Gamow Model for Alpha Decay: The Geiger-Nuttall Law

The tunneling amplitude can be approximated by the WKB formula

, where is the repulsive Coulomb potential energy between the -particle (charge ) and the daughter nucleus (charge ). The energy of the emitted -particle is given by , where is the distance from the center of the nucleus at which the becomes a free particle, while is the approximate radius of the nuclear potential well in which the is originally bound. The integral can be done exactly to give . For , a sufficiently good approximation is , so that . The transition probability per unit time approximates the reciprocal of the half-life for -decay, thus . The Geiger–Nuttall formula introduces two empirical constants to fudge for the various approximations and is commonly written in the form , where , measured in MeV, is often used in nuclear physics in place of .

Snapshots 1 to 3: nuclear potential and alpha wavefunction for three values of energy

References:

[1] Wikipedia, "Geiger–Nuttall Law." http://en.wikipedia.org/wiki/Geiger-Nuttall_law

[2] Wikipedia, "Alpha Decay." http://en.wikipedia.org/wiki/Alpha_decay