As we have seen, the alpha separation energy is negative for heavy nuclei such as uranium, but these nuclei do not immediately decay. In fact, alpha decay mean lives vary from nanoseconds to gigayears. We have seen that the alpha particle can be regarded as trapped by a potential barrier. In order to escape into the environment, the alpha must tunnel through the barrier. This description of alpha decay, which also explains the wide range in lifetimes, was given by Gamow and was oe of the first successes of the new quantum theory (which introduced such counter-intuitive ideas as tunnelling).
Suppose we have a wave packet representing an alpha particle with mass and kinetic energy E impinging on a square potential barrier of height V;SPMgt;E and width . Then the transmission coefficent T is obtained from 2nd year quantum mechanics as
This can be extended to any barrier shape in the form of the WKB approximation:
Here, R and b are the classical turning points of the motion inside and outside the barrier. We may take the barrier to be the sum of a square well nuclear potential of radius R, and a Coulomb potential arising from a charge within R,
We can equate (aproximately) the energy release Q in the alpha decay to the kinetic energy E of the alpha particle and to the potential at the outer classical turning point.
and hence determine b:
Hence the integral over becomes
where and the above expression for b has been used.
For thick barriers ( or ) we can approximate , and hence
The decay constant for alpha decay is thus
The Geiger-Nutall equation is thus recovered. Note the extreme sensitivity of the decay constant on the energy in the above equation.