﻿ Nuclear binding energies explained by shell-to-shell nucleonic interactions
San José State University

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Thayer Watkins
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Nuclear binding energies explained by
shell-to-shell nucleonic interactions

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

#### qIqJF(SIJ)

where qI and qJ are the charges of the nucleons in Shell I and in Shell J, respectively. The average separation distance between the nucleons is denoted as SIJ 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, nInJ. If I is equal to J then the number of interactions is nI(nI-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

#### qIqJF(SIJ)

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(SIJ). 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(SIJ).

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+dIJ) where dIJ 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 dIJ 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

Regression of Binding Energy on the
Interactions within Same Nucleon Type Shells
Shell x Shell Coefficient t-Ratio
N92x92 1.369522519 1.650391709
N63x92 2.66122983 9.672005546
N63x63 -3.236125482 -7.153324704
N41x92 0
N41x63 8.926973227 20.2081161
N41x41 -2.318449003 -2.693539883
N25x92 0
N25x63 0
N25x41 10.02336271 12.50297996
N25x25 3.607280704 2.639640496
N14x92 0
N14x63 0
N14x41 0
N14x25 10.94324306 6.818957218
N14x14 0.966120799 0.24942747
N7x92 0
N7x63 0
N7x41 0
N7x25 0
N7x14 21.31585159 4.41678173
N7x7 0.700953811 0.05328248
N3x92 0
N3x63 0
N3x41 0
N3x25 0
N3x14 0
N3x7 31.14743126 2.148407888
N3x3 19.85164667 0.269256961
N1x92 0
N1x63 0
N1x41 0
N1x25 0
N1x14 0
N1x7 0
N1X3 0.00145 2.45291E-05
63X63 0
41X63 -8.454654798 -9.676399101
41X41 0
25X63 0
25X41 -10.89493197 -1.987752397
25X25 0
14X63 4.068375226 5.21362348
14X41 -1.243179256 -2.286647778
14X25 -1.180021913 -1.189049237
14X14 0.636221022 0.249560308
7X63 0
7X41 0
7X25 -22.27726106 -5.619254055
7X14 -18.06066559 -3.362931381
7X7 -4.488162878 -0.625941075
3X63 0
3X41 0
3X25 0
3X14 0
3X7 -13.99698714 -0.885489684
3X3 -17.2703895 -0.493119308
1x63 0
1x41 2.042876 0.034558523
1x25 0
1x14 0
1x7 0
1x3 19.12690333 0.343189577
C0 28.295674 0.586244577

## Appendix II

Regression of Binding Energy on the
Interactions between Different Nucleon Type Shells
Shell x Shell Coefficient t-Ratio
63xN92 5.392641285 18.18788826
63xN63 5.459407022 0.99911904
63xN41 -17.79613215 -2.385244247
63xN25 0
63xN14 0
63xN7 0
63xN3 0
41xN92 3.375503551 14.6345205
41xN63 -2.889962754 -11.53438779
41xN41 6.917856873 3.830767217
41xN25 -25.20833895 -3.841513614
41xN14 0
41xN7 0
41xN3 23.36429244 0.710379457
25xN92 0
25xN63 13.30724203 22.5224223
25xN41 -4.094898491 -4.591351437
25xN25 2.769570943 2.326275983
25xN14 -17.3826679 -8.356858323
25xN7 0
25xN3 0
14xN92 0
14Xn63 0
14Xn41 17.5472573 10.83205647
14Xn25 -4.880877845 -0.659792463
14Xn14 0.353517785 0.165014049
14Xn7 -18.04503989 -3.882961208
14Xn3 0
7Xn92 0
7Xn63 0
7Xn41 0
7Xn25 37.13616959 2.822521877
7XN14 22.24271459 0.628054406
7xN7 -3.312801787 -0.497058232
7xN3 -19.30529779 -1.223953906
3xN92 0

## Appendix III

Regression Results for q=−2/3 and (1+d)=1.25
Inter coeff t-ratio
C0 28.295674 0.568423845
N1X3 91.97057709 1.394024678
N1x7 0
N1x14 0
N1x25 0
N1x41 1.702799445 0.035549019
N1x63 0
N1x92 19.85309667 0.436889015
N3x3 -14.48018088 -0.58189229
N3x7 43.68935801 2.245240693
N3x14 0
N3x25 0
N3x41 -76.03068543 -1.342897085
N3x63 0
N3x92 -0.221937088 -0.016358404
N7x7 -2.673806032 -0.434025811
N7x14 -8.418036242 -1.867602068
N7x25 -20.69175682 -4.270307771
N7x41 -111.3882562 -3.331241187
N7x63 0
N7x92 -8.237202631 -2.197147771
N14x14 5.495422501 2.666636646
N14x25 -54.38130454 -9.713853777
N14x41 -30.2379953 -20.49320931
N14x63 -6.32595053 -8.643700015
N14x92 10.06845621 9.832638931
N25x25 50.88809535 9.258946485
N25x41 43.8541431 2.623215316
N25x63 0
N25x92 -0.343038632 -0.409895095
N41x41 29.73122539 18.42753552
N41x63 -6.858892708 -9.14004255
N41x92 -3.744947919 -8.135849156
N63x63 9.624229708 10.11141484
N63x92 1.315017946 1.53660688