β‑Radiation and Quark‑Flavour Change (Cambridge AS & A Level 9702)
1. Types of nuclear radiation and nuclide notation
- α‑decay: emission of a helium‑4 nucleus, \(^{4}_{2}\text{He}\) (α‑particle). Heavy, +2 e charge, low penetration.
- β⁻‑decay: a neutron in the nucleus is transformed into a proton with the emission of an electron \(e^-\) and an electron‑antineutrino \(\bar{\nu}_e\).
- β⁺‑decay (positron emission): a proton becomes a neutron while emitting a positron \(e^+\) and an electron‑neutrino \(\nu_e\). (If the Q‑value is too low, the nucleus may instead undergo electron capture.)
- γ‑radiation: emission of a high‑energy photon; no change in the composition of the nucleus, only a loss of excess energy.
Nuclides are written in the standard Cambridge notation A Z X, where A is the mass number (total nucleons) and Z the atomic number (protons). Example: \(^{14}_{6}\text{C}\) (carbon‑14).
2. β‑decay equations and the conserved quantities
For a parent nuclide \(^{A}_{Z}\!X\) the two β‑decays are expressed as
β⁻ decay \(^{A}{Z}\!X \;\longrightarrow\; ^{A}{Z+1}\!Y \;+\; e^- \;+\; \bar{\nu}_e\)
β⁺ decay \(^{A}{Z}\!X \;\longrightarrow\; ^{A}{Z-1}\!Y \;+\; e^+ \;+\; \nu_e\)
These equations satisfy the Cambridge‑required conservation laws:
- Charge: the emitted lepton carries ±1 e, exactly compensating the change in \(Z\) (‑1 for β⁺, +1 for β⁻).
- Mass number (A): unchanged; a neutron ↔ proton conversion does not create or destroy nucleons.
- Lepton number: each lepton is paired with its corresponding (anti)neutrino, giving a total lepton number of 0 before and after the decay.
- Energy (Q‑value): the difference in nuclear binding energy,
\[
Q = \big[M(^{A}{Z}X)-M(^{A}{Z\pm1}Y)\big]c^{2},
\]
is shared between the recoil nucleus, the electron/positron and the (anti)neutrino. Because the (anti)neutrino can take any energy up to \(Q\), the observed β‑spectrum is continuous.
3. Fundamental particles involved in β‑decay
3.1 Quarks
| Flavour | Charge (e) | Mass (MeV c⁻²) |
|---|
| up \(u\) | + \( \tfrac{2}{3}\) | ≈ 2.2 |
| down \(d\) | – \( \tfrac{1}{3}\) | ≈ 4.7 |
| charm \(c\) | + \( \tfrac{2}{3}\) | ≈ 1270 |
| strange \(s\) | – \( \tfrac{1}{3}\) | ≈ 96 |
| top \(t\) | + \( \tfrac{2}{3}\) | ≈ 173 000 |
| bottom \(b\) | – \( \tfrac{1}{3}\) | ≈ 4180 |
Protons and neutrons (the nucleons) are baryons made of the two light flavours only:
- Proton \(p = uud\) (Charge = +1 e)
- Neutron \(n = udd\) (Charge = 0 e)
3.2 Leptons
| Particle | Charge (e) | Lepton number |
|---|
| \(e^-\) | – 1 | +1 |
| \(\nu_e\) | 0 | +1 |
| \(e^+\) | + 1 | –1 |
| \(\bar{\nu}_e\) | 0 | –1 |
4. Weak interaction and quark‑flavour change
The weak force is carried by the massive charged bosons \(W^{\pm}\) (mass ≈ 80 GeV c⁻²). A quark can change its flavour by emitting or absorbing a \(W\) boson:
- \(d \;\longrightarrow\; u + W^-\) (– \( \tfrac{1}{3}\) → + \( \tfrac{2}{3}\); the \(W^-\) carries –1 e)
- \(u \;\longrightarrow\; d + W^+\) (+ \( \tfrac{2}{3}\) → – \( \tfrac{1}{3}\); the \(W^+\) carries +1 e)
The emitted \(W^{\pm}\) boson subsequently decays into a lepton–neutrino pair, preserving charge and lepton number.
5. β⁻ decay – from the quark level to the whole nucleus
- Nuclear equation \(^{A}{Z}\!X \;\longrightarrow\; ^{A}{Z+1}\!Y \;+\; e^- \;+\; \bar{\nu}_e\)
- Quark‑level step \(d \;\longrightarrow\; u + W^-\)
- W‑boson decay \(W^- \;\longrightarrow\; e^- + \bar{\nu}_e\)
- Resulting nucleon change \(udd\;(n) \;\longrightarrow\; uud\;(p)\)
- Energy released Typical Q‑values are a few MeV; the electron and antineutrino share this energy, giving the characteristic continuous β spectrum.
6. β⁺ decay (positron emission) – quark picture
- Nuclear equation \(^{A}{Z}\!X \;\longrightarrow\; ^{A}{Z-1}\!Y \;+\; e^+ \;+\; \nu_e\)
- Quark‑level step \(u \;\longrightarrow\; d + W^+\)
- W‑boson decay \(W^+ \;\longrightarrow\; e^+ + \nu_e\)
- Resulting nucleon change \(uud\;(p) \;\longrightarrow\; udd\;(n)\)
- Energy requirement Because a positron of mass \(me\) is created, the Q‑value must exceed \(2mec^2\) ≈ 1.022 MeV. If this condition is not met, the nucleus decays by electron capture instead.
7. Comparison of quark changes in β‑decay
| Decay | Initial nucleon (quarks) | Quark transition | Final nucleon (quarks) | Leptons emitted |
|---|
| β⁻ | Neutron \(udd\) | \(d \rightarrow u + W^-\) | Proton \(uud\) | \(e^- + \bar{\nu}_e\) |
| β⁺ | Proton \(uud\) | \(u \rightarrow d + W^+\) | Neutron \(udd\) | \(e^+ + \nu_e\) |
8. Key points to remember for the Cambridge syllabus
- The weak interaction changes quark flavour via the exchange of a charged \(W^{\pm}\) boson.
- β⁻ decay: \(d \rightarrow u\) (neutron → proton) + \(e^- + \bar{\nu}_e\).
- β⁺ decay: \(u \rightarrow d\) (proton → neutron) + \(e^+ + \nu_e\).
- In β‑decay the mass number \(A\) is conserved; the atomic number \(Z\) changes by +1 (β⁻) or ‑1 (β⁺).
- Charge, baryon number and lepton number are all conserved in each decay step.
- The Q‑value must be > 1.022 MeV for β⁺ emission; otherwise electron capture occurs.
- The continuous β spectrum arises because the (anti)neutrino carries away a variable amount of the released energy.
- Only the up and down quarks participate in ordinary β‑decay; the heavier flavours (s, c, b, t) appear in more massive hadrons and are not required for the AS/A‑Level curriculum.