Module 6 · Particles and Medical Physics

Fundamental Particles & Isotopes

Specification: OCR A H556 | Sections: 6.7.3, 6.8.1–6.8.3, 6.9.1 | Teaching time: ~8 hours

By the end of this topic you should be able to…

Use ˢX notation and define isotopes; explain why some isotopes are unstable
Classify particles as hadrons or leptons and describe the properties of each
Describe the quark model — up, down, and strange quarks and their antiquarks
Use conservation of charge, baryon number, lepton number, and strangeness in particle interactions
Explain beta decay in terms of quark transformations

Part 1

Isotopes & Nuclear Notation

Every element is defined by its atomic number Z — the number of protons in the nucleus. The mass number A is the total number of nucleons (protons + neutrons). These are written in the standard ˢX notation:

Standard nuclear notation ^A_ZX

For example, carbon-12 is written ¹²₆C — it has 6 protons and 6 neutrons. The number of neutrons N is always A − Z.

Isotopes are atoms of the same element (same Z) with different numbers of neutrons (different A). They have identical chemical properties because their electron configurations are the same, but different nuclear properties — some isotopes are stable while others are radioactive.

Common isotopes

Isotope	Notation	Protons	Neutrons	Stable?
Hydrogen-1	¹₁H	1	0	✓ Stable
Hydrogen-2 (deuterium)	²₁H	1	1	✓ Stable
Hydrogen-3 (tritium)	³₁H	1	2	✗ Radioactive
Carbon-12	¹²₆C	6	6	✓ Stable
Carbon-14	¹⁴₆C	6	8	✗ Radioactive
Uranium-235	²³⁵₉₂U	92	143	✗ Radioactive
Uranium-238	²³⁸₉₂U	92	146	✗ Radioactive

⚡ Key Point

Nuclear stability depends on the neutron-to-proton ratio. For light elements (Z < 20), stable nuclei have roughly equal numbers of protons and neutrons (N/Z ≈ 1). For heavier elements, more neutrons are needed to overcome the electrostatic repulsion between protons (N/Z ≈ 1.5 for the heaviest stable nuclei). Isotopes with too many or too few neutrons are unstable and undergo radioactive decay.

The relative atomic mass of an element is a weighted average of the masses of its naturally occurring isotopes, taking into account their relative abundances.

Part 2

Classifying Subatomic Particles

Subatomic particles are divided into two broad categories: hadrons and leptons. This classification is based on whether particles experience the strong nuclear force.

Leptons

Leptons are fundamental particles — they are not composed of any smaller particles. They do not experience the strong nuclear force. Each lepton has a corresponding antilepton. At A Level, you need to know three leptons and their neutrinos:

Lepton	Symbol	Charge /e	Lepton number L	Antiparticle
Electron	e⁻	−1	+1	e⁺ (positron)
Electron neutrino	νₑ	0	+1	ν̄ₑ
Muon	μ⁻	−1	+1	μ⁺
Muon neutrino	νμ	0	+1	ν̄μ
Tau	τ⁻	−1	+1	τ⁺
Tau neutrino	ντ	0	+1	ν̄τ

Hadrons

Hadrons do experience the strong nuclear force. Unlike leptons, hadrons are not fundamental — they are composed of smaller particles called quarks. Hadrons are subdivided into two groups:

Baryons — made of three quarks (qqq). The proton and neutron are baryons. They have baryon number B = +1.
Mesons — made of a quark–antiquark pair (qq̄). Pions (π) and kaons (K) are mesons. They have baryon number B = 0.

💡 Exam Tip

Every particle has a corresponding antiparticle with the same mass but opposite charge, baryon number, and lepton number. When a particle meets its antiparticle, they annihilate — their combined mass converts into photon energy. Antibaryons (like the antiproton p̄) are made of three antiquarks and have B = −1.

Part 3

The Quark Model

Quarks are fundamental particles that combine to form hadrons. At A Level you need to know three types (called flavours): up, down, and strange. Each has a corresponding antiquark with opposite properties.

Quark	Symbol	Charge /e	Strangeness S	Baryon number B
Up	u	+⅔	0	+⅓
Down	d	−⅓	0	+⅓
Strange	s	−⅓	−1	+⅓
Anti-up	ū	−⅔	0	−⅓
Anti-down	d̄	+⅓	0	−⅓
Anti-strange	s̄	+⅓	+1	−⅓

Quark composition of familiar particles

Particle	Type	Quark composition	Charge check	Baryon number
Proton (p)	Baryon	uud	+⅔ + ⅔ − ⅓ = +1	+1
Neutron (n)	Baryon	udd	+⅔ − ⅓ − ⅓ = 0	+1
π⁺ (pion plus)	Meson	ud̄	+⅔ + ⅓ = +1	0
π⁻ (pion minus)	Meson	ūd	−⅔ − ⅓ = −1	0
K⁺ (kaon plus)	Meson	us̄	+⅔ + ⅓ = +1	0
K⁰ (kaon zero)	Meson	ds̄	−⅓ + ⅓ = 0	0

⚡ Key Point

Notice the pattern: baryons always have B = +1 (three quarks, each with B = +⅓). Mesons always have B = 0 (quark with B = +⅓ plus antiquark with B = −⅓). Antibaryons have B = −1 (three antiquarks). The only hadrons you need to know at A Level are baryons and mesons.

