Know that radioactive decay is a change in an unstable nucleus that can result in the emission of α-particles or β-particles and/or γ-radiation and know that these changes are spontaneous and random

5.1 The Nuclear Model of the Atom

• Structure: a tiny, positively‑charged nucleus (radius ≈ 10⁻¹⁴ m) containing protons (p) and neutrons (n) is surrounded by electrons in orbitals.
• Atomic number (Z) – number of protons; determines the element.
• Mass number (A) – total number of nucleons (protons + neutrons).
• Isotopes: atoms with the same Z but different A (different numbers of neutrons).
• Rutherford scattering experiment (1911): α‑particles from a radioactive source were directed at a thin gold foil. Most passed straight through, but a few were deflected at large angles, proving that the positive charge and most of the mass are concentrated in a small nucleus.

5.2 Radioactivity

5.2.1 Detection of Radioactivity

• Geiger–Müller (GM) counter: a gas‑filled tube that produces a pulse for each ionising event; pulses are counted electronically.
• Count‑rate (c/s): number of pulses recorded per second.
• Background radiation must be measured and subtracted:

  
  Example: background = 20 c s⁻¹, measured = 85 c s⁻¹ → corrected count‑rate = 85 − 20 = 65 c s⁻¹.

• Dead‑time correction (optional for A‑Level): if the GM tube is busy for a short time after each pulse, the true count‑rate R can be estimated from the observed rate r using

  \$R = \frac{r}{1 - r\tau}\$

  where τ is the dead‑time (typically ≈ 10⁻⁵ s).

5.2.2 Three Types of Nuclear Emission

5.2.3 Radioactive Decay

Definition (syllabus wording) – Radioactive decay is a spontaneous, random transformation of an unstable nucleus into a more stable configuration.

• Instability arises from either excess nuclear energy or an unfavourable neutron‑to‑proton (N : Z) ratio.
• The decay of a single nucleus cannot be predicted; only the probability for a large collection can be described.
• During decay the mass number (A) and/or the atomic number (Z) change, producing a different element or a different isotope of the same element.

Balanced Nuclear Equations (nuclide notation)

1. α‑decay – example: Radium‑226

   \$^{226}{88}\mathrm{Ra}\;\rightarrow\;^{222}{86}\mathrm{Rn}+^{4}_{2}\alpha\$

2. β⁻‑decay – example: Carbon‑14

   \$^{14}{6}\mathrm{C}\;\rightarrow\;^{14}{7}\mathrm{N}+e^{-}+\bar\nu_{e}\$

3. β⁺‑decay (supplementary) – example: Fluorine‑18

   \$^{18}{9}\mathrm{F}\;\rightarrow\;^{18}{8}\mathrm{O}+e^{+}+\nu_{e}\$

4. γ‑emission – follows a β decay, example: Cobalt‑60

   \$^{60}{27}\mathrm{Co}\;\rightarrow\;^{60}{28}\mathrm{Ni}^{*}+e^{-}+\bar\nu_{e}\$

   \$^{60}{28}\mathrm{Ni}^{*}\;\rightarrow\;^{60}{28}\mathrm{Ni}+\gamma\$

Randomness of Decay

• The probability that a particular nucleus decays in a very short interval \(dt\) is proportional to \(dt\).
• For a large number \(N\) of undecayed nuclei:

  \$dN = -\lambda N\,dt\$

  where \(\lambda\) is the decay constant (s⁻¹).

• Integrating gives the exponential decay law:

  \$N = N_{0}\,e^{-\lambda t}\$

5.2.4 Half‑Life

• Half‑life (\(t_{1/2}\)) – the time required for half of the original nuclei to decay.
• Relation to the decay constant:

  \$t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}\$

• Half‑life is a statistical property; it does not predict when a particular nucleus will decay.

Typical half‑life calculation (example)

Suppose a sample contains \(N_{0}=8.0\times10^{20}\) atoms of a radionuclide with \(\lambda = 2.0\times10^{-3}\;\text{s}^{-1}\).

1. Half‑life: \(t_{1/2}=0.693/\lambda = 0.693/(2.0\times10^{-3}) = 3.47\times10^{2}\,\text{s}\) (≈ 5.8 min).
2. Number remaining after 10 min (600 s):

   \(N = N_{0}e^{-\lambda t}=8.0\times10^{20}e^{-2.0\times10^{-3}\times600}=8.0\times10^{20}e^{-1.2}\approx 2.6\times10^{20}\) atoms.

5.2.5 Safety, Shielding & Applications

• α‑particles: stopped by a sheet of paper, a few centimetres of air, or the outer dead layer of skin. Use thin plastic or aluminium to collect α emitters safely.
• β‑particles: penetrate skin; require thin metal (Al, Plexiglas) or plastic shielding. Avoid dense metal which can produce bremsstrahlung X‑rays.
• γ‑rays: highly penetrating; require dense, high‑Z materials such as lead, several centimetres of concrete, or thick steel.

Applications (brief, for context):

• Medical imaging – PET scans (β⁺ emitters) and γ‑camera diagnostics.
• Radiotherapy – γ‑rays from Cobalt‑60 or high‑energy β⁻ emitters.
• Carbon dating – β⁻ decay of ¹⁴C (half‑life ≈ 5730 y) for archaeological dating.
• Industrial gauging – γ‑ray sources to measure thickness or density of materials.

Summary Checklist

1. State the syllabus definition of radioactive decay – “spontaneous, random”.
2. Identify the three main types of radiation, giving charge, mass, ionising ability (α > β > γ) and penetrating ability (γ > β > α).
3. Write balanced nuclear equations for α, β⁻, (optional β⁺) and γ emissions.
4. Explain why decay is random and use the decay constant to derive the exponential decay law.
5. Define half‑life, relate it to the decay constant, and perform a half‑life calculation.
6. Describe appropriate shielding for each radiation type.
7. Recall the basic set‑up of a Geiger–Müller counter and how to correct count‑rates for background (and dead‑time, if required).
8. Give at least one practical application of each type of radiation.