Understand Radioactive Decay — Understand E.3 Radioactive decay - IB Physics

SECTION 1

Historical Context & Motivation

In the late 1800s, scientists began noticing that certain materials emitted invisible rays without any external energy source. This mysterious phenomenon, later called radioactivity, challenged the long-held belief that atoms were indivisible and unchanging. Understanding how and why atoms could spontaneously transform opened the door to nuclear physics, medical imaging, carbon dating, and nuclear energy. The story of radioactive decay is one of accidental discoveries, brilliant detective work, and a profound shift in how we view matter itself.

1896 • Becquerel's Accidental Discovery

Henri Becquerel discovered that uranium salts emitted penetrating rays that could fog photographic plates, even without exposure to sunlight. This was the first observation of natural radioactivity.

1898 • The Curies Isolate Polonium & Radium

Marie and Pierre Curie isolated two new radioactive elements — polonium and radium — from pitchblende ore, demonstrating that radioactivity was a property of specific atoms, not a chemical reaction.

1899 • Rutherford Identifies Alpha & Beta Rays

Ernest Rutherford showed that radioactive emissions came in at least two distinct types: heavily ionising alpha (α) rays and more penetrating beta (β) rays.

1900 • Gamma Rays Discovered

Paul Villard identified a third, even more penetrating type of radiation — gamma (γ) rays — which Rutherford later confirmed were high-energy electromagnetic waves.

1902 • Transmutation Theory

Rutherford and Frederick Soddy proposed that radioactive decay involves one element spontaneously transforming into another — a concept called nuclear transmutation. This overturned centuries of belief that elements were permanent.

These discoveries raised a crucial question: if atoms can change on their own, what determines when a particular atom will decay, and what it will become? The IB Physics E.3 topic addresses exactly this: the random, spontaneous nature of radioactive decay, the mathematics of half-life, and the different types of radiation that unstable nuclei can emit.

SECTION 2

Core Principles of Radioactive Decay

Radioactive decay is governed by several fundamental ideas. Before diving into the mathematics, you need a solid grasp of why nuclei decay and what makes the process unique in physics. Unlike most phenomena you have studied so far, radioactive decay is inherently random and spontaneous — you cannot predict exactly when a single atom will decay, only the probability that it will decay within a given time.

Nuclear Instability

A nucleus is unstable when the balance between the strong nuclear force (which holds nucleons together) and the electrostatic repulsion between protons is unfavourable. Such nuclei seek a more stable configuration by emitting particles or energy.

Random & Spontaneous

Radioactive decay is spontaneous — it requires no external trigger. It is also random — you cannot predict which specific nucleus will decay next. However, for large samples, the overall rate of decay is statistically predictable.

Half-Life (t₁/₂)

The half-life is the time taken for half of the undecayed nuclei in a sample to decay. It is constant for a given isotope and is unaffected by temperature, pressure, or chemical state.

Decay Constant (λ)

The decay constant represents the probability per unit time that a given nucleus will decay. A larger λ means a faster rate of decay and a shorter half-life.

Conservation Laws

Every nuclear decay must conserve mass number (A), atomic number (Z), charge, and energy–momentum. These conservation laws let you write and balance nuclear equations.

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KEY TAKEAWAY

Think of radioactive decay like popcorn in a microwave. You cannot predict exactly which kernel will pop next, but you know that, on average, half the unpopped kernels will have popped after a certain amount of time. That predictable 'half-popping time' is the half-life. No amount of shaking the bag or turning up the heat changes which kernel pops next — the process is fundamentally random at the individual level but statistically reliable for large numbers.

SECTION 3

Visualising Radioactive Decay

The graph below illustrates how the number of undecayed nuclei in a sample decreases over time. This characteristic curve is called an exponential decay curve. Notice that the curve never reaches zero — it asymptotically approaches the horizontal axis. Each successive half-life reduces the remaining quantity by exactly half, which is why the curve gets flatter over time but never truly ends.

The curve starts at N₀ (the initial number of undecayed nuclei). After one half-life, N = ½N₀; after two half-lives, N = ¼N₀; and so on. The decay rate (slope) slows as fewer undecayed nuclei remain.

A key feature of the exponential decay curve is that, no matter where you start on the curve, the time it takes for the value to halve is always the same. This is a direct consequence of the decay being a constant-probability process. Each undecayed nucleus has the same probability of decaying in the next second, regardless of how long it has already survived. This is sometimes called the 'memoryless' property of exponential decay.

SECTION 4

Mathematical Framework

The mathematics of radioactive decay follows from one simple idea: the rate at which nuclei decay is proportional to the number of undecayed nuclei remaining. This leads to the exponential decay law and several related equations that you need for IB Physics.

