The number of states within a volume element in phase space is the velume of that element divided by where 2N is the number of dimensions in phase space. A constraint is provided by the conservation of energy. The number of states in a particular volume element is then:
There are thus six degrees of freedom (12 dimensinal phase space).
We note as a preliminary, that the total energy of a particle is
where T is the kinetic energy and the Lorentz factor. This is related to the particle momentum by
It follows that
In the following, we will assume the neutrino has a mass and put .
The phase space volume element is then
where is the angular part. Using
and writing in terms of the energy, this becomes
where the dependence on is stressed.
Substituting for p,
Only the electron is detected; we can thus integrate over the energy of the outgoing neutrino, to obtain
This gives the energy distribution of the emitted electron, i.e. it describes the shape of the beta spectrum. Note that the last factor in the second line gives the correction to the shape for non-zero neutrino mass.
This expression needs to be corrected for the distortion of the electron wave: this is not described accurately by a plane wav, but rather Coulomb waves must be used. This factor is usually cast in the form of the Fermi function (which is really a non-relativistic approximation):
where , is the atomic number of the daughter nucleus, the fine structure constant ( , and
The intergration over will be done below.