San José State University |
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A More Elaborate Statistical Explanation of the Incremental Binding Energies of Nuclides Due to the Formation of Neutron-Neutron Spin Pairs |
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A previous study began the statistical explanation of the incremental binding energies of nuclides due to the formation of neutron-neutron spin pairs. That statistical explanation starts with the proposition that nuclei are held together largely by the spin pairing of the nucleons (protons and neutrons) Such spin pairins are exclusive; m, eaning one neutron can pair with a proton and with one other neutron and no more. The same applies for protons.

There are other nonexclusivee interactions between nucleons but each each such interaction is an order of magnitude smaller than that of a spin pairing.

Therefore It is worthwhile to estimate the increments in binding energy due to the formation of spin pairs and explain their variation in terms of the makeup of the nuclei.

The binding energy due to the formation of a spin pair can be computed as the difference in the incremental binding energy at one point and the average of the value at the two adjacent points, as shown below.

This procedure is not valid near a point where incremental binding energy has a big drop. d near where the incremental binding energy makes a big drop.

Regression Results | |||
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Constant | Regression Coefficient | Coefficient St. Dev. | t Ratio |

C0 | 0 | ||

Neutron Shells | |||

Shell Number | Regression Coefficient | t Ratio | |

2 | 5.70422 | 16.2 | |

3 | 2.34126 | 11.2 | |

4 | 0.88431 | 5.1 | |

5 | 0.36780 | 2.4 | |

6 | 0.15426 | 1.1 | |

7 | 0.15860 | .1.2 | |

8 | -0.38813 | 2.9 | |

Proton Shells | |||

Shell Number | Regression Coefficient | t Ratio | |

1 | 0.26867 | 0.7 | |

2 | 2.14809 | 8.6 | |

3 | 2.30823 | 12.6 | |

4 | 2.36405 | 13.6 | |

5 | 2.54112 | 11.9 | |

6 | 2.42176 | ||

7 | 0.21378 | 18.8 |

The most notable aspect of these regression coefficints is that the values for neutron shell number decrease with the size of the shell number. A graph reveals a surprising regularity.

When logarithms are taken, the equation is approximately

where S is shell number and Y is the regression coefficient for the neutron shell..

In contrast the regression coefficients for proton shells are pretty much constant with with proton shell size except for the end point of the range.

The coefficient of determination (R²) for the above regression equation is reported by Excel as 0.932. This is misleading; a mistake that comes from comparing the variation for the regression equation with the variation about a constant of zero.

Most of the t-ratios of the regression strongly support re the notion that the effect of neutron-neutron spin pair formation is determined by the shell the nucleons are located in.

Adding p and n to the set of explanatory variables improves the coefficient of determination slightly. The regression coeffficient for n is about equal to the negative of the regression coefficient for p; thus suggesting that (n-p) is the proper explanatory variable involving p and n. When (n-p) replaces n in the regression the coefficient of determination changes only slightly but the t-ratio for (n-p) becomes 16.2 in magnitude and that of p becomes 1.4 thus indicating the dependence on p other than through (n-p) is not significantly different from zero at the 95 percent degree of confidence

The effect of a formation of a neutron-neutron spin pair on the incremental binding energy of a neutron varies systematically with the shell number of the number of neutrons of the nuclide under consideration. It varies less systematic with the shell number of the number of protons of the nuclide under consideration, but is roughly constant. There is also a dependence on (n-p), the excess number of neutr

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