EXPONENTIAL DECAY LAW

N = N₀ × e^(−λt)

N = number of undecayed nuclei remaining at time t; N₀ = initial number of undecayed nuclei; λ = decay constant (s⁻¹); t = elapsed time; e ≈ 2.718 (Euler's number).

ACTIVITY (RATE OF DECAY)

A = λN = A₀ × e^(−λt)

A = activity (decays per second, measured in becquerels, Bq); A₀ = initial activity. Activity is proportional to the number of remaining nuclei, so it follows the same exponential curve.

HALF-LIFE & DECAY CONSTANT RELATIONSHIP

t₁/₂ = ln 2 / λ ≈ 0.693 / λ

t₁/₂ = half-life; ln 2 ≈ 0.693. This equation connects the half-life (an experimentally measurable quantity) to the decay constant (a theoretical probability). A short half-life means a large decay constant, and vice versa.

HALF-LIFE COUNTING METHOD

N = N₀ × (½)^(t / t₁/₂)

This is an equivalent form of the decay law that avoids the natural exponential. It is especially convenient when the elapsed time is a whole-number multiple of the half-life. For example, after 3 half-lives, N = N₀ × (½)³ = N₀ / 8.

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IB Exam Tip

The IB data booklet provides the equations N = N₀

e^{−λt}

and t₁/₂ = ln 2 / λ. You are expected to rearrange these fluently. A common exam strategy is to first determine the decay constant λ from a given half-life, and then substitute it into the exponential decay law to find N or A at a specified time.

SECTION 5

Types of Radioactive Decay

Unstable nuclei can decay in several ways depending on what makes them unstable. The three most important types for IB Physics are alpha (α) decay, beta (β) decay, and gamma (γ) emission. Each produces different particles, penetrates matter to different depths, and changes the parent nucleus in a specific way.

The three main types of radioactive decay compared side by side. Alpha decay ejects a helium-4 nucleus (2 protons + 2 neutrons), reducing both A and Z. Beta-minus decay converts a neutron into a proton, emitting an electron and an antineutrino. Gamma emission releases a high-energy photon without changing A or Z.

Property	Alpha (α)	Beta-minus (β⁻)	Gamma (γ)
Particle emitted	⁴₂He nucleus (2p + 2n)	Electron (e⁻) + antineutrino (ν̄_e)	High-energy photon
Change in A	−4	0	0
Change in Z	−2	+1	0
Ionising ability	Very high	Moderate	Low
Penetrating power	Low (stopped by paper)	Medium (stopped by ~mm aluminium)	High (reduced by thick lead/concrete)
Charge	+2e	−1e	0
Deflection in E/B fields	Deflected (low radius)	Deflected (opposite direction)	Not deflected

Summary of properties of alpha, beta-minus, and gamma radiation

SECTION 6

Worked Example

Let's work through a complete problem that combines several concepts from this topic. We will determine how much of a radioactive sample remains after a given time and calculate its activity.

Iodine-131 Decay in Medical Treatment

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Step 1 — Identify Given Values

A hospital uses iodine-131 (¹³¹I) to treat thyroid cancer. The half-life of iodine-131 is t₁/₂ = 8.02 days. A patient receives a sample with an initial activity of A₀ = 3.7 × 10⁸ Bq. We want to find the activity after t = 24.06 days (exactly 3 half-lives).
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Step 2 — Calculate the Decay Constant

Using the relationship λ = ln 2 / t₁/₂, we first convert the half-life to seconds: t₁/₂ = 8.02 × 24 × 3600 = 6.929 × 10⁵ s. Then λ = 0.693 / (6.929 × 10⁵) = 1.000 × 10⁻⁶ s⁻¹. Alternatively, working in days: λ = 0.693 / 8.02 = 0.0864 day⁻¹.

λ ≈ 1.00 × 10⁻⁶ s⁻¹ (or 0.0864 day⁻¹)
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Step 3 — Quick Method: Counting Half-Lives

Since 24.06 days = 3 × 8.02 days = exactly 3 half-lives, we can use the quick method: A = A₀ × (½)³ = A₀ / 8. Therefore A = (3.7 × 10⁸) / 8 = 4.625 × 10⁷ Bq.

A = 4.63 × 10⁷ Bq (after 3 half-lives)
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Step 4 — Verify Using the Exponential Formula

Using A = A₀ × $e^{−λt}$ with λ = 0.0864 day⁻¹ and t = 24.06 days: λt = 0.0864 × 24.06 = 2.079. So A = 3.7 × 10⁸ × $e^{−2.079}$ = 3.7 × 10⁸ × 0.1250 = 4.63 × 10⁷ Bq. This confirms our half-life counting result.

