Modeling Polonium-212 Alpha Half-life
The distance at which the Coulomb potential drops to the level of energy of the observed alpha is
So the width of the barrier is
In addition to the tunneling probability calculated below, the alpha emission rate depends upon how many times an alpha particle with this energy inside the nucleus will hit the walls. The velocity of the alpha can be calculated from
since an alpha at this energy is nonrelativistic. The frequency of hitting the walls is then
Just for comparison purposes, the expected half-life for a single rectangular barrier equal to the peak of the barrier in height will be calculated. The tunneling probability for a rectangular barrier of height 26.2 MeV and width 17.9 fm is
For a given alpha, the probability per second for emission is the product
which may be used in the nuclear decay relationship to obtain the half-life
Obviously this is not a good approximation - it misses by 13 orders of magnitude!!
For a given alpha, the combined tunneling probability per second for emission is the product
which gives half-life
So the model gives a halflife of 0.25 microseconds compared to the experimental halflife of 0.3 microseconds. Not bad! But it must be admitted that this is a fortuitous example. Not all of them agree this well with just a five segment barrier approximation. Krane in Sec 8.4 demonstrates an integral approach to the barrier calculation rather than just a finite number of segments. But even this sophistication does not always give good agreement with the half-life because there are complications of the nuclear environment like non-spherical nuclei. Such quantum mechanical tunneling does however show agreement with the wide range of alpha decay half-lives, spanning over 20 orders of magnitude!
Eisberg & Resnick