Nuclear and Particle Physics

Q: Do I need to learn relativistic equations for this topic?

No. You only need to know the qualitative effect of relativity: fast-moving particles have longer observed lifetimes than stationary particles of the same type. No relativistic calculation formulas are required.

Edexcel International A-Level Physics· 2018 Spec Issue 3, Unit 4 Section 4.5, Statements 111-124· 25 min read

1. Evidence for the Nuclear Model of the Atom★★☆☆☆⏱ 5 min

📘 Definition

Nucleon and Proton Numbers

$A, Z$

The nucleon number $A$ is the total number of protons and neutrons in a nucleus. The proton number $Z$ is the total number of protons, equal to the total positive charge of the nucleus in units of $e$ .

Rutherford's alpha scattering experiment provided the first evidence for the nuclear model, replacing the earlier 'plum pudding' model. Alpha particles were fired at a thin gold foil, with three key observations: 1. Most alpha particles passed through undeflected, 2. A small number were deflected by angles >10°, 3. Very few were deflected by >90°. These observations led to the conclusion that atoms are mostly empty space, with a small, dense, positively charged nucleus at the centre.

📐 Worked Example

Explain why the observation that ~1 in 10,000 alpha particles are deflected by >90° supports the existence of a small, massive, positively charged nucleus.

1
1. Large deflection requires a large repulsive force, which can only be produced by a concentrated positive charge (alpha particles are positive, so repelled by positive nuclei).
2
1. Very few particles are deflected by this angle, so the nucleus must occupy a tiny fraction of the total volume of the atom (~1/10,000 the diameter of the atom).
3
1. The nucleus does not move when hit by an alpha particle, so it must be much more massive than the alpha particle.

Exam tip:

Memorise the three key alpha scattering observations and their corresponding conclusions: these are frequently tested in 3-4 mark structured questions.

2. Particle Accelerators and Charged Particle Motion★★★☆☆⏱ 8 min

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Particle beams are produced via thermionic emission: electrons are released from a heated metal filament, then accelerated by electric fields. Two common accelerator types are linear accelerators (linacs) which use alternating electric fields to accelerate particles in a straight line, and cyclotrons which use magnetic fields to bend particles into a circular path, with electric fields accelerating them on each half-orbit. High energy particles are required to probe nucleon structure because their de Broglie wavelength is comparable to the size of nucleons (~ $1 0^{- 15}$ m), allowing them to resolve internal structure.

🔬 Derivation

Goal:

Derive $r = p / B Q$ for a charged particle moving perpendicular to a uniform magnetic field

Starting from:

Magnetic force on a charged particle: $F = B Q v$ , Centripetal force for circular motion: $F = m v^{2} / r$

1
Equate magnetic force and centripetal force, since the magnetic force provides the centripetal force for circular motion:
2
$BQv = \frac{mv^2}{r}$
3
Cancel $v$ from both sides:
4
$BQ = \frac{mv}{r}$
5
Rearrange for $r$ , and substitute $p = m v$ (momentum):
6
$r = \frac{mv}{BQ} = \frac{p}{BQ}$

Result:

The radius of curvature of a particle's path in a magnetic field is proportional to its momentum, and inversely proportional to magnetic flux density and particle charge.

📐 Worked Example

A proton with momentum $2.4 \times 1 0^{- 19}$ kgm/s travels perpendicular to a uniform magnetic field of flux density 0.50 T. Calculate the radius of its circular path, given the charge of a proton is $1.6 \times 1 0^{- 19}$ C.

1
Use the derived formula $r = p / B Q$ :
2
$r = \frac{2.4 \times 10^{-19}}{0.50 \times 1.6 \times 10^{-19}}$
3
Calculate the result: $r = 3.0$ m

Particle detectors identify particles using two general principles: 1. Ionisation: particles ionise atoms in the detector medium, producing a measurable signal, 2. Deflection: the radius of curvature of a particle's path in a magnetic field is used to calculate its momentum and charge. No detailed engineering of detectors is required.

