What are continuous energy spectra?

The sum of all energy released in a nuclear reaction can be calculated from mass defect, the difference in the mass of the products and the reactants, by the equation #E=m*c^2#.

The amount of mass lost in a particular process of beta is definite, meaning that the sum of kinetic energy of all product particles shall have discrete values.

It is possible to set up a system of equations to solve for the final kinetic energy of the electron:

1. The sum of final kinetic energy of the nucleus and the electron equals to the energy released in the decay;
2. Momentum conserves

This system will yield a finite number of solution (one or two) if the decay produces only two particles: the nucleus and an electron. Hence one might expect to detect electrons that travel at some particular velocity near a collection of nucleus undergoing beta-minus decay.

However, experimental results disagree with the prediction; instead of giving discrete points, plotting kinetic energy against the number of particles possessing that amount of energy will produce a distribution similar to that of a Maxwell-Boltzmann distribution curve.

Both energy and momentum still have to conserve; the release of the antineutrino in beta-plus decays (or neutrino for beta-minus decays) as a third product of beta decays accounts for the continuous energy spectra. For a general beta-minus decay:
#color(white)(X)_Z^A X->#

#color(white)(X)_(Z+1)^A X'+e^(-)+barv#

The sum of kinetic energy of all three products- the daughter nuclei, the electron, and the antineutrino- is a definite value.
#KE_"X'" +KE_(e^-)+KE_(bar v)="Mass Defect"*c^2#

Hence
#KE_(e^-)=Deltam*c^2-KE_"X'" -KE_(bar v)#

The mass of the nucleus is much larger than that of the electron and the antineutrino such that its share of kinetic energy is negligible; the kinetic energy of the antineutrino, however, can vary significantly from #0# all the way to #"Mass Defect"*c^2#- where it takes all the energy from the decay. The presence of a third particle makes it impossible to find unique solutions to the kinetic energy of the electron; similarly, it can take more than one possible value ranging from #0# all the way to #"Mass Defect"*c^2# and therefore gives a continuous energy spectrum.