San José State University

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The Statistical Explanation of the Binding
Energies of Nuclides Based Upon a Four-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 and its standard error of estimate of 3.807 MeV. With an average binding energy of 1072 MeV this standard error of the estimate corresponds to a coefficient of variation of 0.355 of 1 percent. The statistical performance of the three-way classification of shell occupancy is given by a coefficient of determination (R²) is 0.9999492 and the standard error of the estimate is 3.614 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.

Unavoidably there is the desire to see what improvement in statitstical performance can be achieved by a four-way classification of shell occupancy. The result of the effort has more to do with the limitations of the EXCEL regression program than what can be achieved with a four-way classification of shell occupancies. .

The definitions of the shell groups are as follows:

Low shells:
nucleon numbers less than or equal to 28

Middle shell 1:
nucleon number greater than 28
but less than or equal to 50
Middle shell 2:
nucleon number greater than 50
but less than or equal to 82

High shells:
nucleon number greater than 82

,

The shell groups are labeled as {a, b. c. d}. The number of proton-proton, neutron-neutron and neutron-proton spin pairs in shell group Z are denoted as ppZ, nnZ and npZ respectively., The number of interactions between r-type nucleons in shell group X and s-type nucleons in shell group 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
const 4.740766827 1.1
Nucleon Spin Pairs
ppa 0.552524716 0.3
nna 3.090603665 1.7
npa 10.76096446 7.8
ppb 0 NC
nnb 0.928986584 0.3
npb 0 NC
ppc 1.179607898 0.5
nnc -14.54784345 -4.9
npc 0 NC
ppd 1.229116256 0.4
nnd 0.270373168 0.11970957
npd 0 NC
Nucleon Interactions
papa -0.463133595 -4.1
pana 0.532384109 4.8
papb 0 NC
panb -15.31654115 -13.8
papc -0.068630184 -1.4
panc 0.699997808 13.3
papd 0 NC
pand 0 NC
nana -0.122212364 -1.1
napb 15.934174 14.4
nanb 0 NC
napc 0 NC
nanc 0 NC
napd 0 NC
nand 0 NC
pbpb 0 NC
pbnb -0.256501101 -10.7
pbpc 0 NC
pbnc 0.01398365 0.4
pbpd 0 NC
pbnd -1.60131086 -0.3
nbnb 0 NC
nbpc 0 NC
nbnc 0 NC
nbpd 0 NC
nbnd 0 NC
pcpc -0.560200182 -17.9
pcnc 0.411798925 14.0
pcpd -0.205964463 -3.2
pcnd 0.278807801 14.5
ncnc -0.278198698 -10.7
ncpd 0 NC
ncnd 1.250799455 0.3
pdpd -0.461717838 -5.7
pdnd 0.258942574 7.5
ndnd -0.177111873 -13.2

NC = Not Computed

The zeroes are for the variables for which the regression program did not compute a coefficient. As a result of the 21 of the 48 variables being left out of the analysis the statistical performance of regression is decidedly inferior to those of the two-way and three-way classifications. The coefficient of determination for the above regression is only 0.9982513, a far cry from the 0.9999434 and 0.9999492 for the two-way and three-way classifications. The standard error of the estimate for the above regression is 21.2 MeV, which corresponds to a coefficient of variation of about 2 percent.

Conclusions

The analysis cannot be pushed to a four-way classification of shell occupancy because of the limitations of the regression program available.


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