Radioactivity

Radioactivity

Info: Board Coverage AQA Paper 2 | Edexcel CP3 | OCR (A) Paper 2 | CIE P4

Explore the simulation above to develop intuition for this topic.

Intuition

Radioactivity is nature’s way of stabilising unstable nuclei: Imagine a crowded room where people are packed too tightly — some will logically move to less crowded areas. Similarly, nuclei with too many or too few neutrons relative to protons are unstable and will spontaneously emit particles to reach a more stable configuration. This emission is radioactivity.

Why it matters: Radioactivity powers the Sun, enables medical imaging and cancer treatment, provides energy through nuclear power, and allows carbon dating of ancient artefacts. Understanding it explains why some materials are dangerous and others are harmless.

The key insight: Radioactive decay is fundamentally random — you cannot predict when a specific nucleus will decay, only the probability that it will decay within a given time. This is why we use half-life (the time for half the nuclei to decay) rather than exact decay times. The randomness is not due to our ignorance — it’s a fundamental property of quantum mechanics.

Definition. Radioactivity is the spontaneous emission of radiation from unstable atomic nuclei.

1. Nuclear Structure

The atom consists of a nucleus containing protons and neutrons (collectively called nucleons), surrounded by electrons.

Notation. A nuclide is written as $\prescript{A}{Z}\mathrm X$ where $A$ is the mass number (nucleon number) and $Z$ is the atomic number (proton number). The neutron number is $N = A - Z$.

Isotopes are atoms of the same element (same $Z$) with different mass numbers (different $N$).

2. Types of Radioactive Decay

Alpha Decay ($\alpha$)

Definition. An alpha particle ($\alpha$) is a helium nucleus ($\prescript{4}{2}\mathrm{He}$), Emitted during alpha decay.

$\prescript{A}{Z}\mathrm X \to \prescript{A-4}{Z-2}\mathrm Y + \prescript{4}{2}\alpha$

Conservation checks:

• Mass number: $A = (A-4) + 4$ $\checkmark$
• Atomic number: $Z = (Z-2) + 2$ $\checkmark$

Alpha particles are highly ionising but poorly penetrating (stopped by paper or a few cm of air).

Beta Decay ($\beta^-$)

Definition. A beta particle ($\beta^-$) is an electron emitted when a neutron converts to a Proton in beta-minus decay.

A neutron converts to a proton, emitting an electron and an antineutrino:

$\prescript{A}{Z}\mathrm X \to \prescript{A}{Z+1}\mathrm Y + \prescript{0}{-1}\beta^- + \bar{\nu}_e$

Conservation checks:

• Mass number: $A = A + 0$ $\checkmark$
• Atomic number: $Z = (Z+1) + (-1)$ $\checkmark$
• Charge: $0 = (+1) + (-1) + 0$ $\checkmark$

The antineutrino was postulated by Fermi (1934) to conserve energy and momentum — the observed Electron energy spectrum is continuous, implying a third particle carries away the remaining energy.

Penetration: Beta particles are moderately ionising, stopped by a few mm of aluminium.

Beta-Plus Decay ($\beta^+$)

Definition. A beta-plus particle ($\beta^+$) is a positron emitted when a proton converts to a Neutron in beta-plus decay.

A proton converts to a neutron, emitting a positron and a neutrino:

$\prescript{A}{Z}\mathrm X \to \prescript{A}{Z-1}\mathrm Y + \prescript{0}{+1}\beta^+ + \nu_e$

This only occurs in proton-rich nuclei.

Gamma Decay ($\gamma$)

Definition. A gamma ray ($\gamma$) is high-energy electromagnetic radiation emitted during Nuclear transitions.

After alpha or beta decay, the daughter nucleus is often in an excited state. It de-excites by Emitting a gamma ray (high-energy photon):

$\prescript{A}{Z}\mathrm X^* \to \prescript{A}{Z}\mathrm X + \gamma$

No change in $A$ or $Z$. Gamma rays are weakly ionising but highly penetrating (stopped by thick Lead or concrete).

