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Second Differences of Binding Energy
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 ProtonsNuclides 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.,

IBEn = ½[BE(n, p) − BE(n-2, p)]
and
IBEp = ½[BE(n, p) − BE(n, p-2)]

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

What the Data Were Expected to Show

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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