via $\ln N = \ln N_0 - \lambda t$ — straight line graphs of $\ln N$ vs $t$ give gradient $-\lambda$ .

Figure 4 — Linearising the decay law: ln N against t

Plotting ln N against t gives a straight line of gradient −λ and intercept ln N₀. This is the standard way to get λ from experimental data.

$A = \lambda N$ .
$N = N_0 e^{-\lambda t}$ .
$t_{1/2}\lambda = \ln 2$ .

See the full worked example for radioactive decay →

Quick recap

Decay = spontaneous + random.
$A = \lambda N$ ; $N = N_0 e^{-\lambda t}$ .
Half-life inversely related to $\lambda$ .

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

Spontaneous AND random.
$A = \lambda N$ .
$t_{1/2} = \ln 2 / \lambda$ .

How it’s examined

Radioactive decay on every Paper 4 in 2022-2024. Most-tested: half-life calculation (6 marks), $\ln N$ vs $t$ graph (6 marks).

Sources: Cambridge International A Level Physics 9702 syllabus (2025-2027); 9702 Examiner Reports 2022-2024; 9702/42 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

Take this whole topic with you

Step-by-step worked examples — Radioactive decay

Step-by-step solutions to past-paper-style questions on radioactive decay, written exactly the way a tutor would explain them at the board.

1Decay constant from half-life (4 marks)
Extended• Adapted from 9702/42 May/Jun 2024• half-life
Question
A radioisotope has half-life 5.0 days. Find decay constant in s $^{-1}$ . (4 marks)
Step-by-step solution
1. Step 1
  $t_{1/2} = 5.0 \times 86400 = 4.32 \times 10^{5}$ s.
2. Step 2
  $\lambda = \ln 2 / t_{1/2}$ .
  $\lambda = 0.693 / 4.32 \times 10^{5} \approx 1.6 \times 10^{-6} \text{ s}^{-1}$
Answer
$\lambda \approx 1.6 \times 10^{-6}$ s $^{-1}$ .
2Activity over time (5 marks)
Extended• activity
Question
Initial activity 800 Bq, half-life 12 h. Find activity after 36 h. (5 marks)
Step-by-step solution
1. Step 1
  Three half-lives. $A = 800 \times (1/2)^3 = 100$ Bq.
Answer
$A = 100$ Bq.
3Remaining nuclei (5 marks)
Extended• exponential
Question
$N_0 = 10^{12}$ , $\lambda = 0.10$ /s. Find $N$ at $t = 30$ s. (5 marks)
Step-by-step solution
1. Step 1
  $N = N_0 e^{-\lambda t}$ .
  $N = 10^{12} \times e^{-3.0} \approx 5.0 \times 10^{10}$
Answer
$N \approx 5.0 \times 10^{10}$ .

Key Formulae — Radioactive decay

The formulae you need to memorise for radioactive decay on the Cambridge International A Level 9702 paper, with every variable defined in plain English and a note on when to use it.

Activity
$A = \lambda N$
When to use
Rate of decay. Units: becquerel (Bq) = 1 decay/s.
Exponential decay
$N = N_0 e^{-\lambda t}$
When to use
Number of undecayed nuclei vs time. Also applies to A and count rate R.
Half-life
$t_{1/2} = \dfrac{\ln 2}{\lambda}$
When to use
Time for half the nuclei to decay.

Key Definitions and Keywords — Radioactive decay

Definitions to memorise and the exact keywords mark schemes credit for radioactive decay answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9702 sitting.

Decay constant
Examiner keyword
Probability per unit time that a given nucleus will decay. Units: s $^{-1}$ .
Activity
Examiner keyword
Number of decays per unit time. Units: becquerel (Bq).
Half-life
Examiner keyword
Time taken for activity (or number of nuclei) to fall to half its initial value. Independent of starting amount.
Spontaneous and random
Examiner keyword
Decay is SPONTANEOUS (unaffected by external conditions) and RANDOM (cannot predict which nucleus decays when).

Common Mistakes and Misconceptions — Radioactive decay

The traps other students keep falling into on radioactive decay questions — taken from recent Cambridge International A Level 9702 examiner reports and mark schemes — and how to avoid them.

✕Mixing $\ln 2$ and $\log 2$
9702 Examiner Reports 2022-2024
Why it happens
Calculator confusion.
How to avoid it
Use $\ln 2 = 0.693$ (natural log). $t_{1/2}\lambda = 0.693$ .
✕Forgetting random AND spontaneous keywords
Why it happens
Only stating one.
How to avoid it
Mark scheme rewards both 'random' AND 'spontaneous'.

Radioactive decay — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Radioactive decay

Short Notes - Radioactive decay

Detailed Study Notes

At a glance

What you’ll learn

Spontaneous and random decay

Decay law and half-life

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Decay constant from half-life (4 marks)

2Activity over time (5 marks)

3Remaining nuclei (5 marks)

Activity

Exponential decay

Half-life

Decay constant

Activity

Half-life

Spontaneous and random

✕Mixing ln⁡2\ln 2ln2 and log⁡2\log 2log2

✕Forgetting random AND spontaneous keywords

Radioactive decay — frequently asked questions

Radioactive decay

Short Notes - Radioactive decay

Detailed Study Notes

At a glance

What you’ll learn

Spontaneous and random decay

Decay law and half-life

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Decay constant from half-life (4 marks)

2Activity over time (5 marks)

3Remaining nuclei (5 marks)

Activity

Exponential decay

Half-life

Decay constant

Activity

Half-life

Spontaneous and random

✕Mixing ln⁡2\ln 2ln2 and log⁡2\log 2log2

✕Forgetting random AND spontaneous keywords

Radioactive decay — frequently asked questions

✕Mixing $\ln 2$ and $\log 2$

✕Mixing $\ln 2$ and $\log 2$