Info: Board Coverage AQA Paper 2 | Edexcel CP3 | OCR (A) Paper 2 | CIE P4 | Requirement | AQA | Edexcel | OCR (A) | CIE | | ---------------------------- | ----------- | ----------- | ----------- | -------- | | Alpha decay equations | Yes | Yes | Yes | Yes | | $\beta^-$ decay equations | Yes | Yes | Yes | Yes | | $\beta^+$ decay equations | Required | Required | Not in core | Yes | | Neutrino / antineutrino | Qualitative | Qualitative | Qualitative | Detailed | | Spontaneous vs random nature | Required | Required | Required | Required |

3. Exponential Decay Law

Derivation of $N = N_0 e^{-\lambda t}$

Definition. The decay constant $\lambda$ is the probability per unit time of a nucleus decaying.

$\boxed{\lambda = \frac{\ln 2}{t_{1/2}}}$

The probability that any single nucleus decays in a small time interval $dt$ is $\lambda\,dt$.

If there are $N$ nuclei, the number decaying in $dt$ is:

$dN = -\lambda N\,dt$

(The minus sign indicates $N$ decreases.)

$\frac{dN}{dt} = -\lambda N$

Separating variables and integrating:

$\int_{N_0}^{N}\frac{dN}{N} = -\int_0^t\lambda\,dt$

$\ln N - \ln N_0 = -\lambda t$

$\ln\left(\frac{N}{N_0}\right) = -\lambda t$

$\boxed{N = N_0 e^{-\lambda t}}$

Activity

Definition. Activity $A$ is the rate of decay of a radioactive sample: $A = \lambda N = -\frac{dN}{dt}$.

$\boxed{A = \lambda N = -\frac{dN}{dt}}$

Expanding using the exponential decay law:

$A = \lambda N_0 e^{-\lambda t} = A_0 e^{-\lambda t}$

Definition. The becquerel is the SI unit of activity; 1 Bq = 1 decay per second.

$\boxed{1\,\mathrm{Bq} = 1\,\mathrm{decay s}^{-1}}$

Half-Life

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

$\boxed{N(t) = N_0 \cdot 2^{-t/t_{1/2}}}$

Setting $N = N_0/2$:

$\frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}}$

$\frac{1}{2} = e^{-\lambda t_{1/2}}$

$\ln 2 = \lambda t_{1/2}$

$\boxed{t_{1/2} = \frac{\ln 2}{\lambda}}$

Intuition. After one half-life, half remain. After two, a quarter. After $n$ half-lives, $N = N_0/2^n$. The decay is exponential — the activity is always proportional to the number of Remaining nuclei, so it decreases exponentially.

Radiation Detection

Scintillation Detector

A scintillation detector uses a crystal (e.g. Sodium iodide) that emits flashes of light when Radiation passes through it. The light pulses are converted to electrical signals by a Photomultiplier tube. Scintillation detectors are more sensitive than GM tubes and can distinguish Between different types of radiation by the intensity of the light pulse (pulse-height analysis), Making them useful for identifying specific isotopes.

Half-Life Measurement Techniques

Graphical Method

To determine half-life experimentally:

1. Measure the corrected count rate (subtracting background) at regular time intervals.
2. Plot a graph of corrected count rate (or activity) against time.
3. Find the time for the count rate to halve. This is $t_{1/2}$.
4. Alternatively, plot $\ln A$ vs $t$; the gradient equals $-\lambda$.

The logarithmic method is more accurate because it uses all data points, not just a single halving:

$\ln A = \ln A_0 - \lambda t$

The gradient of the straight line gives $-\lambda$And then $t_{1/2} = \ln 2 / \lambda$.

Practical Considerations

• For short half-lives (seconds to minutes), readings can be taken in real time with a GM tube and data logger.
• For long half-lives (years to millennia), direct measurement is impractical. Instead, measure a known mass of the isotope and use $A = \lambda N$ where $N$ is calculated from the mass and Avogadro’s number.