Exam tip:

You are required to derive $r = p / B Q$ from first principles: make sure you can reproduce the derivation above exactly for exam questions.

3. Mass-Energy Equivalence and Particle Units★★★☆☆⏱ 7 min

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📘 Definition

Mass-Energy Equivalence

$∆ E = c^{2} ∆ m$

Mass and energy are interchangeable: a mass defect $∆ m$ corresponds to an energy change $∆ E = c^{2} ∆ m$ . For Unit 4, this is only applied to matter-antimatter creation and annihilation events.

Example:

When an electron and positron annihilate at rest, their total mass is converted into energy carried by two gamma photons.

📐 Worked Example

Calculate the minimum energy of each gamma photon produced when an electron and positron annihilate at rest. Give your answer in MeV. Mass of an electron = $9.11 \times 1 0^{- 31}$ kg, $e = 1.60 \times 1 0^{- 19}$ C, $c = 3.00 \times 1 0^{8}$ m/s.

1
1. Calculate total mass defect: total mass = $2 m_{e} = 2 \times 9.11 \times 1 0^{- 31} = 1.822 \times 1 0^{- 30}$ kg
2
1. Calculate total energy released: $∆ E = c^{2} ∆ m = (3.00 \times 1 0^{8})^{2} \times 1.822 \times 1 0^{- 30} = 1.64 \times 1 0^{- 13}$ J
3
1. Convert to eV: divide by $1.60 \times 1 0^{- 19}$ : $1.64 \times 1 0^{- 13} /1.60 \times 1 0^{- 19} = 1.025 \times 1 0^{6}$ eV = 1.025 MeV
4
1. Divide by 2 for each photon: minimum energy per photon = 0.51 MeV (2 significant figures)

Common particle units you need to be able to convert between: 1. Energy: MeV (mega-electronvolt) = $1 0^{6}$ eV, GeV = $1 0^{9}$ eV, 2. Mass: MeV/ $c^{2}$ or GeV/ $c^{2}$ , which come from rearranging $∆ E = c^{2} ∆ m$ to $m = E / c^{2}$ . A qualitative relativistic effect you need to know is that fast-moving particles have longer observed lifetimes than stationary particles of the same type: no relativistic calculation equations are required.

Exam tip:

You will be given the conversion factor $1$ eV = $1.60 \times 1 0^{- 19}$ J on the data sheet, so you do not need to memorise it, but make sure you can convert between SI units and particle units correctly.

4. Standard Model and Particle Interaction Conservation★★★★☆⏱ 7 min

The standard quark-lepton model categorises particles into three groups: 1. Baryons: made of 3 quarks, e.g. proton (uud), neutron (udd), baryon number = +1, 2. Mesons: made of 1 quark and 1 antiquark, e.g. $π^{+}$ pion, baryon number = 0, 3. Leptons: fundamental particles, e.g. electron, electron neutrino, lepton number = +1. Every particle has a corresponding antiparticle with opposite charge, baryon number and lepton number. Symmetry in the standard model predicted the existence of the top quark decades before it was experimentally detected.

📘 Definition

Particle Interaction Conservation Laws

For any particle interaction to be allowed, three quantities must be conserved (equal on both sides of the equation): 1. Total charge, 2. Total baryon number, 3. Total lepton number. Antiparticles have negative values for baryon and lepton number.

📐 Worked Example

Test if the following interaction is allowed: $p + n \to π^{+} + π^{0}$ . Proton: charge +1, baryon number +1, lepton number 0. Neutron: charge 0, baryon number +1, lepton number 0. $π^{+}$ : charge +1, baryon number 0, lepton number 0. $π^{0}$ : charge 0, baryon number 0, lepton number 0.

1
1. Check charge conservation: left total = +1 + 0 = +1, right total = +1 + 0 = +1 ✔️
2
1. Check baryon number conservation: left total = +1 + +1 = +2, right total = 0 + 0 = 0 ❌
3
1. Conclusion: interaction is not allowed, since baryon number is not conserved.

Exam tip:

When completing particle equations, always check all three conservation laws to find the missing particle's properties. Quark flavour changes and weak interaction Feynman diagrams are out of scope, so you do not need to study these.

