44  Deep Inside the Atom

44.1 Syllabus inquiry question

  • What is matter made of at the most fundamental level?
Feynman Insight

From The Feynman Lectures on Physics, Vol I, Chapter 2:

The question “What are things made of?” has been asked for millennia. We now know the answer: matter is made of quarks and leptons, interacting through exchange of force-carrying bosons. Everything you see—tables, chairs, stars—is built from just a handful of truly fundamental particles.

44.2 Learning Objectives

  • Describe the Standard Model of matter
  • Identify quarks, leptons, and bosons
  • Explain particle accelerator evidence for quarks
  • Analyse conservation laws in particle interactions
  • Apply relativistic momentum to particle physics

44.3 Content

44.3.1 The Standard Model

The Standard Model classifies all known fundamental particles:

Category Examples Role
Quarks up, down, charm, strange, top, bottom Make up hadrons (protons, neutrons)
Leptons electron, muon, tau, neutrinos Fundamental particles (don’t feel strong force)
Gauge Bosons photon, W±, Z⁰, gluon Carry the fundamental forces
Higgs Boson Higgs Gives particles mass

44.3.2 Interactive: Standard Model Overview

44.3.3 Quarks

Six types (flavours) of quarks exist in three generations:

Generation Quark Charge Mass (approx)
1st Up (u) +2/3 e 2 MeV/c²
1st Down (d) -1/3 e 5 MeV/c²
2nd Charm (c) +2/3 e 1.3 GeV/c²
2nd Strange (s) -1/3 e 95 MeV/c²
3rd Top (t) +2/3 e 173 GeV/c²
3rd Bottom (b) -1/3 e 4.2 GeV/c²
Quark Confinement

Quarks are never observed in isolation. They always combine to form: - Baryons (3 quarks): proton (uud), neutron (udd) - Mesons (quark + antiquark): pion (uđ or dū)

44.3.4 Interactive: Quark Composition

44.3.5 Verifying Quark Charges

Proton charge calculation: - 2 up quarks: 2 × (+2/3) = +4/3 - 1 down quark: 1 × (-1/3) = -1/3 - Total: +4/3 - 1/3 = +1 ✓

Neutron charge calculation: - 1 up quark: 1 × (+2/3) = +2/3 - 2 down quarks: 2 × (-1/3) = -2/3 - Total: +2/3 - 2/3 = 0 ✓

44.3.6 Leptons

Generation Lepton Charge Mass
1st Electron (e⁻) -1 0.511 MeV/c²
1st Electron neutrino (νₑ) 0 ~0
2nd Muon (μ⁻) -1 106 MeV/c²
2nd Muon neutrino (νμ) 0 ~0
3rd Tau (τ⁻) -1 1.78 GeV/c²
3rd Tau neutrino (ντ) 0 ~0
Leptons vs Quarks

Leptons don’t feel the strong nuclear force. That’s why electrons orbit nuclei instead of being absorbed into them.

44.3.7 Gauge Bosons (Force Carriers)

Force Boson Mass Range
Electromagnetic Photon (γ) 0 Infinite
Weak W⁺, W⁻, Z⁰ ~80-90 GeV/c² ~10⁻¹⁸ m
Strong Gluon (g) 0 ~10⁻¹⁵ m
Gravity Graviton (?) 0 Infinite
Why Weak Force is Weak

The W and Z bosons are massive. By Heisenberg’s uncertainty principle, massive particles can only exist briefly, so the force has extremely short range.

44.3.8 Particle Accelerators

Why we need accelerators:

  1. de Broglie wavelength: To probe small structures, need high momentum (short λ) \[\lambda = \frac{h}{p}\]

  2. E = mc²: To create massive particles, need high energy \[E_{min} = mc^2\]

  3. Relativistic effects: At high speeds, momentum increases dramatically \[p = \gamma m_0 v\]

44.3.9 Interactive: Accelerator Energy vs Particle Mass

44.3.10 Evidence for Quarks

Deep inelastic scattering (1968): - Electrons fired at protons at SLAC accelerator - Some electrons scattered at large angles - Similar to Rutherford’s alpha scattering! - Implied point-like constituents inside protons

