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Gamow Model for Alpha Decay: The Geiger-Nuttall Law

Alpha emission is a radioactive process involving two nuclei X and Y, which has the form , the helium-4 nucleus being known as an alpha particle. All nuclei heavier than Pb () exhibit alpha activity. Geiger and Nuttall (1911) found an empirical relation between the half-life of alpha decay and the energy of the emitted alpha particles. Using more recent data, the Geiger–Nuttall law can be written , where is in seconds, in MeV, and is the atomic number of the daughter nucleus. The observed range of half-lives is huge, varying from years for to sec for . We limit our consideration to even-even nuclei. Slightly different values of the parameters pertain when odd or nuclei are involved.
George Gamow in 1928, just two years after the invention of quantum mechanics, proposed that the process involves tunneling of an alpha particle through a large barrier. The barrier is created by the Coulomb repulsion between the alpha particle and the rest of the positively charged nucleus, in addition to breaking the strong nuclear forces acting on the alpha particle. Gurney and Condon independently proposed a similar mechanism. A plot of the nuclear potential also shows the alpha-particle wavefunction . The amplitude of the transmitted wave is highly magnified

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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
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