The general equation of α decay
is as follows: AZXN
+ 42He2 + Q,
I have labeled the table of nuclides below to show my choice of not quite typical examples of α decay. I have selected these examples partly because I wanted to demonstrate that the recoil of the daughter nucleus caused by α emission depends on the mass of the daughter.
Since parent nuclei are at rest before the decay, phrasing momentum conservation is simple:
This implies that the products move in opposite directions and their speeds (v) and masses (m) are inversely proportional:
Phrasing energy conservation is also very simple:
which means that the daughter nucleus and the α particle share the decay energy (Q) released in the process in the form of kinetic energy. The parent had no kinetic energy; the kinetic energy of the daughter, Edaughter, is called the recoil energy; Eα is referred to as alpha energy. The ratio of the kinetic energies is:
As the squares of momenta are equal for the decay products due to momentum conservation, kinetic energies and masses are also inversely proportional for the decay products:
I had another reason for including 8Be among the examples: the rapid 2α decay of 8Be brakes down the helium burning step of nucleosynthesis, making this nucleus very special. The decayability of 8Be is discussed below .
|The vertical axes of the figures below show the binding energy per nucleon (B/A) in different units. The table to the left shows B/A values for light (A < 20) stable nuclides as a function of the mass number (A). The figure to the right compares the stabilities of the isobaric nuclides for A = 8 that are not represented in the first diagram at all with each other and with the stability of 4He.|
|The above diagram has two remarkable features: (1) the extreme stability of 4He in comparison with its neighbors; (2) none of the elements have a stable isotope with mass number 8. In other words, there is no stable nuclide with mass number A = 8, or, which is the same: there is no stable nucleus composed of 8 nucleons. (The ratio „B/A” is in parentheses on the vertical axis because it is actually the mass equivalent of the binding energy per nucleon.)||The above curve binds the points of a mass parabola that belongs to the isobar A = 8. It is obvious that the winner in the bunch is 8Be. However, the binding energy per nucleon for 4He (red line) is higher by ~11 keV. For eight nucleons this adds up to ~92 keV, i.e. this is the extra amount of binding energy if 8 nucleons form 2 4He nuclei rather than a single 8Be nucleus. This Q = 92 keV is the driving force of the 2α decay of 8Be.|
Geiger and Nuttall originally discovered log-log linearity between the decay constant and the range of α rays. Due to the relationship between range and energy (more energetic particles can penetrate deeper in a given substance), the decay constant (or the half life which is inveresely proportional to it) and the alpha energy also yield a linear log-log plot. Such graphs are called Geiger–Nuttall plots.
The Geiger–Nuttall plots of the data of different decay series (which are slightly different) show that the half life of alpha decay very strongly depends on the alpha-energyl. A mere 50% increase in alpha energy causes ~15-decimal drop in half life.
The strong energy dependence is explained by quantummechanical tunnel effect. Note the slope of the potential curve in the gif animation. Remember that E = - grad V. This helps to understand why the Coulomb barrier acts as a barrier both for incoming α particles (Rutherford scattering) and outgoing α particles (α decay).
The above animation is a slightly modified version of the one I found on a German site .
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