Beta decay and the neutrino

In the 1920s, measurements of beta decay seemed to show energy disappearing. Every decay along a given branch releases the same energy, yet the emitted electrons came out with a whole range of energies. In 1930 Pauli proposed that an undetected particle was carrying off the rest; it was finally detected in 1956. This lesson follows that argument: how conservation of energy and momentum predicted the (anti)neutrino.

The puzzle: electrons with a range of energies

For a particular decay branch, the parent and daughter nuclear states are fixed, so the energy released, Q, is fixed too. A nuclide can have several branches leading to different daughter energy levels; compare electrons within one branch. It comes from the decrease in rest mass.

Alpha decay behaves as you would expect from that: alpha particles from one two-body decay branch have the same kinetic energy. Beta-minus decay does not. If you plot the number of electrons against their kinetic energy, you get a continuous spectrum. It starts at zero, rises to a peak at a fairly low energy and falls to zero at a maximum kinetic energy that is practically equal to Q. Most electrons have much less kinetic energy than Q.

Two products would share the energy in only one way

Suppose a nucleus at rest decayed into just two products: the daughter nucleus and the electron.

• Conservation of momentum: the total momentum stays zero, so the two products move off in opposite directions with equal momenta.
• Conservation of energy: their kinetic energies add up to Q.

These two conditions fix both kinetic energies, so every electron from that decay would have the same kinetic energy, just below Q. The daughter’s recoil is already part of this calculation; it cannot absorb a different amount of energy each time. That is what happens in a two-body alpha decay branch, and it is why that branch gives one alpha energy.

The beta spectrum shows that this two-product picture is wrong. A momentum imbalance between the electron and daughter would also require another particle: the two visible momenta alone would not add to zero.

A third particle carries away the difference

Rather than give up conservation of energy and momentum, Pauli proposed that a third particle is emitted. In beta-minus decay it is the antineutrino, ν bar. It has no charge and a rest mass so small that it can be neglected here, and it interacts so weakly with matter that early detectors could not register it.

With three products, energy and momentum can be shared in many different ways:

where E_(k,e) is the electron’s kinetic energy, E_(ν bar) the antineutrino’s energy and E_(k,D) the daughter’s recoil kinetic energy. The three momenta add up to zero, but they need not lie along one line. When recoil energy is neglected, the electron energy approaches Q as the antineutrino takes almost no energy; in most decays the antineutrino takes a large share.

The daughter nucleus is thousands of times heavier than the electron, so its recoil kinetic energy is tiny and can usually be neglected in the energy account. Its momentum cannot be neglected: a heavy particle can carry a large momentum with very little kinetic energy.

With the recoil energy neglected, the electron and antineutrino energies add up to Q. For an illustrative decay with Q = 0.80 MeV:

Fixed total energy, variable electron energy

A straight line falls from an antineutrino energy of 0.80 MeV at zero electron kinetic energy to zero at an electron kinetic energy of 0.80 MeV. The two energies have an approximately fixed sum.

Scroll across the figure to read all labels.

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What to noticeEnergy-sharing approximation, not measured data or an electron-count spectrum. Antineutrino rest mass and the small daughter recoil energy are neglected. The plotted endpoints are approximate limits; exact endpoints include recoil. Momentum still includes the daughter nucleus.

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Figure data: Fixed total energy, variable electron energy

Electron kinetic energy (MeV)	Antineutrino energy (MeV)
0.0	0.8
0.1	0.7
0.2	0.6
0.3	0.5
0.4	0.4
0.5	0.3
0.6	0.2
0.7	0.1
0.8	0.0

Figure data: Fixed total energy, variable electron energy

Electron kinetic energy (MeV)	Antineutrino energy (MeV)
0.0	0.8
0.1	0.7
0.2	0.6
0.3	0.5
0.4	0.4
0.5	0.3
0.6	0.2
0.7	0.1
0.8	0.0

This graph shows how the energy is shared in any one decay: an electron with 0.20 MeV leaves 0.60 MeV for the antineutrino. It is not the beta spectrum, which shows how many electrons have each energy.