When the incremental binding energies of neutron pairs are plotted against the number of proton pairs the relationships are linear over a range corresponding to the proton shells. For example, consider the case below:

The slope of the relationship corresponds to the binding energy associated with the interaction of the proton pairs with the last, in this case the 25th, neutron spin pair in the nuclide.

The same applies to the relationship between the incremental binding energies of neutron pairs and and proton pairs, as is illustrated below.

Furthermore when the incremental binding energies of neutron pairs are plotted versus the number of neutron pairs therre is also a more or less linear relationship, only that it has a negative slope.

The sharp drop after 41 neutron pairs corresponds to the magic number of 82 neutrons. The magic numbers correspond to the complete filling of a nucleon shell.

Magic Numbers and Shell Occupancies

The conventional nucleonic magic numbers are {2, 8, 20, 28, 50, 82, 126}. These are based upon the number of stable nuclides. Those numbers also show up in terms of sharp drops in the levels of incremental binding energies. However 6 and 14 also show up in terms of drops in incremental binding energy. If the numbers corresponding to filled shells are taken to be {2, 6, 14, 28, 50, 82, 126} then there is a simple algorithm for their generation. The occupancies of the shells for this sequence are {2, 4, 8, 14, 22, 32, 44}. These numbers divided by 2 are {1, 2, 4, 7, 11, 16, 22}. Consider the sequence {0, 1, 2, 3, 4, 5, 6} and take the cumulative sums to get {0, 1, 3, 6, 10, 15, 21}. When 1 is added to each of these numbers the result is {1, 2, 4, 7, 11, 16, 22}, which is just the occupancies of the shells divided by 2, or equivalently the occupancies of the shells in terms of the number of nucleonic spin pairs. If sequence is extended one more level the occupancy of the next shell is 29 spin pairs or 58 nucleons. This would make the next magic number 126+58=184 nucleons or 92 spin pairs. In the work that follows the maximum occupancies of the shells in terms of spin pairs is taken to be {1, 2, 4, 7, 11, 16, 22, 29}.

The Binding Energies Due to
Nucleonic Spin Pair Interactions

The binding energy of a nuclide can be expressed as

BE = BEPP + BEPN + BENN

where BEPP is the binding energy due to the interaction of proton spin pairs with other proton spin pairs and similarly for BEPN and BENN. The interaction binding energy of two nucleon pairs is a function of the shell numbers they are in.

The interaction binding energy due to a nucleon in Shell I with a nucleon in Shell J is

q_Iq_JF(S_IJ)

where q_I and q_J are the charges of the nucleons in Shell I and in Shell J, respectively. The average separation distance between the nucleons is denoted as S_IJ and F(S) is the function that gives the strong force as a function of separation distance.

The total binding energy due to the interaction of nucleons in Shell I with those in Shell J is then the number of such interactions times the binding energy amount for interaction, given above. If Shell I is different from Shell J then the number of interactions is equal to the product of the numbers of nucleons in the two shells, n_In_J. If I is equal to J then the number of interactions is n_I(n_I-1).

The Stages of the Project

• Stage 1: The analysis is limited to nuclides with both an even number of protons and an even number of neutrons. This is to avoid the complication of the binding energy associated with the formation of neutron-neutron spin pairs and proton-proton spin pairs. This reduces the number of observations from 2931 to 738.

  Although the above selection of observations eliminates the effects of proton-proton and neutron-neutron spin pairs there are still the effects of neutron-proton spin pairs in the data. The coupling of a neutron-neutron spin pair with a proton-proton spin pair produces an alpha module; i.e., a chain of the form -n-p-p-n-, or equivalently, -p-n-n-p-. These alpha modules can account for a substantial portion of the binding energy of a nuclide. The number of alpha modules in a nuclide is simply the minimum of the number of neutron spin pairs and the number of proton spin pairs. The regression of binding energies on the number of alpha modules gives the following results

  BE = 37.34663 + 37.10123α

  with a coefficient of determination (R²) of 0.9795 and a t-ratio for the coefficient of 187.5. The standard error of the estimate is 72.6 MeV.

  Of course with there being zero alpha modules the binding energy should be zero. The regression without a constant is

  BE = 38.18455α

  with a standard error of the estimate of 74.4 MeV and a t-ratio for the coefficient of 432.9.

• Stage 2: The tabulation of the number nucleons in each shell for each of the 738 nuclides.
• Stage 3: The computation of the number of interactions between the nucleons in the six proton shells and the seven neutron shells.
• Stage 4: The regression of binding energies on the 102 variables of the number of interactions. Microsoft's EXCEL regression program will handle a maximum of about sixty indiependent variables.
• Stage 5: The analysis and interpretation of the results.

The Interpretation of the Results

For the interactions between proton pairs and neutron pairs and between neutron pairs and neutron pairs The regression coefficient should represent

q_Iq_JF(S_IJ)

Thus if the nucleon charge of a proton is taken as unity and that of neutron as q then those for the pairs would be 2 and 2q, respectively. The regression coefficient for the interactions of proton pairs in the I-th shell with neutron pairs in the J-th shell would be 2*2q=4q while those for neutron pair interaction would be 4q²F(S_IJ). Therefore the ratio of the coefficient for neutron-neutron interactions to the proton-neutron interactions should be q.

Likewise the square of the coefficent for the proton-neutron interaction to the coefficient for the neutron-neutron interactions should be F(S_IJ).

The proton pair interactions are affected by the electrostatic repulsion between protons. The ratio of the electrostatic force to the strong force is dependent upon the separation distance between the proton pairs. The distances from the I-th proton shell to some shell J may be taken to be equal to the distance from he I-th neutorn shell to J. Under this assumption the values of the ratio of the electrostatic force between proton pairs to that of the strong force can be calculated. If those ratios are found to be positive numbers then that would be a confirmation that the nature of the strong force between proton is of the same character as the electrostatic force between protons; namely repulsion.

Regression Results

Since EXCEL can accept only about sixty independent variable it is not possible at this point to get the regression coefficients of the full model. However in lieu of the full model results regressions were run on subsets of the independent variables. When binding energy was regressed on the proton pair to proton pair interaction and neutron pair to neutron pair interaction the result obtained are the ones shown Appendix I. There are many cases in which EXCEL did not include a variable in the regression that was supposed to be there. This is because of multicollinearity; i.e. one variable is equal to a linear combination of the other variables in the regression.

About the only meaningful bit of information to come out of that regression is that its coefficient of determination (R²) is 0.991258. This is impressive but a far cry from the values of 0.99998 and so forth obtained for related models.

If the binding energies are regressed on the interactions between proton shells and neutron shells the coefficient of determination is slightly higher at 0.991628. The results for this regression are given in Appendix II. Again there is no particular pattern and there is no reason for expecting there to be.

If the required regression could have been carried out and the model vindicated it have had coefficients for the interactions in definite ratios. If the nucleonic charge of the neutron relative to that of the proton is q then the interactrion of the proton spin pairs in shell I with the neutron spin pairs in shell J is proportional to q whereas the interaction of the neutron spin pairs in shell I with the neutron spin pairs in shell J is proportional to q². If the electrostatic repulsion between protons were not operating the interaction of the proton spin pairs in shell I with proton spin pairs in shell J would be proportional to 1, but because of the electrostatic repulsion it is proportional to (1+d_IJ) where d_IJ is a function of the distance between the spin pairs in shell I and shell J.

A regression was carried out using q equal to −2/3 and all the d_IJ equal to 0.25. The results as displayed in Appendix III. The coefficient of determination for this regression is 0.99068.

(To be continued.)

Appendix I

Appendix II

Appendix III