Part 4

Conservation Laws & Beta Decay

In all particle interactions and decays, the following quantities are always conserved (the total before = total after):

Charge — measured in units of the elementary charge e
Baryon number B — baryons: +1, antibaryons: −1, everything else: 0
Lepton number L — leptons: +1, antileptons: −1, everything else: 0
Strangeness S — conserved in the strong interaction but can change by ±1 in the weak interaction

Beta-minus decay

In β⁻ decay, a neutron inside a nucleus converts into a proton, emitting an electron and an electron antineutrino:

Beta-minus decay (overall) n → p + e⁻ + ν̄ₑ

At the quark level, a down quark transforms into an up quark via the weak interaction:

Beta-minus decay (quark level) d → u + e⁻ + ν̄ₑ

Beta-plus decay

In β⁺ decay, a proton converts into a neutron, emitting a positron and an electron neutrino:

Beta-plus decay (overall) p → n + e⁺ + νₑ

At the quark level, an up quark transforms into a down quark:

Beta-plus decay (quark level) u → d + e⁺ + νₑ

Question: A nucleus of carbon-14 undergoes beta-minus decay. Write the nuclear equation and verify charge and baryon number conservation.

Identify the process: β⁻ decay means a neutron → proton + electron + antineutrino.

Write the equation: ¹⁴₆C → ¹⁴₇N + e⁻ + ν̄ₑ

Check charge: Before = +6. After = +7 + (−1) + 0 = +6. ✓

Check baryon number: Before = +1. After = +1 + 0 + 0 = +1. ✓

Question: A K⁻ meson (s̄u) decays into a μ⁻ and a μ antineutrino. Verify charge, lepton number, and strangeness conservation.

Write the equation: K⁻ → μ⁻ + ν̄μ

Check charge: Before = −1. After = (−1) + 0 = −1. ✓

Check lepton number: Before = 0. After = +1 + (−1) = 0. ✓

Check strangeness: Before = +1 (s̄ quark). After = 0. Strangeness is not conserved — this is a weak interaction decay, which is allowed to change strangeness by ±1. ✓

⚠️ Common Mistake

Students often forget that the electron antineutrino (ν̄ₑ) is emitted in β⁻ decay, and the electron neutrino (νₑ) is emitted in β⁺ decay. Without the neutrino/antineutrino, lepton number would not be conserved. Remember: the bar over the ν means it's an antiparticle with L = −1.

Interactive

Quark Builder

Select quarks to build a baryon (3 quarks) or meson (quark + antiquark). The tool checks conservation laws and identifies the particle.

Select quarks:

u (+⅔)

d (−⅓)

s (−⅓)

ū (−⅔)

d̄ (+⅓)

s̄ (+⅓)

Your particle:

Slot 1

Slot 2

Slot 3

Analysis

Total charge

—

Baryon number

—

Strangeness

—

Select quarks to build a particle

Mode: Baryon

Slots filled: 0/3

Known particles: proton, neutron, π⁺, π⁻, K⁺, K⁰, Σ⁺, Σ⁰, Σ⁻, Ξ⁰, Ξ⁻

Interactive

Isotope Builder

Adjust the number of protons and neutrons to build an isotope and check its stability.

Protons (Z)6

Neutrons (N)6

¹²₆C

Carbon-12

✓ Stable

Mass number (A)

N/Z ratio

1.00

Nuclear binding

Stable

Decay mode

None

Check your understanding

Knowledge Check

State the quark composition of (a) a proton and (b) a neutron. Verify the charge of each using the quark charges.

3 marks

Proton: uud [1 mark]
Neutron: udd [1 mark]
Charge check: Proton = +⅔ + ⅔ − ⅓ = +1 ✓; Neutron = +⅔ − ⅓ − ⅃ = 0 ✓ [1 mark]

Explain the difference between a hadron and a lepton. Give one example of each.

3 marks

Hadrons experience the strong nuclear force; leptons do not [1 mark]
Hadrons are composite (made of quarks); leptons are fundamental [1 mark]
Example hadron: proton, neutron, pion; Example lepton: electron, muon, neutrino [1 mark]

What is an isotope? Explain why isotopes of the same element have identical chemical properties but different nuclear stability.