Key evidence: 1. Scattering patterns consistent with 3 point charges per proton 2. Charges of +2/3 and -1/3 (not +1) 3. Only ~50% of proton momentum in quarks (rest in gluons)

44.3.11 Conservation Laws in Particle Physics

Quantity Always Conserved? Example
Energy Yes E = mc² conversions
Momentum Yes Collision products
Charge Yes Beta decay: n → p + e⁻ + ν̄
Baryon number Yes Proton stability
Lepton number Yes Neutrino production
Strangeness Strong only Kaon production

44.3.12 Particle Interactions

Example: Beta decay at quark level \[n \rightarrow p + e^- + \bar{\nu}_e\]

At quark level: \[d \rightarrow u + W^- \rightarrow u + e^- + \bar{\nu}_e\]

A down quark converts to an up quark by emitting a W⁻ boson, which then decays to an electron and antineutrino.

44.3.13 Interactive: Beta Decay Mechanism

44.3.14 Relativistic Momentum

At speeds approaching c, classical momentum fails:

\[p = \gamma m_0 v = \frac{m_0 v}{\sqrt{1 - v^2/c^2}}\]

Speed γ Classical p Relativistic p
0.1c 1.005 0.1 m₀c 0.1005 m₀c
0.5c 1.155 0.5 m₀c 0.577 m₀c
0.9c 2.294 0.9 m₀c 2.065 m₀c
0.99c 7.089 0.99 m₀c 7.018 m₀c
Why Nothing Reaches c

As v → c, momentum → ∞. An infinite force would be required to accelerate to c. This is why particles with mass can never reach the speed of light.

44.3.15 Antimatter

Every particle has an antiparticle with: - Same mass - Opposite charge - Opposite quantum numbers

Particle Antiparticle Symbol
Electron Positron e⁺
Proton Antiproton p̄
Up quark Anti-up ū
Neutrino Antineutrino ν̄

Pair production: \(\gamma \rightarrow e^- + e^+\) (photon → matter + antimatter)

Annihilation: \(e^- + e^+ \rightarrow 2\gamma\) (matter + antimatter → energy)

44.4 Worked Examples

44.4.1 Example 1: Quark composition

Identify the quark content of π⁻ (pion minus) given it has charge -1.

Solution:

  1. Pions are mesons (quark + antiquark)

  2. To get charge -1, we need d + ū̄ wait, that’s wrong…

  3. Actually: d has charge -1/3, and ū (anti-up) has charge -2/3

  4. d + ū: (-1/3) + (-2/3) = -1 ✓

Answer: π⁻ = dū

44.4.2 Example 2: Relativistic momentum

Calculate the momentum of an electron travelling at 0.95c.

Solution:

  1. Calculate γ: \(\gamma = 1/\sqrt{1 - 0.95^2} = 1/\sqrt{0.0975} = 3.20\)

  2. \(p = \gamma m_0 v = 3.20 \times 9.11 \times 10^{-31} \times 0.95 \times 3.0 \times 10^8\)

  3. \(p = 8.3 \times 10^{-22}\ \text{kg·m/s}\)

44.4.3 Example 3: Minimum energy for particle creation

What minimum energy is needed to create a muon-antimuon pair? (m_μ = 106 MeV/c²)

Solution:

  1. Need to create 2 muons: total mass = 2 × 106 = 212 MeV/c²

  2. Minimum energy = 2m_μc² = 212 MeV

  3. Convert to joules: \(E = 212 \times 10^6 \times 1.6 \times 10^{-19} = 3.4 \times 10^{-11}\) J

44.4.4 Example 4: de Broglie wavelength at high speed

Calculate the de Broglie wavelength of a proton travelling at 0.8c.

Solution:

  1. \(\gamma = 1/\sqrt{1 - 0.64} = 1/0.6 = 1.667\)

  2. \(p = \gamma m_0 v = 1.667 \times 1.67 \times 10^{-27} \times 0.8 \times 3.0 \times 10^8\)

  3. \(p = 6.68 \times 10^{-19}\) kg·m/s

  4. \(\lambda = h/p = 6.63 \times 10^{-34}/6.68 \times 10^{-19} = 9.9 \times 10^{-16}\) m

This is smaller than a proton (~10⁻¹⁵ m), so can probe nuclear structure.