A = 4.63 × 10⁷ Bq ✓ (confirmed)
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Step 5 — Interpret the Result

After 24 days, only 1/8 (12.5%) of the original activity remains. This means the radiation dose to the patient decreases significantly over about three weeks, which is important for treatment planning and patient safety. After 10 half-lives (≈ 80 days), the remaining activity would be less than 0.1% of the original.

SECTION 7

Applications, Strengths & Limitations

Radioactive decay is not just a theoretical concept — it underpins many technologies and dating methods. However, the mathematical model has certain assumptions and limitations that you should be aware of for the IB exam and for critical thinking about real-world applications.

Application	How Decay Is Used	Key Isotope
Carbon dating	Ratio of ¹⁴C to ¹²C in organic remains reveals age (up to ~50 000 years)	Carbon-14 (t₁/₂ ≈ 5730 yr)
Medical imaging (PET)	Positron-emitting tracers are injected; gamma rays from annihilation are detected	Fluorine-18 (t₁/₂ ≈ 110 min)
Smoke detectors	Alpha particles ionise air between plates; smoke disrupts the current	Americium-241 (t₁/₂ ≈ 432 yr)
Nuclear power	Fission of heavy nuclei (triggered, not spontaneous) releases energy; waste management uses decay curves	Uranium-235, Plutonium-239
Cancer treatment	Targeted radiation destroys tumour cells; short half-life limits exposure to healthy tissue	Iodine-131 (t₁/₂ ≈ 8 days)

Real-world applications of radioactive decay

Strengths & Limitations of the Decay Model

Strengths	Limitations
Highly accurate for large sample sizes (Avogadro-scale numbers)	Cannot predict when a specific individual nucleus will decay
Half-life is unaffected by external conditions (temperature, pressure, chemical bonding)	For very small samples, statistical fluctuations become significant
Exponential model matches experimental data extremely well	Model assumes a pure sample of one isotope — real samples may contain decay chains
Simple mathematics (exponential functions) makes calculations accessible	Does not explain the quantum mechanical mechanism of tunnelling (covered in more advanced theory)

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KEY TAKEAWAY

The exponential decay model works beautifully for large populations of atoms — much like how insurance companies can predict the number of claims per year even though they cannot predict which individual will file one. The strength of the model lies in statistics: the more atoms you have, the more reliable the prediction. The model breaks down only when sample sizes become tiny, or when you try to say something about a single nucleus.

SECTION 8

Connection to Advanced Theory

The E.3 treatment of radioactive decay sits at the intersection of classical nuclear physics and quantum mechanics. While the IB course focuses on the statistical description (exponential decay law, half-life), the underlying mechanism involves quantum phenomena that you may encounter in further study or in the IB Higher Level option.

Concept in E.3 (Standard)	Advanced Extension
Decay is random and spontaneous	Quantum tunnelling explains alpha decay — the alpha particle has a probability of 'tunnelling' through the nuclear potential barrier even though classically it does not have enough energy to escape.
Activity A = λN	The decay constant λ is derived from Fermi's Golden Rule in quantum mechanics, connecting the transition rate to the matrix element and density of states.
Beta decay emits an electron and antineutrino	The weak nuclear force mediates beta decay via W and Z bosons. The continuous energy spectrum of beta particles was historically the evidence for the neutrino's existence.
Decay chains (series of decays)	Bateman equations describe the time evolution of multiple linked isotopes in a decay chain, leading to concepts like secular equilibrium.

How E.3 concepts connect to more advanced nuclear and quantum physics

Understanding the quantum tunnelling model of alpha decay is a beautiful example of how quantum mechanics explains something that classical physics cannot. In classical physics, an alpha particle inside the nucleus does not have enough kinetic energy to overcome the electrostatic potential barrier. However, quantum mechanics tells us there is a small but non-zero probability that the particle can appear on the other side of the barrier. This probability determines the decay constant λ and, ultimately, the half-life of the isotope.

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Looking Ahead

If you continue to study physics at university, you will learn to derive the decay constant from first principles using quantum mechanics. For the IB exam, focus on applying the exponential decay law, understanding the three decay types, and interpreting half-life graphs and calculations with confidence.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why the rate of radioactive decay decreases over time, even though the probability of decay for each individual nucleus remains constant.

Show answer

The rate of decay (activity) is given by A = λN. While the decay constant λ does not change, the number of undecayed nuclei N decreases over time as nuclei decay. Since activity is proportional to N, the activity decreases as the sample shrinks. Each surviving nucleus still has the same probability of decaying in the next second, but there are fewer of them to contribute to the total count. This is why the exponential decay curve gets flatter over time — the slope (rate) diminishes as N falls.