3 marks

Isotopes are atoms of the same element with the same number of protons but different numbers of neutrons [1 mark]
Chemical properties depend on electron configuration, which is the same for all isotopes of an element [1 mark]
Nuclear stability depends on the neutron-to-proton ratio; different neutron numbers change this ratio, making some isotopes unstable [1 mark]

Exam Practice

Exam-Style Questions

(a) A Σ⁺ baryon has a strangeness of −1 and charge +1. Deduce its quark composition. [3 marks]

(b) The Σ⁺ decays via the weak interaction into a proton and a pion (π⁰). State the quark composition of π⁰ and verify that charge and baryon number are conserved in this decay. [4 marks]

8 marks

(a)

Σ⁺ is a baryon, so three quarks [1 mark]
Strangeness −1 means one strange quark (s) [1 mark]
Need charge +1 with s (−⅓): remaining two quarks must give +4/3, so two up quarks: uus [1 mark]

(b)

π⁰ has quark composition uū or dd̄ [1 mark]
Charge: before = +1 (Σ⁺); after = +1 (p) + 0 (π⁰) = +1 ✓ [1 mark]
Baryon number: before = +1 (Σ⁺); after = +1 (p) + 0 (π⁰) = +1 ✓ [1 mark]
Full equation: uus → uud + uū (or dd̄) [1 mark]

(c)

Strangeness changes from −1 (Σ⁺) to 0 (p + π⁰); this is allowed because the decay is via the weak interaction, which can change strangeness by ±1 [1 mark]

Sodium-22 (²²₁₁Na) undergoes beta-plus decay.

(a) Write the full nuclear equation for this decay. [2 marks]

(b) Describe what happens at the quark level during this decay. [2 marks]

7 marks

(a)

²²₁₁Na → ²²₁₀Ne + e⁺ + νₑ [2 marks]

(b)

An up quark in the proton converts to a down quark via the weak interaction [1 mark]
Quark equation: u → d + e⁺ + νₑ [1 mark]

(c)

Charge: Before = +11; After = +10 + (+1) + 0 = +11 ✓ [1 mark]
Baryon number: Before = +22; After = +22 + 0 + 0 = +22 ✓ [1 mark]
Lepton number: Before = 0; After = (−1) + (+1) = 0 ✓ [1 mark]

A student states: "A positron is a type of lepton with the same mass as an electron but with a positive charge."

(a) Is this statement correct? Justify your answer. [2 marks]

(b) State the lepton number of a positron. [1 mark]

(c) When a positron and an electron meet, they annihilate each other. Write an equation for this annihilation and state what is produced. [2 marks]

5 marks

(a)

Yes, the statement is correct [1 mark]
The positron (e⁺) is the antiparticle of the electron (e⁻); it is a lepton with identical mass (0.511 MeV/c²) and opposite charge (+1e) [1 mark]

(b)

Lepton number of positron = −1 (it is an antilepton) [1 mark]

(c)

e⁻ + e⁺ → 2γ (two photons) [1 mark for correct equation]
The rest mass energy of both particles is converted into photon energy (gamma rays) [1 mark]

The table shows four hypothetical particles with their quark compositions. For each, determine the charge, baryon number, and strangeness. State whether each is a baryon, meson, or neither.

A: uds B: ss̄ C: uuu D: ud̄

[6 marks]

6 marks

Particle	Charge	Baryon number	Strangeness	Type
A: uds	+⅔ − ⅓ − ⅓ = 0	+1	−1	Baryon (Λ⁰)
B: ss̄	−⅓ + ⅓ = 0	0	−1 + 1 = 0	Meson (η-type)
C: uuu	+⅔ + ⅔ + ⅔ = +2	+1	0	Baryon (Δ⁺⁺)
D: ud̄	+⅔ + ⅓ = +1	0	0	Meson (π⁺)

[6 marks: 1.5 marks per particle for correct Q, B, S, and type]

Topic Summary

Isotopes

Same Z, different N. Stability depends on N/Z ratio. Radioactive isotopes decay via α, β⁻, or β⁺ emission.

Particle Classification

Leptons: fundamental, no strong force (e⁻, νₑ, μ⁻, νμ). Hadrons: composite, strong force. Baryons (qqq, B=+1) and mesons (qq̄, B=0).

Quark Properties

u: +⅔, s=0, B=+⅓. d: −⅓, s=0, B=+⅓. s: −⅓, s=−1, B=+⅓. Antiquarks: flip all signs.

Conservation Laws

Charge, baryon number, and lepton number are always conserved. Strangeness is conserved in strong interactions but can change in weak interactions.

Key Equations & Relationships

p = uud

n = udd

π⁺ = ud̄

K⁺ = us̄

β⁻: d → u + e⁻ + ν̄ₑ

β⁺: u → d + e⁺ + νₑ