Fundamental particles

1. Overview

The Standard Model of particle physics classifies all known matter into two categories: fundamental particles, which possess no internal structure, and composite particles, which are formed by the interaction of fundamental constituents. Matter is primarily composed of quarks and leptons. Quarks are unique because they carry fractional elementary charges and are never found in isolation; they combine to form hadrons, such as the protons and neutrons that constitute the atomic nucleus. In contrast, leptons, including the electron and the neutrino, do not experience the strong nuclear force and exist as independent fundamental entities. Understanding these particles is crucial for describing the weak nuclear force interactions that drive nuclear beta ($\beta$) decay, where the identity of a quark changes flavour to facilitate the transformation of nucleons.

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Key Definitions

• Fundamental Particle: A particle that has no internal structure and cannot be divided into smaller constituents.
• Quark: A fundamental particle that carries a fractional elementary charge and experiences the strong nuclear force.
• Antiquark: The antimatter counterpart of a quark, possessing the same mass but opposite electric charge.
• Hadron: A composite particle made of quarks held together by the strong nuclear force. Hadrons are not fundamental.
• Baryon: A type of hadron consisting of three quarks ($qqq$). Protons and neutrons are the most stable baryons.
• Meson: A type of hadron consisting of one quark and one antiquark ($q\bar{q}$).
• Lepton: A fundamental particle that does not experience the strong nuclear force.
• Neutrino ($\nu$): A fundamental lepton with zero electric charge and a rest mass that is very small (nearly zero).
• Weak Nuclear Force: The fundamental interaction responsible for flavour change in quarks during beta decay.

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Content

3.1 The Six Flavours of Quarks

There are six distinct types, or "flavours," of quarks. In the Cambridge 9702 syllabus, you must be able to recall all six names and their respective charges. These are often grouped into three "generations" of increasing mass.

Quark Flavour	Symbol	Charge ($Q$) in terms of $e$	Charge in Coulombs ($C$)
Up	$u$	$+\frac{2}{3}e$	$+1.07 \times 10^{-19} \text{ C}$
Down	$d$	$-\frac{1}{3}e$	$-0.53 \times 10^{-19} \text{ C}$
Strange	$s$	$-\frac{1}{3}e$	$-0.53 \times 10^{-19} \text{ C}$
Charm	$c$	$+\frac{2}{3}e$	$+1.07 \times 10^{-19} \text{ C}$
Top	$t$	$+\frac{2}{3}e$	$+1.07 \times 10^{-19} \text{ C}$
Bottom	$b$	$-\frac{1}{3}e$	$-0.53 \times 10^{-19} \text{ C}$

Properties of Antiquarks: For every quark flavour, there exists a corresponding antiquark. Antiquarks are represented by a bar over the symbol (e.g., $\bar{s}$ is the anti-strange quark).

• Mass: Identical to the quark.
• Charge: Exactly opposite to the quark.
  • Example: The anti-up ($\bar{u}$) has a charge of $-\frac{2}{3}e$.
  • Example: The anti-bottom ($\bar{b}$) has a charge of $+\frac{1}{3}e$.

3.2 Quark Composition of Nucleons

Protons and neutrons are not fundamental; they are baryons composed of up ($u$) and down ($d$) quarks.

1. The Proton ($p$):

• Composition: $uud$ (two up quarks, one down quark).
• Charge Calculation: $$Q_p = (+\frac{2}{3}e) + (+\frac{2}{3}e) + (-\frac{1}{3}e) = +1e$$
• Conclusion: The proton has a net positive charge of $+1.60 \times 10^{-19} \text{ C}$.

2. The Neutron ($n$):

• Composition: $udd$ (one up quark, two down quarks).
• Charge Calculation: $$Q_n = (+\frac{2}{3}e) + (-\frac{1}{3}e) + (-\frac{1}{3}e) = 0$$
• Conclusion: The neutron is electrically neutral.

3.3 Classification of Hadrons

Hadrons are particles composed of quarks. They are divided into two groups based on the number of quarks they contain.