5. Common Pitfalls

Wrong move:

Confusing nucleon number (A) with proton number (Z) when calculating baryon number

Why:

Baryon number depends only on particle type, not total nucleon count: all baryons have baryon number +1 regardless of A.

Correct move:

Assign baryon number +1 to all baryons, -1 to antibaryons, and 0 to all other particles, regardless of nucleon number.

Wrong move:

Using $∆ E = c^{2} ∆ m$ for nuclear binding energy or fission/fusion calculations in Unit 4

Why:

Binding energy, nuclear decay and fission/fusion are out of scope for Unit 4, and are only tested in Unit 5.

Correct move:

Only apply mass-energy equivalence to matter-antimatter creation and annihilation events in Unit 4 questions for this topic.

Wrong move:

Using relativistic momentum equations to derive or use $r = p / B Q$

Why:

Relativistic equations are explicitly out of scope for this topic, and the derivation only uses classical circular motion and magnetic force.

Correct move:

Derive $r = p / B Q$ by equating magnetic force $B Q v$ to classical centripetal force $m v^{2} / r$ , and use classical momentum $p = m v$ for all calculations unless told otherwise.

Wrong move:

Checking separate lepton family numbers (electron, muon, tau) for conservation

Why:

For Edexcel IAL Unit 4, you only need to check total lepton number conservation, not separate family lepton numbers.

Correct move:

Assign lepton number +1 to all leptons, -1 to all antileptons, and 0 to all other particles, then sum total lepton number across the interaction.

Wrong move:

Describing detailed detector engineering components like drift chambers or calorimeters

Why:

Only the general principles of ionisation and deflection for particle detection are required, no technical details of detector design are in scope.

Correct move:

Explain detectors identify particles by measuring ionisation of the detection medium and deflection in electric/magnetic fields to find charge and momentum.

6. Quick Reference Cheatsheet

Concept	Key Formula / Rule	Exam Use
Nucleon/Proton Number	$A$ = total nucleons, $Z$ = total protons	Identify nuclide composition, calculate nuclear charge
Alpha Scattering	3 key observations → small dense positive nucleus	Explain Rutherford experiment conclusions
Charged Particle in B-Field	$r = p / B Q$	Calculate track radius, find particle momentum/charge
Mass-Energy Equivalence	$∆ E = c^{2} ∆ m$	Calculate energy from matter-antimatter annihilation/creation
Unit Conversions	1 eV = $1.60 \times 1 0^{- 19}$ J, 1 MeV/ $c^{2}$ = $1.78 \times 1 0^{- 30}$ kg	Convert between SI and particle physics units
Conservation Laws	Charge, baryon number, lepton number all conserved	Test allowed interactions, complete particle equations

7. Frequently Asked

Do I need to learn relativistic equations for this topic?

No. You only need to know the qualitative effect of relativity: fast-moving particles have longer observed lifetimes than stationary particles of the same type. No relativistic calculation formulas are required.

Can I use $∆ E = c^{2} ∆ m$ for nuclear binding energy questions in Unit 4?

No. Mass-energy equivalence for binding energy, fission and fusion is reserved for Unit 5. In Unit 4, only apply this formula to matter-antimatter creation and annihilation events.

What conservation laws do I use to test if a particle interaction is allowed?

You must check three separate conservation rules: total charge, total baryon number, and total lepton number. All three must be equal on both sides of the interaction for it to be allowed.

Going deeper

What's Next

Now that you have mastered Nuclear and Particle Physics for Edexcel IAL Physics Unit 4, you are ready to apply this knowledge to past paper questions and consolidate your understanding of Unit 4 content. This topic is frequently tested alongside fields and circular motion in multi-step extended response questions, so make sure you are comfortable combining these concepts to solve problems. Practice interpreting particle track diagrams, which often appear in 4-6 mark questions requiring you to calculate momentum or identify particles from track curvature. Once you have completed all Unit 4 topics, you can move on to Unit 5 content, including nuclear decay and energy.