9478 / 2027
Nuclear Physics overview

Lesson 8 of 10

Missing energy and momentum

  • I can explain how energy and momentum conservation in beta decay support the inference of an additional neutral particle.Syllabus

    Syllabus 9478, 20(o). Explain how energy and momentum conservation in beta decay support the inference of an additional neutral particle.

Make a guess. You are not marked.

Beta-minus particles from one nuclide come out with a range of kinetic energies, not one fixed energy. What did this suggest?

  1. The nucleus emits electrons from different shells of the atom.
  2. Energy is not conserved in nuclear decay, so the amounts vary.
  3. A second particle, the antineutrino, carries away the rest of the energy.
Show the answer

A second particle, the antineutrino, carries away the rest of the energy.

Each decay releases the same total energy. The varied electron energies showed that another, hard-to-detect particle shares it: the antineutrino.

Beta decay gives evidence for an additional particle through both the spread of electron energies and the momentum that is missing from an electron-and-daughter account.

Test the two-product prediction

Consider a parent nucleus initially at rest and a fixed decay branch to a specified daughter state. If only an electron and that daughter were produced, momentum conservation would require equal and opposite momenta. Together with the fixed energy available, this would fix the electron's energy.

Instead, beta electrons from that branch have a continuous range of energies up to an endpoint. The energy varies because it is shared with another emitted particle as well as the recoiling daughter. The extra particle is uncharged and difficult to detect; in beta-minus decay it is an antineutrino.

Electron energies form a continuous spread

Electron energies form a continuous spreadA schematic electron kinetic-energy spectrum occupies a continuous range from zero to an endpoint E-max. The horizontal coordinate is kinetic energy divided by E-max. The vertical quantity is a qualitative count density with no numerical normalisation. A separate dashed vertical guide at the endpoint identifies the one fixed electron energy predicted by an ideal two-body decay from a parent at rest on a fixed branch. It is not a second measured finite-width distribution and its guide height is not a count. The continuous spectrum is evidence that another participant shares the available energy and momentum.Electron count density(schematic vertical scale)Two-body prediction00.51Electron kinetic energy / EmaxThe endpoint guide is not a count.

A fixed two-body decay of a parent initially at rest would give a fixed electron energy for that branch. The observed spread motivates an additional energy carrier while preserving conservation. This schematic adds no independent event energy or isotope-specific endpoint.

Translate momenta head to tail to test the complete account

Translate momenta head to tail to test the complete accountThe parent is initially at rest, so total momentum is zero. Three translated vectors use an equal horizontal and vertical scale in one common momentum unit, p-zero. From the starting point O, the electron vector goes three units right. From its tip, the daughter vector goes three units left and four units down. The missing vector then goes four units up, returning to O and closing the triangle. Thus the vector sum is zero. These momentum vectors are translated without rotation; they are not spatial particle trajectories. No separate kinetic-energy values, masses or real isotope are attached to this illustrative event.Parent initially at restTranslated momentum vectorspe = (3, 0)p0pd(-3, -4)p0pmissing(0, 4)p0OThe final tip returns to O.+x+ype+pd+pmissing= 0p0 is one common momentum unit.These arrows are not trajectories.

The electron and daughter momenta alone leave a downward resultant. An additional upward momentum closes the account. An unobserved neutral carrier can conserve energy and momentum; the diagram does not justify abandoning either conservation law.

The spectrum illustrates a continuous electron-energy distribution for a fixed branch; the endpoint guide marks the two-product prediction, not a measured narrow peak. The separate vector example closes the momentum account with an additional particle. Its arbitrary momentum scale is not a calibrated event from the spectrum, and the translated vectors are not trajectories.

Supplied momentum account

Find the missing vector

The parent is at rest. The supplied electron and daughter momenta in a planar example, in units of a common momentum p0, are:

pelectron = (3, 0)p0
pdaughter = (-3, -4)p0
Sum = (0, -4)p0

The total initial momentum is zero, so the missing particle must carry (0, +4)p0. Adding this vector closes the account. Adding only the two momentum magnitudes would discard their directions and give the wrong result.

The energy distribution and momentum imbalance support an additional carrier. They are not a reason to abandon energy or momentum conservation. Balancing nucleon number and charge alone would not have revealed the missing energy and momentum.

Check your understanding

For one beta-decay branch of a parent initially at rest, electrons have a continuous energy spread and the measured electron-plus-daughter momentum does not close the account. What inference retains conservation?

  1. An additional neutral particle carries energy and momentum.
  2. The electron is an orbital electron, so its energy need not enter the nuclear account.
  3. Energy and momentum cease to be conserved in nuclear processes.
Show the answer

An additional neutral particle carries energy and momentum.

A third participant permits variable sharing of the available energy and completes the vector momentum account. In beta-minus decay this carrier is an antineutrino.