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Neutrino In Beta Decay (Conservation of Energy & Momentum)

Key idea: Explain why beta particles have a continuous range of kinetic energies and how the antineutrino preserves energy and momentum in beta-minus decay.

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Apply conservation laws to nuclear equations and beta decay, including antineutrino evidence.

1. Definitions (Must Know)

  • Beta-minus (β⁻) decay: a neutron in the nucleus changes into a proton and emits an electron and an electron antineutrino:
    • n → p + e⁻ + ν bar ₑ
  • Electron antineutrino, ν bar ₑ: an electrically neutral, extremely low-mass particle that interacts very weakly with matter; it carries energy and momentum in β⁻ decay.
  • Conservation laws in nuclear processes (relevant here):
    • nucleon number (mass number) conserved
    • charge conserved
    • energy and momentum conserved

2. Key Ideas (What Earns Marks)

  • Beta particles are emitted with a continuous range of kinetic energies.
  • If only the daughter nucleus + electron were produced, energy and momentum would predict a single electron energy (for a given decay).
  • The remaining energy and momentum are carried by a third emitted particle: the electron antineutrino.
  • In nuclear equations for β⁻ decay:
    • A stays the same
    • Z increases by 1

3. Detailed Explanations

A. The “beta energy problem”

In many decays, the beta particle is not emitted with a fixed kinetic energy. Instead, measurements show a spectrum from near 0 up to some maximum value.

This is a problem if we assume a two-body decay:

parent nucleus → daughter nucleus + e⁻

Because for a two-body decay, conservation of energy and momentum would lead to a single (fixed) electron energy.

B. Antineutrino as the solution

To satisfy conservation of energy and momentum, the decay must produce three particles:

parent nucleus → daughter nucleus + e⁻ + ν bar ₑ

The available energy is shared between:

  • the electron (beta particle)
  • the electron antineutrino
  • (a tiny amount) the recoil kinetic energy of the daughter nucleus

The electron therefore has a range of energies because the energy sharing varies from one decay to another.

C. Writing nuclear equations for β⁻ decay

General form: ^A_ZX → ^A_(Z + 1)Y + e⁻ + ν bar ₑ

Checks:

  • nucleon number: A is unchanged
  • charge: Z = (Z + 1) + (-1)

4. Common Mistakes

  • Forgetting that β⁻ emission increases proton number by 1.
  • Leaving out the antineutrino when explaining the continuous beta spectrum.
  • Writing charge conservation incorrectly in the nuclear equation.

5. Exam Tips

  • Mark-scheme phrasing: “the continuous beta spectrum shows that the available energy is shared; the antineutrino carries energy and momentum so both are conserved.”
  • In syllabus calculations, the antineutrino rest mass is normally negligible compared with nuclear mass scales.
  • Use antineutrino for β⁻ decay and neutrino for β⁺ decay.

6. Worked Examples

Modelled example 1

Balanced nuclear equation (β⁻ decay)

Core

Problem

Write the nuclear equation for the β⁻ decay of carbon-14, including the electron antineutrino.
Study the worked solution

  1. Track nucleon number

    Method

    A remains 14.

    Reason

    A neutron changes into a proton, so the total number of nucleons is unchanged.

    Working

    A_daughter = 14

  2. Track charge

    Method

    The daughter proton number is 7.

    Reason

    The nucleus gains one proton while the emitted electron carries charge -1.

    Working

    6 = 7 + (-1)

  3. Write the full products

    Method

    The daughter is nitrogen-14, accompanied by an electron and electron antineutrino.

    Reason

    The antineutrino carries variable energy and momentum while having zero charge and nucleon number.

    Working

    ¹⁴₆C → ¹⁴₇N + e⁻ + ν bar ₑ

Common misconception 2

Why isn’t the electron energy fixed?

Find and correct the mistake

Learner claim

A learner says every carbon-14 decay has the same available energy, so the emitted electron must always have the same kinetic energy; the measured continuous beta spectrum therefore violates energy conservation. Diagnose the claim.

Try this before viewing the solution

View solution step by step

  1. Keep total available energy fixed

    Method

    Each decay has a fixed total energy available to its products, apart from the same parent-state conditions.

    Reason

    Energy conservation applies to the complete system, not to the electron alone.

    Working

    Q = Kₑ + K_(ν bar) + K_recoil

  2. Include all three products

    Method

    The electron, antineutrino and daughter recoil share the energy and momentum.

    Reason

    Beta-minus decay is a three-body process.

    Working

    daughter nucleus + e⁻ + ν bar ₑ

  3. Explain the spectrum

    Method

    Different allowed sharing from event to event gives a continuous range of electron kinetic energies.

    Reason

    The antineutrino and recoil carry the remainder, so total energy and momentum remain conserved.

    Working

    variable Kₑ with fixed total Q

7. Mind Stretchers

Mind stretcher 1: Example: Two-body vs three-body decay argumentExtension

Explain (qualitatively) why a two-body decay would give a fixed electron kinetic energy, but a three-body decay gives a range.

Show Answer

In a two-body decay, conservation of momentum fixes the magnitudes of the momenta of the two products (they must be equal and opposite). With fixed product masses, this fixes their kinetic energies.

In a three-body decay, momentum can be shared among three particles in many different ways, so the electron momentum (and therefore kinetic energy) can vary continuously.

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027