Gamow Model for Alpha Decay: The Geiger-Nuttall Law
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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.
Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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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