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and the Relative Strong Force Charge of the Neutron Compared to that of the Proton |
Previous studies (1 and 2) developed the argument that the second differences in binding energies are approximately equal to the interactive binding energy of the last nucleon of a type with the next to last nucleon of the same type. This proposition has been tested and now this study will compare the second differences of the binding energies of neutrons and protons to shed some light on the matter of the relative strong force charge of a neutron compared to that of a proton.
| The Computation of the Second Differences in Binding Energies | |||||||
|---|---|---|---|---|---|---|---|
| Nuclides with 10 Protons | Nuclides with 10 Neutrons | ||||||
| Number of Neutrons | Binding Energy | 1st Diff IBEp | 2nd Diff | Number of Protons | Binding Energy | 1st Diff IBEn | 2nd Diff |
| (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | ||
| 4 | 4 | 69.99 | |||||
| 5 | 5 | 88.191 | 18.201 | ||||
| 6 | 97.325 | 6 | 110.753 | 22.562 | 4.361 | ||
| 7 | 112.9 | 15.575 | 7 | 123.865 | 13.112 | -9.45 | |
| 8 | 132.1535 | 19.2535 | 3.6785 | 8 | 139.807 | 15.942 | 2.83 |
| 9 | 143.7805 | 11.627 | -7.6265 | 9 | 147.80136 | 7.99436 | -7.94764 |
| 10 | 160.644859 | 16.864359 | 5.237359 | 10 | 160.644859 | 12.843499 | 4.849139 |
| 11 | 167.40597 | 6.761111 | -10.103248 | 11 | 163.0762 | 2.431341 | -10.412158 |
| 12 | 177.76991 | 10.36394 | 3.602829 | 12 | 168.5776 | 5.5014 | 3.070059 |
| 13 | 182.97053 | 5.20062 | -5.16332 | 13 | 168.703 | 0.1254 | -5.376 |
| 14 | 191.836 | 8.86547 | 3.66485 | 14 | 172.004 | 3.301 | 3.1756 |
| 15 | 196.02 | 4.184 | -4.68147 | 15 | 171.18 | -0.824 | -4.125 |
| 16 | 201.6 | 5.58 | 1.396 | 16 | 171.4 | 0.22 | 1.044 |
| 17 | 203.01 | 1.41 | -4.17 | ||||
| 18 | 206.89 | 3.88 | 2.47 | ||||
| 19 | 208.2 | 1.31 | -2.57 | ||||
| 20 | 212.1 | 3.9 | 2.59 | ||||
| 21 | 211.5 | -0.6 | -4.5 | ||||
| 22 | 213.3 | 1.8 | 2.4 | ||||
The first differences are also called the incremental binding energies of neutrons (IBEn) and of protons (IBEp). Here are the patterns.
The sawtooth pattern of odd-even fluctuations have to do with the binding energy resulting from the formation of nucleon pairs. This phenomenon makes the second differences quite erratic.
Nevertheless the ratios can be computed and they are displayed below.
| Nucleon Number | 2nd Diff Neutrons | 2nd Diff Protons | Ratio |
| 8 | 2.83 | 3.6785 | 0.769335327 |
| 9 | -7.94764 | -7.6265 | 1.042108438 |
| 10 | 4.849139 | 5.237359 | 0.925874854 |
| 11 | -10.412158 | -10.103248 | 1.030575316 |
| 12 | 3.070059 | 3.602829 | 0.852124539 |
| 13 | -5.376 | -5.16332 | 1.041190552 |
| 14 | 3.1756 | 3.66485 | 0.86650204 |
| 15 | -4.125 | -4.68147 | 0.88113349 |
| 16 | 1.044 | 1.396 | 0.747851003 |
The ratios are less than one for the even numbered cases and generally greater than one for the odd numbered cases. It is clear that the odd-even fluctuation distorts the pattern so much that it is hard to identify it. One way of eliminating the distortion is to take the first differences over increments of two nucleons; i.e.,
The graphs of the resulting patterns are shown below.
