understand that α-particles have discrete energies but that β-particles have a continuous range of energies because (anti)neutrinos are emitted in β-decay

Topic: Atoms, Nuclei and Radiation

Learning objective

Explain why α‑particles have discrete energies whereas β‑particles show a continuous energy spectrum, and describe the essential role of the (anti)neutrino in β‑decay. In doing so you will also be able to:

• write the correct decay notation for α, β⁻, β⁺ and γ radiation,
• state that nucleon number (A) and charge (Z) are conserved,
• identify isotopes, isotones and antiparticles,
• calculate Q‑values for nuclear decays,
• relate the shape of the observed energy spectra to the number of massive particles produced in the decay.

1. The nuclear atom model

• Rutherford α‑scattering experiment (1911) – A thin gold foil was bombarded with α‑particles. Most passed straight through, but a small fraction were deflected at large angles.

  
  Interpretation: the atom must contain a tiny, dense, positively‑charged centre (the nucleus) that can produce strong Coulomb repulsion. This nucleus occupies only ~10⁻⁴ of the atomic volume.

• Simple nuclear model – The atom consists of:

  • a nucleus containing Z protons (charge + e) and N neutrons (neutral),
  • electrons in orbitals outside the nucleus.

  The mass of the atom is essentially the mass of the nucleus because neutrons and protons are ~2000 times heavier than electrons.

• Distinguishing A and Z:

  • A = total number of nucleons = Z + N (mass number).
  • Z = number of protons = atomic number = charge of the nucleus (in units of e).

2. Isotopes and nuclear notation

• An isotope (or nuclide) has the same Z (same element) but a different A (different number of neutrons).
• Standard notation: ^{A}_{Z}X where X is the chemical symbol, A the mass number and Z the atomic number.
• Example: ^{14}{6}C (carbon‑14) and ^{12}{6}C (carbon‑12) are isotopes of carbon.

3. Conservation laws in nuclear decay

For every nuclear transformation the following quantities are conserved:

• Mass number (A): total nucleons before = after.
• Atomic number (Z): total charge before = after.
• Energy and momentum: shared among all decay products.

4. Decay notation – worked example (α‑decay)

Uranium‑238 decays by α‑emission:

\$^{238}{92}\!U \;\longrightarrow\; ^{234}{90}\!Th \;+\; \alpha\$

• Mass number: 238 → 234 + 4 (conserved).
• Atomic number: 92 → 90 + 2 (conserved).

5. Types of nuclear radiation

5.1 α‑decay – discrete energy emission

• General reaction: ^{A}{Z}X → ^{A-4}{Z-2}Y + \alpha
• Q‑value (energy released):

  \$Q = \bigl[M(^{A}{Z}X) - M(^{A-4}{Z-2}Y) - M_{\alpha}\bigr]c^{2}\$

• Only two massive particles are produced (α‑particle and recoiling daughter nucleus). Two‑body kinematics fixes the kinetic energy of each particle:

  \$T{\alpha} = \frac{Q\,M{Y}}{M{Y}+M{\alpha}} \approx Q\;(1-\frac{M{\alpha}}{M{Y}})\$

• Result: a sharp line (discrete) in the α‑particle energy spectrum, typically 4–9 MeV.

5.2 β⁻ decay (electron emission) – continuous spectrum

• Reaction: ^{A}{Z}X → ^{A}{Z+1}Y + e^{-} + \bar{\nu}_{e}
• Q‑value (ignoring the tiny recoil):

  \$Q = \bigl[M(^{A}{Z}X) - M(^{A}{Z+1}Y)\bigr]c^{2}\$

• Three‑body final state (electron, antineutrino, recoil nucleus) → the total kinetic energy Q can be shared in infinitely many ways:

  \$Q = T{e} + T{\bar{\nu}} + T_{\text{recoil}}\$

• The (anti)neutrino can take any kinetic energy from 0 up to ≈ Q, so the electron kinetic energy Tₑ varies continuously from 0 to Q (minus a negligible recoil term). This produces the observed continuous β⁻ spectrum.

5.3 β⁺ decay (positron emission) – continuous spectrum

• Reaction: ^{A}{Z}X → ^{A}{Z-1}Y + e^{+} + \nu_{e}
• Q‑value (including the rest‑mass of the created positron‑electron pair):

  \$Q = \bigl[M(^{A}{Z}X) - M(^{A}{Z-1}Y) - 2m_{e}\bigr]c^{2}\$

• Again a three‑body decay (positron, neutrino, recoil) → the positron kinetic energy is continuous from 0 up to Q − 2 mₑc².
• Charge conservation: Z → (Z − 1) + (+1) (the positron), so the total charge is unchanged.

5.4 γ‑radiation – photon emission

• Reaction: ^{A}{Z}X^{*} → ^{A}{Z}X + \gamma ( * denotes an excited nuclear state ).
• No change in A or Z; only the nuclear energy level changes.
• Photon energy equals the difference between the two nuclear levels:

  \$E{\gamma}=E{\text{initial}}-E_{\text{final}}\$

• Because the photon is mass‑less, the nucleus recoils only slightly; the emitted γ‑ray has a fixed energy – a sharp line in the γ‑spectrum.

6. Comparison of α, β⁻, β⁺ and γ decay

7. Why the (anti)neutrino is essential in β‑decay

The continuous β‑spectrum observed in the early 20th century seemed to violate energy conservation if only an electron (or positron) were emitted. In 1930 Wolfgang Pauli proposed a neutral, very low‑mass particle – the neutrino (or antineutrino) – to carry away the “missing’’ energy and momentum. Inclusion of this third particle makes the three‑body decay compatible with the conservation laws and explains the continuous energy distribution.

8. Summary points

• α‑decay: two‑body final state → fixed kinetic energy for the α‑particle → discrete line spectrum.
• β⁻ and β⁺ decay: three‑body final state (lepton + (neutrino/antineutrino) + recoil) → variable sharing of the Q‑value → continuous β‑spectra.
• γ‑decay: photon emission from an excited nucleus; A and Z unchanged; photon energy is discrete.
• Conservation of A and Z holds for all nuclear transformations.
• Understanding the number of massive particles in the final state allows you to predict whether the emitted radiation will have a discrete or continuous energy spectrum – a key skill for IGCSE/A‑Level exam questions.