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of Adding and Subtracting Neutrons and Protons |
Let BE(n,p) be the binding energy of a nuclide as a function of the number of neutrons n and the number of protons p which it contains. The alpha nuclides are the ones in which n and p are equal and are an even number so that they could contain an integral number of alpha particles. This means that n=p=2a where a is the number of alpha particles.
The effect of subtracting one neutron from each alpha nuclide is then
The effect on binding energy of adding one neutron is
The difference of these two quantities is then
The analogous quantity for the subtraction and addition of a proton to each alpha nuclide is
The graphs of these two quantities, DNBE and DPBE, as functions of the number of alpha particles a are shown below.
The close correspondence of the shapes is remarkable. The points where the curves show a sharp drop at 7, 10 and 14 alpha particles at where the numbers of neutrons and the numbers of protons are equal to 14, 20 and 28, all magic numbers.
Although the curves virtually coincide over some ranges the values, as shown below, are not identical.
| Number of Alpha Particlels | DNBE (MeV) | DPBE (MeV) |
| 1 | 21.779527 | 21.46329 |
| 2 | 17.44012 | 17.23372 |
| 3 | 14.013356 | 13.775493 |
| 4 | 11.527132 | 11.520412 |
| 5 | 10.412158 | 10.103248 |
| 6 | 9.42156 | 9.20142 |
| 7 | 8.83683 | 8.70614 |
| 8 | 6.58743 | 6.4008 |
| 9 | 6.64809 | 6.4646 |
| 10 | 7.2433 | 7.2785 |
| 11 | 7.0353 | 6.7697 |
| 12 | 6.015 | 5.752 |
| 13 | 5.7824 | 5.4998 |
| 14 | 6.4711 | 6.394 |
| 15 | 4.6627 | 4.771 |
| 16 | 5.07 | 5.52 |
| 17 | 5.3 | 5.51 |
| 18 | 5.4 | 5.27 |
| 19 | 4.675 | 4.16 |
| 20 | 4.9 | 4.6 |
| 21 | 5.0 | 4.7 |
| 22 | 4.8 | 4.3 |
| 23 | 4.3 |
The implication of the close correspondence of the two curves is that each twist and turn is of significance in the physics of nuclei.
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