The values of the second differences computed from the first differences displayed in the graphs are given in the following table.
| Nucleon Number | 2nd Diff Neutrons | 2nd Diff Protons | Ratio |
| 9 | -1.974 | -2.55882 | 0.77144934 |
| 10 | -1.1945705 | -1.5492505 | 0.771063492 |
| 11 | -2.4329445 | -2.7815095 | 0.874684951 |
| 12 | -3.2502095 | -3.6710495 | 0.885362483 |
| 13 | -0.7802455 | -1.1529705 | 0.676726334 |
| 14 | -0.749235 | -1.1002 | 0.680998909 |
| 15 | -0.50831 | -0.4747 | 1.070802612 |
| 16 | -1.642735 | -1.5405 | 1.066364817 |
If two particles have strong force charges of q1 and q2 the force between them would be proportional to the product of their charges q1q1 and so would the potential energy associated with their interaction. If the strong force charge of a proton is taken to be unity and that of a neutron is denoted as q then the ratio of the binding energy associated with neutron-neutron interactions to that of proton-proton interactions would be equal to q². Previous studies found q to be negative and less than unity in magnitude. That much of the previous studies is generally confirmed by the above results. Previous studies however found q to be equal to −2/3 or −3/4 which would mean the ratio should be 4/9=0.444. or 0.5625. This was not confirmed. However the analysis of second differences found only that they should be approximately equal to the interaction binding energy of the last two particles. Therefore there could be a significant margin of error in the ratios.
The average of the ratios in the above table is 0.850.
The same procedure applied to the nuclides with 18 protons and with 18 neutrons gives the data in the following table.
| Nucleon Number | 2nd Diff Neutrons | 2nd Diff Protons | Ratio |
| 15 | -0.78 | -1.9096 | 0.408462505 |
| 16 | -2.2395 | -2.76612 | 0.809617804 |
| 17 | -1.2644 | -1.587575 | 0.796434814 |
| 18 | -0.90875 | -1.188585 | 0.764564587 |
| 19 | -1.97615 | -2.256425 | 0.875788028 |
| 20 | -1.7077 | -1.978695 | 0.863043572 |
| 21 | -1.0953 | -1.2299 | 0.890560208 |
| 22 | -0.98433 | -1.28925 | 0.763490401 |
| 23 | -0.24982 | -0.359 | 0.695877437 |
| 24 | -0.21927 | -0.47 | 0.466531915 |
The average of the ratios is 0.733.
For the case of 24 nucleons the results are:
| Nucleon Number | 2nd Diff Neutrons | 2nd Diff Protons | Ratio |
| 21 | -1.14 | -1.2829 | 0.888611739 |
| 22 | -0.6725 | -0.9101 | 0.738929788 |
| 23 | -0.3985 | -0.8608 | 0.46294145 |
| 24 | -0.8955 | -1.1225 | 0.797772829 |
| 25 | -1.2855 | -1.5413 | 0.834036203 |
| 26 | -1.66695 | -1.9735 | 0.844666836 |
| 27 | -0.66015 | -0.9975 | 0.661804511 |
| 28 | -0.4804 | -0.7415 | 0.647875927 |
| 29 | -0.66125 | -0.975 | 0.678205128 |
| 30 | -1.16015 | -1.185 | 0.979029536 |
The average of the ratios is 0.753.
Finally for the case of 32 nucleons the results are:
| Nucleon Number | 2nd Diff Neutrons | 2nd Diff Protons | Ratio |
| 29 | -1.355 | -1.28165 | 1.057230913 |
| 30 | -1.09 | -1.52825 | 0.713234091 |
| 31 | -0.625 | -0.9947 | 0.62833015 |
| 32 | -0.525 | -0.7285 | 0.720658888 |
| 33 | -1.34 | -1.4305 | 0.936735407 |
| 34 | -1.15 | -1.56 | 0.737179487 |
| 35 | -0.4765 | -0.775 | 0.61483871 |
The average of the ratios is 0.773.
When the results of the four cases are displayed together it is clear that the ratio of the effect of neutron-neutron strong force interaction is less than that of proton-proton interaction. This means that the strong force charge of a neutron is less than that of a proton.
(To be continued.)
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