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Neutron Pairs Within Nuclei |
This is an examination of the binding energies of nuclides which could contain an integral number of alpha particles plus a fixed number of neutron pairs. It is based upon the presumption that within a nucleus the neutrons and protons form alpha particles whenever possible. The formation of an alpha particle entails a binding energy of 28.295 million electron volts (MeV). The binding energy from the formation of alpha particles accounts for most of the binding energies of nuclides. If BE is the binding energy of a nuclide and α is the number of alpha particles it contains then the excess binding energy XSBE is given by
Below is shown the plot of the XSBE for the nuclides which could contain an integral number of alpha particles, hereafter referred to as the alpha nuclides.
A bent line in which the bends come at the points where the number of neutrons (and the number of protons) is equal to the magic numbers of {2, 6, 14, 28}. The statistical fit is near perfect, as shown by the coefficient of determination (R²) for the regression of 0.999169.
The plot of the excess binding energies of the nuclides which could contain an integral number of alpha particles plus an additional neutron pair is shown below.
The bends in this case are not so sharp as in the case of the alpha nuclides. This may be because the effects of transitions to a new neutron shell and a new proton shell is spread over two changes in the number of alpha particles.
At this point it is important to note that there is a question of which is the crucial variable, the number of neutrons or the number of alpha particles? A comparison of the excess binding energies for the alpha nuclides and the alpha+2neutron cases is shown below.
A larger picture of the differences is useful at this point.
The difference is a sequence of nearly linear functions. This is described as a piecewise linear function. Note that the jumps in the level occur after 3, 7 and 14 alpha particles, which correspond to the magic numbers of neutrons and protons of 6, 14 and 28.
Now a comparison can be made between the XSBE of the alpha+2neutron nuclides and that of the alpha+4neutron nuclides.
The relationship is also piecewise linear. The differences are plotted versus the number neutrons in the alpha particles of the nuclides because this makes the transitions occur at the magic numbers of 6, 14, 28 and 50. The number of neutrons in the alpha particles is identical with the number of protons. Those transition points are equivalent to transitions at 3, 7, 14 and 25 alpha particles.
The differences between the excess binding energies for the alpha+6neutron nuclides and those for the alpha+4neutron nuclides show a similar pattern.
However the first transition appears to occur jointly for 3 and 4 alpha particles. This corresponds to 6 and 8 neutrons. The numbers 8 and 20 are also nuclear magic numbers, but in a different category from the {2, 6, 14, 28, 50, 82, 126} sequence. A comparison of the differences for the last two cases is of interest.
The pattern for the differences of the XSBE for the Alpha+8neutron nuclides compared to those of the Alpha+6neutron nuclides is similar, but with the transitions coming at 4 and 10 alpha particles and thus at 8 and 20 neutrons as well as at 14, 28 and 50.
For the differences between the XSBE of the Alpha+10neutron nuclides and those of the Alpha+8neutron nuclides the pattern is similar but with the transition at 14 neutrons disappearing and one occurring at 16.
The case for the Alpha+12neutron nuclides and the Alpha+10neutron continues the pattern and the trend in the changes in the pattern.
Likewise the pattern continues for the Alpha+14neutron nuclides compared to the Alpha+12neutron nuclides.
The pattern continues at least throughout the rest of the cases but with fewer and fewer nuclides. The next magic number after 50 is 82, which corresponds to 41 alpha particles. Only the comparison of the Alpha+42neutron nuclides and the Alpha+40neutron nuclides is shown now.
At this point in the analysis one can only say that the operative variable determining the effect of additional neutron pairs is either the number of alpha particles or the number of protons. But a similar analysis concerning the effect of additional proton pairs concludes that the operative variable is either the number of alpha particles or the number of neutrons. Thus the joint conclusion is that the operative variable is the number of alpha particles in the nuclide. To confirm this, here is the effect of adding a neutron pair and a single proton to the alpha nuclides. Here the number of protons is not equal to half the number of alpha particles.
The transitions occur after the magic numbers for alpha particles, which correspond to magic numbers of neutrons and protons in the alpha particles, not the total numbers of either neutrons or protons.
It is not plausible that the operative variable would be the number of alpha particles that could be but are not formed. It would have to be the number that are actually formed. Thus the analysis provides definite evidence for the alpha particle structure of nuclides; i.e., whenever possible the nucleons of a nucleus form alpha particles. In other words, a nucleus is composed of alpha particles along with possisbly either neutron pairs or proton pairs and a singleton nucleon of the other type.
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