describe the changes to quark composition that take place during β– and β+ decay

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 \(\bar{u}_e\).
• β⁺‑decay (positron emission): a proton becomes a neutron while emitting a positron \(e^+\) and an electron‑neutrino \(u_e\). (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

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

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

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

1. Nuclear equation \(^{A}_{Z}\!X \;\longrightarrow\; ^{A}_{Z+1}\!Y \;+\; e^- \;+\; \bar{u}_e\)
2. Quark‑level step \(d \;\longrightarrow\; u + W^-\)
3. W‑boson decay \(W^- \;\longrightarrow\; e^- + \bar{u}_e\)
4. Resulting nucleon change \(udd\;(n) \;\longrightarrow\; uud\;(p)\)
5. 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

1. Nuclear equation \(^{A}_{Z}\!X \;\longrightarrow\; ^{A}_{Z-1}\!Y \;+\; e^+ \;+\; u_e\)
2. Quark‑level step \(u \;\longrightarrow\; d + W^+\)
3. W‑boson decay \(W^+ \;\longrightarrow\; e^+ + u_e\)
4. Resulting nucleon change \(uud\;(p) \;\longrightarrow\; udd\;(n)\)
5. Energy requirement Because a positron of mass \(m_e\) is created, the Q‑value must exceed \(2m_ec^2\) ≈ 1.022 MeV. If this condition is not met, the nucleus decays by electron capture instead.

7. Comparison of quark changes in β‑decay

8. Key points to remember for the Cambridge syllabus

1. The weak interaction changes quark flavour via the exchange of a charged \(W^{\pm}\) boson.
2. β⁻ decay: \(d \rightarrow u\) (neutron → proton) + \(e^- + \bar{u}_e\).
3. β⁺ decay: \(u \rightarrow d\) (proton → neutron) + \(e^+ + u_e\).
4. In β‑decay the mass number \(A\) is conserved; the atomic number \(Z\) changes by +1 (β⁻) or ‑1 (β⁺).
5. Charge, baryon number and lepton number are all conserved in each decay step.
6. The Q‑value must be > 1.022 MeV for β⁺ emission; otherwise electron capture occurs.
7. The continuous β spectrum arises because the (anti)neutrino carries away a variable amount of the released energy.
8. 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.