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and Nucleon Pairs Within Nuclei |
Consider nuclides made up of alpha particles and a fixed number of either neutron pairs or proton pairs. A proposition is developed elsewhere that implies that the interaction binding energy of the last alpha particle with the last nucleon pair is equal to the cross difference of the excess binding energy with respect to the number of alpha particles and the number of nucleonic pairs of a specific type for changes within one nuclear shell. Let the excess binding energy of a nuclide containing α alpha particles and n neutron pairs be designated as X(α, n). Then the incremental excess binding energy for the n-th neutron pair is
The cross difference the increment in this incremental binding energy with respect to the number of alpha particles; i.e.,
Here is the graph of excess binding energies of the alpha nuclides and the alpha+2neutron nuclides.
The difference is the incremental excess binding energy due to the addition of one neutron pair to the nuclides made up entirely of alpha particles, the alpha nuclides.
An enlarged version of the differences is shown below.
The form of this relationship indicates there are shells of alpha particles and the points of transition between shells, 3, 7 and 14, correspond to 6, 14 and 28 neutrons and protons, the so-called magic numbers.
The increments within a shell with respect to the number of alpha particles are then the interaction energies of the last alpha particle with the neutron pair. These are shown below.
The vertical black lines indicate the dividing points between the alpha particle shells.
The positive values indicate an attraction between the neutron pair and the alpha particles.
The analogous graph for a proton pair is shown below.
The generally negative values of the interaction excess binding energies reflect the repulsion between an alpha particle and a proton pair. This repulsion is not only due to the repulsion betwen them due to the electrostatic (Coulomb) force but due to the nuclear stong force as well.
A shell-by-shell comparison of the pattern for the proton pair with that for the neutron pair reveals a remarkable similarity. That similarity can be made more striking by a scatter diagram of the two sets of data.
The relationship is almost linear but a quadratic function gives a better fit. The regression equation for that quadratic relationship is
The coefficient of determination (R²) for the regression is 0.96919.
The effect of a second added neutron pair is computed by subtracting the excess binding energy of the alpha+2neutron nuclides from those of the alpha+4neutron nuclides having the same number of neutrons. This is shown in comparison with the effect of the first neutron pair in the graph below.
The comparison of the interaction excess binding energies is given in the graph below.
The effects of the first and second proton pair added to the alpha nuclides is shown below.
The comparison of the effects of the third proton pair with that of the second is shown below.
In this case there is a sharper increase after 10 alpha particles as well as after 7 and 14. Ten alpha particles corresponds to 20 neutrons and 20 protons, twenty being a magic number.
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