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on the Nuclides Composed Entirely of Alpha Modules and One Neutron Spin Pair |
The details of the alpha module model of nuclear structure are given elsewhere. Briefly the basis for it is that whenever possible a nucleon (neutron or proton) forms a spin pair with one other nucleon of the same type and with one nucleon of the opposite type. This leads to chains composed of modules of the form -n-p-p-n- (or equivalently -p-n-n-p-). These chains form rings which rotate in four modes; as a vortex ring, rotation about an axis through the center of the ring and perpendicular to its plane and flipping rotation about two diameters of the ring. These rotations take place so rapidly that the alpha module ring dynamically appears as a sphere with the nucleons smeared uniformly throughout a spherical shell.
The occupancies of the spherical shells are basically what were found by Maria Goeppert Mayer and Hans Jensen and labeled nuclear magic numbers. The Goeppert Mayer and Jensen magic numbers were based upon the numbers of stable isotopes and isotones. Their values were (2, 8, 20, 28, 50, 82, 126). The patterns of incremental binding energies justify a modified sequence of (2, 6, 14, 28, 50, 82, 126). The numbers 8 and 20 instead represent the filling of subshells within shells.
Here is a graph of the binding energies of nuclides which could be composed of alpha modules plus one neutron spin pair. They will be referred to as α+1nn nuclides.
The linearity of the display is impressive. A regression of the binding energy BE on the number of apha modules #α gives
The coefficient of determination (R²) for this equation is 0.9978 and the standard error of the estimate is 12.48276 MeV. The t-ratio for the regression coefficient is 104.4.
As impressive as is this statistical performance it can be improved considerably. First consider the graph of the incremental binding energies.
This graph reveals that there are slight changes in the slope of the relationship at critical values of #α. One critical value is #α equal to 14, which corresponds to 28 neutrons and 28 protons. Twenty eight is a magic number corresponding to the filling of a shell. Another critical number is at #α equal to 4, which corresponds to 8 neutrons and 8 protons. Another is for #α equal to 7 which corresponds to 14 neutrons and 14 protons. The changes in slope are not so sharp as is the case with the alpha nuclides.
When the regression equation involves changes of slope at 4 and 14 the result is
where δ(z) is the ramp function and is equal to z if z>0 and zero otherwise.
The coefficient of determination for this equation is 0.99986 and the standard error of the estimate is 3.299 MeV. The average value of BE is 427.3 MeV so the standard error of the estimate of 3.299 MeV corresponds to a coefficient of variation of 0.7 of 1 percent. However the t-ratio for the δ(#α-2) is only −1.17\ so the effect of that variable is not significantly different from zero at the 95 percent level of confidence.
The inclusion of δ(#α-7) raises the R² value to 0.99991 and reduces the standard error of the estimate to 2.74 MeV and the coefficient of variation to 0.225 of 1 percent.
The regression results are
The t-ratio for the coefficient of the δ(#α-4) variable is however only −1.86 so it is not significantly different from 0 at the 95 percent level of confidence.
The binding energies of the alpha+1nn nuclides are overwhelmingly explained by a bent-line function of the number of the number of alpha modules. The critical points correspond generally to the magic numbers of a nuclear shell model.
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