The transition probability per unit time in nuclear -decay is given by the golden rule:

where is the density of final states. The matrix element includes a summation over all possible angular momentum states of the leptons, over all final spin states of the nucleus and an average over the spin orientations of the initial nucleus. For convenience, the nucleus is assumed to be infinitely heavy and therefore carries no energy but contributes the necessary recoil momentum for the leptons [Hor-75]. The density of final states is:

where and are the number of states available to the
-particle and the neutrino in the momentum intervals **p** to
**p+dp** and **q** to **q+dq** respectively and is the
total energy carried by the leptons:

From the energy-momentum relations of the leptons, one has:

and

Neglecting the neutrino mass, the transition probability becomes:

where

and

The matrix element is given by equation () with
instead of **G**. One rewrites the matrix element as:

where the lepton matrix element is given by equation and the hadron matrix element generally comprises a vector and an axial vector contributions:

where **V** is the vector necessary to construct .
With equation (), the transition
probability is:

where the factor , known as the Fermi Function, includes among other things, correction factors such as screening effects by atomic electrons and final state Coulomb interaction effects which are particularly important at low energies and for high-Z nuclei. By integration the transition probability over the energy spectrum, the decay rate is:

with

The product **ft** of the Fermi integral function **f** and the decay half
live is independent of kinematic effects and involves only
the matrix element:

where **K** is a product of fundamental constants. In cgs units,

Mon Jan 15 14:57:06 MET 1996