Problem Set

4. Decay Chains

Info: Board Coverage AQA Paper 2 | Edexcel CP3 | OCR (A) Paper 2 | CIE P4 Definition. A decay chain (or radioactive series) is a sequence of radioactive decays where each Daughter nuclide is itself radioactive, continuing until a stable nuclide is reached.

The Uranium-238 Decay Series

The most important natural decay chain begins with uranium-238 and ends at stable lead-206 after 14 Decays (8 alpha and 6 beta-minus):

Verification of net change: 8 alpha decays reduce $A$ by $8 \times 4 = 32$ and $Z$ by $8 \times 2 = 16$. 6 beta-minus decays increase $Z$ by $6 \times 1 = 6$ and leave $A$ unchanged. Net: $A$ decreases by 32 ($238 - 206 = 32$ $\checkmark$) and $Z$ decreases by 10 ($92 - 82 = 10$ $\checkmark$).

Other natural decay chains include:

• Thorium-232 series $\to$ $^{208}\mathrm{Pb}$ (6 alpha, 4 beta-minus)
• Uranium-235 series $\to$ $^{207}\mathrm{Pb}$ (7 alpha, 4 beta-minus)

Secular Equilibrium

When a long-lived parent nuclide (very large $t_{1/2}$) decays through a chain of shorter-lived Daughters, a state of secular equilibrium is established after several daughter half-lives. In Equilibrium:

$\boxed{\lambda_1 N_1 = \lambda_2 N_2 = \lambda_3 N_3 = \cdots}$

Each daughter has the same activity as the parent. This has practical importance in radiological Protection: the activity of radon-222 (a daughter of $^{238}\mathrm U$) in buildings is sustained By the essentially constant uranium in the ground beneath.

5. Applications of Radioactivity

Carbon-14 Dating

6. Case Studies

Chernobyl (1986)

The Chernobyl disaster released approximately $5.2 \times 10^{18}$ Bq of radioactive material into The atmosphere. The most significant isotopes released were:

• Iodine-131 ($t_{1/2} = 8.04$ days): Caused thyroid cancer, particularly in children who drank milk from cows that had grazed on contaminated grass. The short half-life meant the acute danger passed within months.
• Caesium-137 ($t_{1/2} = 30.2$ years): More persistent contaminant. Caesium behaves chemically like potassium and was taken up by plants and animals, entering the food chain. The 30 km exclusion zone around the reactor remains in place partly because of $^{137}\mathrm{Cs}$ contamination.

The different half-lives illustrate a key principle: short-lived isotopes produce intense but brief Radiation hazards, while long-lived isotopes create lower but persistent contamination.

Fukushima Daiichi (2011)

The Fukushima disaster, triggered by a tsunami following the Tohoku earthquake, released significant Quantities of radioactive isotopes including $^{131}\mathrm I$, $^{134}\mathrm{Cs}$ ($t_{1/2} = 2.06$ years), and $^{137}\mathrm{Cs}$.

Key differences from Chernobyl:

• Most radioactive material was released into the Pacific Ocean rather than the atmosphere, leading to rapid dilution of water-soluble isotopes.
• $^{134}\mathrm{Cs}$ has a short half-life (2.06 years) and decayed rapidly; $^{137}\mathrm{Cs}$ remains the primary long-term concern.
• The disaster highlighted the importance of cooling systems for spent fuel and the vulnerability of nuclear plants to extreme natural events.

Both disasters demonstrate that understanding half-lives is critical for predicting contamination Timelines and informing public health responses.

Additional Problems

Cross-References

• Nuclear Energy: Shows how radioactive decay and nuclear reactions release energy through mass-energy conversion.
• Quantum Physics: Explains the quantum mechanical basis for radioactive decay and nuclear stability.
• Half-Life: Explores the mathematical treatment of decay rates and activity in more depth.
• Radiation Safety: Covers practical applications and safety considerations when working with radioactive materials.