Baryons ($qqq$):

• Consist of three quarks.
• Examples: Protons ($uud$), Neutrons ($udd$), and Sigma particles (e.g., $\Sigma^+ = uus$).
• Antibaryons ($\bar{q}\bar{q}\bar{q}$): Consist of three antiquarks. An antiproton ($\bar{p}$) has the composition $\bar{u}\bar{u}\bar{d}$ and a charge of $-1e$.

Mesons ($q\bar{q}$):

• Consist of one quark and one antiquark.
• Examples:
  • Pion plus ($\pi^+$): Composition $u\bar{d}$. Charge: $(+\frac{2}{3}e) + (+\frac{1}{3}e) = +1e$.
  • Kaon neutral ($K^0$): Composition $d\bar{s}$. Charge: $(-\frac{1}{3}e) + (+\frac{1}{3}e) = 0$.
• Mesons are inherently unstable and are often produced in high-energy particle collisions.

3.4 Leptons

Leptons are a family of fundamental particles that do not experience the strong nuclear force. This explains why electrons are found in shells around the nucleus rather than bound within it.

• Electron ($e^-$): Charge = $-1e$. Stable and fundamental.
• Positron ($e^+$): The antiparticle of the electron. Charge = $+1e$.
• Neutrinos ($\nu$):
  • They have no charge and negligible mass.
  • They interact only via the weak nuclear force and gravity, making them extremely difficult to detect.
  • In beta decay, we specifically encounter the electron neutrino ($\nu_e$) and the electron antineutrino ($\bar{\nu}_e$).

3.5 Quark Changes in Beta Decay

Beta decay occurs when a quark within a nucleon changes its flavour. This process is mediated by the weak nuclear force.

$\beta^-$ Decay (Electron Emission): Occurs in neutron-rich nuclei. A neutron transforms into a proton.

• Nucleon Equation: $n \to p + e^- + \bar{\nu}_e$
• Quark Change: A down quark changes into an up quark.
• Quark Equation: $d \to u + e^- + \bar{\nu}_e$
• Charge Check: $-\frac{1}{3}e \to +\frac{2}{3}e + (-1e) + 0 = -\frac{1}{3}e$ (Charge is conserved).

$\beta^+$ Decay (Positron Emission): Occurs in proton-rich nuclei. A proton transforms into a neutron.

• Nucleon Equation: $p \to n + e^+ + \nu_e$
• Quark Change: An up quark changes into a down quark.
• Quark Equation: $u \to d + e^+ + \nu_e$
• Charge Check: $+\frac{2}{3}e \to -\frac{1}{3}e + (+1e) + 0 = +\frac{2}{3}e$ (Charge is conserved).

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Worked Example 1 — Identifying an Unknown Baryon

Question: A baryon known as the Omega-minus ($\Omega^-$) particle is composed entirely of strange quarks. (a) State the quark composition of the $\Omega^-$ particle. (b) Calculate the electric charge of the $\Omega^-$ particle in terms of $e$.

Solution:

1. Identify the particle type: A baryon consists of three quarks.
2. Determine composition: Since it is composed only of strange ($s$) quarks, the composition is $sss$.
3. Recall the charge of a strange quark: $Q_s = -\frac{1}{3}e$.
4. Sum the charges: $$Q_{total} = Q_s + Q_s + Q_s$$ $$Q_{total} = (-\frac{1}{3}e) + (-\frac{1}{3}e) + (-\frac{1}{3}e)$$ $$Q_{total} = -1e$$ Answer: (a) $sss$; (b) $-1e$

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Worked Example 2 — Quark Transformation in Decay

Question: During a specific nuclear reaction, a particle with quark composition $uds$ decays into a particle with composition $uus$, an electron, and an antineutrino. (a) Identify the type of decay occurring. (b) Show that charge is conserved in this decay.

