San José State University

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The Statistical Explanation of the Binding
Energies of Nuclides Based Upon a Three-Way
Classification of Nucleon Shell Occupancies

A regression model which is an outgrowth of the Alpha Module Model of nuclear structure preforms very well in explaining the binding energies of 2931 nuclides. It explains 99.98 percent of the variation in binding energy of the nuclides based upon the numbers of the three types of nucleon spin pairs and the three types of nucleonic (strong force) interactions. The statistical performance of the model is improved when the pair formations and interactions take into account the shells the nucleons are in. The regression program used cannot take into account all of the different shells, but two-way and three-way classifications of the shells of neutrons and protons is within its capability.

The two-way classification is low shell (28 or less nucleons) and high shell (more than 28). The performance of that model is given at Two-way Shell Classification. The statistical performance of that model is given by its coefficient of determination (R²) of 0.9999434, its standard error of estimate of and its coefficient of variation of 0.355 of 1 percent. This webpage is for reporting the statistical performance of the three-way classification of shell occupancy.

The definitions of the variables are as follows:

Low shells:
neutron number n and proton number p less than or equal to 28

Middle shells:
neutron number n and proton number p greater than 28
but less than or equal to 82

High shells:
neutron number n and proton number p greater than 82

,

The neutron number n and proton number p for a nuclide are converted into six numbers, (nL, nM, nH) and (pL, pM, pH), where

0<nL≤28
0<nM≤54
0<nH
 
0<pL≤28
0<pM≤54
0<pH

The number of neutron-neutron and proton-proton spin pairs in shell class Z are denoted as nZ%2 and pZ%2, respectively., The number of neutron-proton spin pairs in shell class Z is denoted as min(nZ, pZ).

The number of interactions between r nucleons in shell class X and s nucleons in shell class Y is denoted as rXsY and is equal to the product of rX and sY unless r and s are the same and X and Y are the same. In that case the number of interactions between nucleons r in shell class X is ½rX(rX-1).

Regression Results
(MeV)
Variable coeff t-ratio
nL%2 14.08787099 57.1
nM%2 2.359108542 5.7
nH%2 2.026470382 5.3
pL%2 19.30174475 80.7
pM%2 2.674346977 8.1
pH%2 1.759614648 3.0
min(nL,pL) 1.158396585 11.3
min(nM,pM) 7.751156523 26.8
min(nH,pH) NC NC
nLnL -0.860992298 -61.6
nLnM -0.284753431 -20.3
nLnH NC NC
nLpL 1.141099288 88.1
nLpM -0.39316782 -1.8
nLpH NC NC
nMnM -0.181841372 -43.8
nMnH -0.070930798 -18.3
nMpL 0.52424779 39.8
nMpM 0.331411421 108.6
nMpH -0.14614752 -23.1
nHnH -0.189722762 -92.2
nHpL NC NC
nHpM 0.318017218 113.9
nHpH 0.279176858 49.6
pLpL -1.397678761 -83.4
pLpM 0.310109501 1.5
pLpH NC NC
pMpM -0.587996667 -140.2
pMpH NC NC
pHpH -0.481625584 -36.0
Const -17.23566507 -24.0

NC = Not Computed

The effects of the three types of spin pair formations are all positive and but of a higher order of magnitude for the lower shells compared with the middle and higher shels. The effects of the interactions of like nucleons are generally negative indicating that the force between like nucleons is repulsion. Due to the vagaries of the regression program the coefficients for some variable were not computed. The effects of the interactions of the unlike nucleons are generally positive indicating that the force of the interaction is attraction. The exceptions have large t-ratios so there is no chance that these anomalies are due only to chance.

It would be expected that the effects of interactions of nucleons in the same shell would be greater than for the interactions of the same type in different shells and that is generally but not always the case.

The coefficient of determination (R²) for the regression is 0.9999492 and the standard error of the estimate is 3.614 MeV. With an average binding energy of 1076 MeV this standard error of the estimate corresponds to a coefficient of variation of 0.00336, i.e., 0.336 of 1 percent. This is only a marginal improvement over the performance of the two-way classification.

If the nucleonic (strong force) charge of the proton is 1 and that of a neutron is q then, ignoring the electrostatic repulsion between protons, the interactions of the types pp, nn and np in the same shell class should be proportional to 1, q² and q, respectively. When the electrostatic repulsion of protons is taken into account the interaction between protons should be proportional to (1+d) where d is a positive value representing the magnitude of the electrostatic repulsion between protons relative to the nucleonic (strong force) repulsion. Therefore the ratios of the coefficients should give

cnn/cnp = q
cnp/cpp = q/(1+d)
cnn/cpp = q²/(1+d)

When this formulas are applied to the coefficients for the lower shell class the results are:

q = −-0.754528819
q/(1+d) = −0.816424574
q²/(1+d) = 0.61601587
and hence
q/(1+d)½ = −0.784866785

These are significantly higher than the estimate of q as −2/3.

The values based upon the coefficients for the middle shell class are

q = −0.548687704
q/(1+d) = −0.563628061
q²/(1+d) = 0.309255787
and hence
q/(1+d)½ = −0.556107711

These are significantly lower than the estimate of q as −2/3.

The values based upon the coefficients for the higher shell class are

q = −0.336429539
q/(1+d) = −1.170889115
q²/(1+d) =0.393921686
and hence
q/(1+d)½ = −0.627631807

These are erratic compared to the estimate of q as −2/3.

The average of the three above estimates of q is −0.546548688

Conclusions

The regression model which is derived from the Alpha Module Model of nuclear structure explains nearly all of the variation in the binding energies of nuclides. The three-way classification of neutrons and protons used involved dividing points at 28 and 82 nucleons.

The signs and magnitudes of the regression coefficients are generally consistent with the model and the estimate of the nucleonic (strong force) charge of the neutron relative to that of the proton is compatible with the previous estimate of involving a different sign and a magnitude less than one.


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