San José State University

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The Statistical Explanation of the Binding
Energies of Nuclides Based Upon a Two-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 interactions do not 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 classifications of the shells of neutrons and protons is within its capability.

The two-way classification is low shell (50 or less nucleons) and high shell (more than 50). The definitions of the variables are as follows: Let x and y be the numbers of neutrons and protons, respectively, in a nuclide. Then

low shell neutrons n=x if x≤50 and n=50 if x>50
high shell neutrons N=0 if x≤50 and N=x−50 if x>50
low shell protons p=y if y≤50 and p=50 if y>50
high shell protons P= 0 if y≤50 and P=y−50 if y>50

The dividing point of 50 was chosen because it is approximately one half the range of proton numbers and it is a nuclear magic number. A nuclear magic number represents the number of nucleons corresponding to filled nucleon shells. In the case of 50 the first five nucleon shells are filled. The sixth shell is filled when there are 82 nucleons and the seventh when there are 126 nucleons. The proton numbers do not reach the 126 level but the neutron numbers go beyond 126 into the eighth shell.

The Regression Results

Regression Results
(MeV)
Variable Coefficient t-Ratio
nn pairs 6.815948009 38.4
pp pairs 9.527180938 46.3
np pairs 11.63570751 70.6
nn -0.115342185 -13.4
nN -0.193262197 -40.2
np 0.29564005 32.3
nP -0.28970563 -114.0
NN -0.194470269 -97.3
Np 0.353182032 73.1
NP 0.285528864 87.0
pp -0.600758081 -59.7
pP 0 NA
PP -0.539989357 -93.4
c0 -22.43079821 -38.54

The effects of the three types of spin pair formations are all positive and the same order of magnitude. The effects of the interactions of like nucleons; nn, nN, NN, pp and PP; are negative indicating that the force between like nucleons is repulsion. The exception is pP. The regression program did not compute the coefficient for this variable. The effects of three out of the four interactions of the unlike nucleons; np, Np and NP; are all positive indicating that the force of the interaction is attraction. The exception is nP and with a t-ratio of 114 there is no chance that this anomaly is 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. The regression results are mixed on this matter.

The coefficient of determination (R²) for the regression is 0.999923 and the standard error of the estimate is 4.43 MeV. With an average binding energy of 1072 MeV this standard error of the estimate corresponds to a coefficient of variation of 0.0041, i.e., 0.41 of 1 percent.

If the spin pair formations are classified as to whether they occur in lower shells or upper shells the coefficient of determination is raised to 0.9999398 and the standard error of the estimate is reduced to 3.93 MeV and the coefficient of variation to 0.37 of 1 percent. But two of the coefficients for the regression were not computed.

If the dividing point between upper and lower shells is made at the next higher magic number, 82, the coefficient of determination is lowered to 99.9918. If it is lowered to the next lower magic number of 28 the coefficient of determination is raised to 99.99434 and all of the coefficients are computed, as is shown below. The standard error of the estimate for this regression is 3.807 MeV so the coefficient of variation is 0.355 of 1 percent.

Regression Results
(MeV)
Variable Coefficient t-Ratio
nn pairs 9.174127076 27
pp pairs 14.46930047 42.4
np pairs 6.738574611 23.1
NN pairs 2.428214428 8.1
PP pairs 2.997505519 9.7
NP pairs 8.674341133 30.5
nn -0.384874975 -15.2
nN -0.127872432 -8.8
np 0.621286749 23.9
nP -1.692394545 -7.3
NN -0.204284538 -125.5
Np 0.418860473 30.3
NP 0.30230735 119.6
pp -0.933713395 -34.1
pP 1.596289079 7.1
PP -0.56698556 -136.3

Generally the signs and magnitudes of coefficients fulfill expectations, but they are some anomalies. The coefficients for the interactions nn and pp are negative indicating a repulsion. The interaction np is positive, indicating an attraction. Likewise the interactions between like nucleons in the upper shells, NN and PP, are negative and NP is positive. The anomalies are the signs of some of the interactions between nucleons in different shells. The coefficient for the interaction between protons in the lower shells and neutrons in the upper shells, Np, is positive as is expected for the attraction between unlike nucleons. Likewise the sign of the coefficient for the interaction between neutrons is the two shell levels, nN, is negative, as would be expected for a repulsion between nucleons of the same type. But the interaction represented by nP appears to be a repulsion. The t-ratio for this interaction is relatively small.

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 pp, nn and np should be proportional to 1, q² and q, respectively. And likewise for PP, NN and NP. 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 the results are:

q = -0.384874975/0.621286749 = −0.619480417
q/(1+d) = 0.621286749/(-0.933713395) = −0.66539342
q²/(1+d) = -0.384874975/(-0.933713395) = 0.412198194
and hence
q/(1+d)½ = −0.64202663

These are not contradictory of the estimate of q as −2/3.

The values based upon the coefficients for PP, NN and NP are

q = −0.675751147
q/(1+d) = −0.533183507
q²/(1+d) = 0.360299367
and hence
q/(1+d)½ = −0.60024942

Again these are compatible with the estimate of q as −2/3.

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 best two-way classification of neutrons and protons is into lower shells of 28 or fewer nucleons and higher shells of more than 28 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 −2/3.


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