Solution:

1. Analyze the change: The $uds$ particle becomes $uus$.
   • The $d$ quark has changed into a $u$ quark.
   • The $s$ and the other $u$ remain unchanged.
2. Identify the decay: A $d \to u$ transition accompanied by an electron ($e^-$) and an antineutrino ($\bar{\nu}_e$) is $\beta^-$ decay.
3. Calculate initial charge ($Q_i$): $$Q_i = Q_u + Q_d + Q_s = (+\frac{2}{3}e) + (-\frac{1}{3}e) + (-\frac{1}{3}e) = 0$$
4. Calculate final charge ($Q_f$): $$Q_f = (Q_u + Q_u + Q_s) + Q_{electron} + Q_{antineutrino}$$ $$Q_f = [(+\frac{2}{3}e) + (+\frac{2}{3}e) + (-\frac{1}{3}e)] + (-1e) + 0$$ $$Q_f = (+1e) + (-1e) + 0 = 0$$
5. Conclusion: Since $Q_i = Q_f$, charge is conserved.

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Key Equations & Constants

Quantity / Particle	Symbol / Composition	Charge	Status
Elementary Charge	$e$	$1.60 \times 10^{-19} \text{ C}$	Data Sheet
Up, Charm, Top	$u, c, t$	$+\frac{2}{3}e$	Memorise
Down, Strange, Bottom	$d, s, b$	$-\frac{1}{3}e$	Memorise
Proton	$uud$	$+1e$	Memorise
Neutron	$udd$	$0$	Memorise
Baryon	$qqq$	Sum of 3 quarks	Memorise
Meson	$q\bar{q}$	Sum of $q$ and $\bar{q}$	Memorise
$\beta^-$ Quark Change	$d \to u + e^- + \bar{\nu}_e$	Conserved	Memorise
$\beta^+$ Quark Change	$u \to d + e^+ + \nu_e$	Conserved	Memorise

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Common Mistakes to Avoid

• ❌ Wrong: Labeling a proton or neutron as a "fundamental particle." ✓ Right: Protons and neutrons are hadrons (specifically baryons); they have internal quark structures. Only quarks and leptons are fundamental.
• ❌ Wrong: Forgetting the "bar" or the sign change for antiquarks. ✓ Right: An anti-down quark ($\bar{d}$) has a charge of $+\frac{1}{3}e$. Always flip the sign of the corresponding quark.
• ❌ Wrong: Confusing the composition of mesons and baryons. ✓ Right: Baryons = 3 quarks; Mesons = 1 quark + 1 antiquark. A particle with 2 quarks ($qq$) does not exist in the Standard Model.
• ❌ Wrong: Omitting the neutrino or antineutrino in decay equations. ✓ Right: $\beta^-$ decay always produces an antineutrino ($\bar{\nu}_e$). $\beta^+$ decay always produces a neutrino ($\nu_e$).
• ❌ Wrong: Assuming all quarks have the same mass. ✓ Right: While the syllabus doesn't require memorising masses, you should know that different flavours have different masses (e.g., the Top quark is much heavier than the Up quark).

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Exam Tips

1. The "Fractional Charge" Check: If you are asked to suggest a quark composition for a particle with a specific charge, always ensure your fractional charges ($1/3$ and $2/3$) add up to an integer ($0, \pm 1, \pm 2$). Hadrons never have fractional net charges.
2. Beta Decay Logic: If you forget which quark changes in $\beta^-$ decay, remember the nucleon change: $n \to p$. A neutron ($0$) becomes a proton ($+1$). To become more positive, a down quark ($-\frac{1}{3}$) must be replaced by an up quark ($+\frac{2}{3}$).
3. Defining Fundamental Particles: When asked for a definition, use the phrase "no internal structure". This is a specific keyword in Cambridge mark schemes.
4. Lepton Properties: If asked why an electron is a lepton, state that it is a fundamental particle that does not feel the strong nuclear force.
5. Show Your Working: In charge calculation questions, don't just write the answer. Write out the sum: $(+\frac{2}{3}e) + (-\frac{1}{3}e) + ...$ to ensure you pick up method marks even if you make a simple arithmetic error.
6. Antimatter Notation: Ensure the bar over the symbol for an antiquark is clear. $\bar{u}$ is an anti-up quark; $u$ is an up quark. Mixing these up will result in a total charge error.