1.4 β‑decay at quark level (required learning outcome)
β⁻ decay: a down quark changes into an up quark, emitting an electron and an antineutrino
$$d \;\rightarrow\; u + e^{-} + \bar{u}_{e}$$
At the nucleon level: $n \rightarrow p^{+}+e^{-}+\bar{u}_{e}$.
β⁺ decay (positron emission): an up quark changes into a down quark, emitting a positron and a neutrino
$$u \;\rightarrow\; d + e^{+} + u_{e}$$
At the nucleon level: $p^{+} \rightarrow n + e^{+} + u_{e}$.
2. Structure of the Atom
Nucleus
Contains $Z$ protons and $N$ neutrons (collectively called nucleons).
Radius ≈ $1\times10^{-15}\,\text{m}$ (1 fm); about 10 000 times smaller than the whole atom.
Mass number $A = Z + N$.
Holds > 99.9 % of the atomic mass.
Electron cloud
Electrons occupy *orbitals* that, for an introductory model, are represented by concentric shells $K$, $L$, $M$, … with principal quantum numbers $n = 1,2,3,\dots$.
Maximum capacity of a shell: $2n^{2}$ electrons (e.g. $K$ holds 2, $L$ holds 8, $M$ holds 18).
Atomic radius is typically $0.1$–$0.3\,$nm (1 Å = $10^{-10}$ m).
3. Nuclear Notation, Isotopes & Conservation Laws
The Cambridge notation for a nuclide is $^{A}_{Z}\text{X}$, where:
$\text{X}$ = chemical symbol,
$Z$ = atomic number (number of protons),
$A$ = mass number ($Z+N$).
In **all nuclear reactions** the following are conserved:
Charge (Z) – the total number of protons before and after the reaction is the same.
Nucleon number (A) – the total number of protons + neutrons is unchanged.
Isotope
Notation
$Z$ (protons)
$N$ (neutrons)
$A$ (mass number)
Carbon‑12
$^{12}_{6}\text{C}$
6
6
12
Carbon‑14
$^{14}_{6}\text{C}$
6
8
14
Uranium‑235
$^{235}_{92}\text{U}$
92
143
235
4. Types of Radioactive Decay
4.1 Alpha (α) decay
Emission of a $^{4}_{2}\alpha$ particle (2 p + 2 n).
$$^{A}_{Z}\text{X}\;\rightarrow\;^{A-4}_{Z-2}\text{Y}+^{4}_{2}\alpha$$
Low penetrating power (stopped by a sheet of paper); high ionising power.
Example (exam style): A sample contains $2.0\times10^{6}$ atoms of a radionuclide. After 10 days the activity is measured to be $5.0\times10^{5}$ disintegrations s⁻¹. If the initial activity was $1.0\times10^{6}$ disintegrations s⁻¹, calculate the half‑life.
Determine $\lambda$ from $A=\lambda N$: $\lambda = A/N_{0}=1.0\times10^{6}/2.0\times10^{6}=5.0\times10^{-1}\,\text{s}^{-1}$ (note: convert days to seconds if required).
Half‑life $t_{1/2}=0.693/\lambda\approx1.39\,$s (illustrative – the numbers can be adapted for a realistic half‑life).
6. Mass Defect and Binding Energy
The mass of a nucleus is slightly less than the sum of the masses of its separate nucleons. The difference is the mass defect $\Delta m$.
$$
\Delta m = Zm_{p}+Nm_{n}-m_{\text{nucleus}}
$$
Using Einstein’s relation $E=mc^{2}$, the corresponding binding energy $E_{b}$ is
$$
E_{b}= \Delta m\,c^{2}
$$
Typical binding energies are a few MeV per nucleon; the curve of $E_{b}/A$ explains why:
Heavy nuclei release energy by fission (splitting into fragments with higher $E_{b}/A$).
Light nuclei release energy by fusion (joining to form a nucleus with higher $E_{b}/A$).
7. Nuclear Reactions and Applications
Fission – a heavy nucleus splits, emitting neutrons and ≈ 200 MeV per event.
$$^{235}_{92}\text{U}+^{1}_{0}n\;\rightarrow\;^{141}_{56}\text{Ba}+^{92}_{36}\text{Kr}+3^{1}_{0}n+\text{energy}$$
Positron Emission Tomography (PET) – uses β⁺ emitters such as $^{18}_{9}\text{F}$.
Radiotherapy – high‑energy γ‑rays from $^{60}_{27}\text{Co}$ or electron beams.
Diagnostic X‑rays – produced by bremsstrahlung when high‑energy electrons are decelerated in a target.
Industrial & safety – Geiger–Müller counters, cloud chambers, shielding calculations and neutron activation analysis all rely on the concepts above.
8. Practical Skills (AO3)
Cloud‑chamber experiment – Visualise α‑ and β‑tracks.
Fill a sealed container with alcohol vapour and place a cold plate (dry ice) on top.
Introduce a weak α/β source (e.g., a thoriated lantern mantle).
Observe straight, thick α‑tracks and thinner, more erratic β‑tracks.
Record track length and density; discuss how ionisation density relates to penetrating power.
Always wear lab coat, gloves, and eye protection when handling radioactive material.
Use a calibrated Geiger–Müller tube; record background counts and subtract them from sample counts.
To determine a half‑life experimentally, plot $\ln(N)$ versus time – the slope equals $-\lambda$.
9. Key Points to Remember
The nucleus contains almost all the atomic mass but occupies only ~10⁻⁵ of the atomic volume.
Protons define the element ($Z$); neutrons affect stability and give rise to isotopes.
Electrons balance the nuclear charge in a neutral atom and occupy discrete shells following the $2n^{2}$ rule.
α, β⁻, β⁺ and γ radiations differ in composition, charge, penetrating power and ionising ability.
In every nuclear reaction, both charge ($Z$) and nucleon number ($A$) are conserved.
Radioactive decay follows an exponential law; the half‑life is a characteristic constant for each radionuclide.
Mass defect and binding energy explain why energy is released in fission (heavy nuclei) and fusion (light nuclei).
Understanding nuclear processes underpins applications in medicine, industry, energy generation and radiation safety.
10. Suggested Classroom Diagram
Simple schematic of an atom: a central sphere (the nucleus) containing red dots (protons) and blue dots (neutrons); concentric circles labelled $K$, $L$, $M$, … with the correct number of yellow dots (electrons) for a chosen element (e.g., sodium, $Z=11$).
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