Question 2: BASIC CALCULATION

A sample of strontium-90 has a half-life of 28.8 years. Calculate the decay constant λ in s⁻¹. If a sample initially contains 5.0 × 10²⁰ atoms of Sr-90, what is the initial activity?

Show answer

First, convert the half-life to seconds: t₁/₂ = 28.8 × 365.25 × 24 × 3600 ≈ 9.087 × 10⁸ s. Then λ = ln 2 / t₁/₂ = 0.693 / (9.087 × 10⁸) ≈ 7.63 × 10⁻¹⁰ s⁻¹. The initial activity is A₀ = λN₀ = (7.63 × 10⁻¹⁰)(5.0 × 10²⁰) = 3.8 × 10¹¹ Bq. This is about 380 GBq, which is an extremely high activity — strontium-90 is a significant concern in nuclear waste because of its relatively long half-life combined with high initial activity.

Question 3: INTERMEDIATE

A radioactive isotope has a half-life of 6.0 hours. At time t = 0, a Geiger counter records an activity of 4800 Bq from the sample. (a) What will the activity be after 18 hours? (b) How long will it take for the activity to fall to 300 Bq?

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(a) 18 hours = 3 half-lives, so A = 4800 × (½)³ = 4800 / 8 = 600 Bq. (b) We need A = A₀ × (½)^(t/t₁/₂), so 300 = 4800 × (½)^(t/6). Dividing: 300/4800 = 1/16 = (½)⁴. Therefore t/6 = 4, giving t = 24 hours. Alternatively, using the exponential form: λ = 0.693/6 = 0.1155 h⁻¹; 300 = 4800 ×

e^{−0.1155t}

; ln(300/4800) = −0.1155t; −2.773 = −0.1155t; t = 24.0 hours. Both methods give the same answer. The key insight is recognising that 300 is exactly 1/16 of 4800, corresponding to 4 half-lives.

Question 4: APPLIED

An archaeologist finds a wooden artefact in which the ratio of carbon-14 to carbon-12 is 25% of the ratio found in living organisms. The half-life of carbon-14 is 5730 years. Estimate the age of the artefact. Explain one assumption you must make for this estimate to be valid.

Show answer

If the ¹⁴C/¹²C ratio is 25% of the living value, then the fraction of ¹⁴C remaining is 0.25 = (½)ⁿ, where n is the number of half-lives. Since 0.25 = (½)², we have n = 2 half-lives. The age is t = 2 × 5730 = 11 460 years. A key assumption is that the ratio of ¹⁴C to ¹²C in the atmosphere has remained approximately constant over the past 11 000+ years. If cosmic ray intensity (which produces ¹⁴C in the atmosphere) varied significantly, the initial ¹⁴C content of the wood would differ from what we assume, making the date inaccurate. Calibration curves using tree-ring data help correct for this.

Question 5: CRITICAL THINKING

A student argues: 'After two half-lives, all of the radioactive nuclei in a sample will have decayed — half decay in the first half-life, and the other half decay in the second.' Identify the error in this reasoning. Then explain why, in theory, a radioactive sample never fully decays to zero. Discuss whether this has any practical significance.

Show answer

The student's error is treating the half-life as if it applies to the original total each time. In reality, after the first half-life, half of N₀ remains. The second half-life halves the remaining quantity (not the original), so N = N₀/4 — one quarter remains, not zero. After each successive half-life, the remaining amount is halved again: N₀/8, N₀/16, and so on. Mathematically, the exponential function

e^{−λt}

approaches zero asymptotically but never equals zero, so the model predicts the sample never fully decays. In practice, however, since matter is quantised into individual atoms, eventually you reach the last undecayed atom, and once it decays, the sample truly has zero activity. For macroscopic samples, a common rule is that after about 10 half-lives (when ~0.1% remains), the activity is negligible for practical purposes. This distinction between the continuous mathematical model and the discrete atomic reality is a good example of model limitations.

SUMMARY

Lesson Summary

Summary

Radioactive decay is the spontaneous and random transformation of an unstable nucleus into a more stable one, emitting particles or energy. The three main types are alpha decay (emitting ⁴₂He, reducing A by 4 and Z by 2), beta-minus decay (converting a neutron to a proton, emitting e⁻ and ν̄, increasing Z by 1), and gamma emission (releasing a high-energy photon with no change to A or Z). These decay types differ in their ionising ability, penetrating power, and behaviour in electric and magnetic fields.

The number of undecayed nuclei follows the exponential decay law: N = N₀ $e^{−λt}$ , where λ is the decay constant. The half-life (t₁/₂ = ln 2 / λ) is the time for half the remaining nuclei to decay. Activity (A = λN) also decays exponentially. These equations apply to carbon dating, medical diagnostics, nuclear energy, and waste management — real-world contexts that frequently appear on the IB exam.