44.4.5 Example 5: Conservation in particle decay

Verify conservation of charge, baryon number, and lepton number in beta decay: n → p + e⁻ + ν̄ₑ

Solution:

Property Before After Conserved?
Charge 0 +1 + (-1) + 0 = 0 ✓
Baryon number 1 1 + 0 + 0 = 1 ✓
Lepton number 0 0 + 1 + (-1) = 0 ✓

(Antineutrino has lepton number -1)

44.5 Common Misconceptions

Common Misconceptions

  • Misconception: Quarks can exist on their own like electrons. Correction: Quarks are never observed in isolation due to confinement. They always form hadrons (baryons or mesons).

  • Misconception: The proton is fundamental like the electron. Correction: Protons are made of quarks (uud). Electrons are fundamental—they have no internal structure.

  • Misconception: Photons have mass because they carry energy. Correction: Photons are massless. E = mc² applies to rest mass. For photons, E = hf = pc.

  • Misconception: The Higgs gives particles their mass directly. Correction: The Higgs field gives particles mass. The Higgs boson is an excitation of this field—it’s evidence for the field, not the source of mass.

  • Misconception: Antimatter is science fiction. Correction: Antimatter is routinely produced in accelerators and used in medicine (PET scans use positrons). It’s just hard to store because it annihilates on contact with matter.

44.6 Practice Questions

44.6.1 Easy (2 marks)

State the quark composition of a proton and calculate its total charge.

  • Proton = uud (1)
  • Charge = 2(+2/3) + 1(-1/3) = +4/3 - 1/3 = +1 (1)

Answer: uud; charge = +1

44.6.2 Medium (4 marks)

Calculate the relativistic momentum of an electron travelling at 0.90c. Compare this to the classical momentum.

  • \(\gamma = 1/\sqrt{1 - 0.81} = 1/\sqrt{0.19} = 2.29\) (1)
  • Classical: \(p_{class} = m_0 v = 9.11 \times 10^{-31} \times 0.90 \times 3.0 \times 10^8 = 2.46 \times 10^{-22}\) kg·m/s (1)
  • Relativistic: \(p = \gamma m_0 v = 2.29 \times 2.46 \times 10^{-22} = 5.6 \times 10^{-22}\) kg·m/s (1)
  • Relativistic momentum is 2.29× larger (1)

Answer: p = 5.6 × 10⁻²² kg·m/s; this is 2.3× the classical value

44.6.3 Hard (5 marks)

Deep inelastic scattering experiments fire electrons at protons. Explain how this provides evidence for quarks, and calculate the minimum de Broglie wavelength needed to probe structures of size 10⁻¹⁵ m.

  • Large-angle scattering indicates point-like constituents (like Rutherford) (1)
  • Scattering pattern consistent with three charged particles per proton (1)
  • Measured charges consistent with +2/3 and -1/3 (not +1) (1)
  • For \(\lambda < 10^{-15}\) m, need \(p > h/\lambda\) (1)
  • \(p > 6.63 \times 10^{-34}/10^{-15} = 6.63 \times 10^{-19}\) kg·m/s; requires relativistic speeds (1)

Answer: Large-angle scattering indicates point-like charges of +2/3 and -1/3 inside protons. Need λ < 10⁻¹⁵ m, requiring p > 6.6 × 10⁻¹⁹ kg·m/s.

44.7 Multiple Choice Questions

Test your understanding with these interactive questions:

44.8 Summary

Key Takeaways
  • Standard Model: quarks, leptons, and gauge bosons
  • 6 quarks (u, d, c, s, t, b) with fractional charges
  • Proton = uud (+1 charge); Neutron = udd (0 charge)
  • 6 leptons (e, μ, τ and their neutrinos)
  • Force carriers: photon (EM), W/Z (weak), gluon (strong)
  • Quarks confined—always in hadrons (baryons or mesons)
  • Relativistic momentum: \(p = \gamma m_0 v\)
  • Deep inelastic scattering proved quarks exist

44.9 Self-Assessment

Check your understanding:

After studying this section, you should be able to: