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Incremental Binding Energies of Nuclides Due to the Formation of Neutron-Neutron Spin Pairs |
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Background
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.
pin 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 where the incremental binding energy makes a big drop.
The Excel regression does not compute a coefficient for a variable if it finds something wrong with the dataset with respect to that variable, but it does not reveal the nature of the problem.
| Regression Results | |||
|---|---|---|---|
| Constant | Regression Coefficient | Coefficient St. Dev. | t Ratio |
| C0 | 1.05526 | 0.24147 | 4.4 |
| Neutron Shells | |||
| Shell Number | Regression Coefficient | Coefficient St. Dev. | t Ratio |
| 2 | 6.76750 | 0.3037 | 22.3 |
| 3 | 3.30170 | 0.2005 | 16.5 |
| 4 | 1.65964 | 0.16675 | 10.0 |
| 5 | 0.87382 | 0.14855 | 5.9 |
| 6 | 0.34280 | 0.14007 | 2.4 |
| 7 | -0.05920 | 0.13714 | .-0.4 |
| 8 | -0.29307 | -0.13237 | -2.2 |
| Proton Shells | |||
| Shell Number | Regression Coefficient | Coefficient St. Dev. | t Ratio |
| 1 | -1.9324 | 0.32307 | -6.0 |
| 2 | 0 | ||
| 3 | 0.2675 | 0.16399 | 1.6 |
| 4 | 0.50282 | 0.18318 | 2.7 |
| 5 | 0.82990 | 0.19560 | 4.2 |
| 6 | 0.82517 | 0.20217 | 4.1 |
| 7 | 0.60369 | 0.21378 | 2.8 |
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.
The regression constant C0 must be included in the values for each shell number. When this done and logarithms taken, the equation is approximately
where S is shell number and Y is the regression coefficient for the neutron shell..
The coefficient of determination (R²) for the above regression equation is 0.569. While this value is not overwhelming most of the t-ratios strongly support 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 to 0.612. The regression coeffficient for n is about equal to the negative of the regression coefficient for p; thus suggesting that (p-n) is the proper explanatory variable involving p and n. when (p-n) replaces n in the the coefficient of determination changes only slightly but the t-ratio for (p-n) becomes 16.2 and that of p becomes 1.4 thus indicating the dependence on p other than through (p-n) is not significantly different from zero at the 95 percent degree of confidence
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