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The Statistical Explanation of the
Binding Energies of Nuclides

Nuclei are made up neutrons and protons but their measured masses are less than the masses of their constituent nucleons. The mass deficits when expressed in energy units through the Einstein formula of E=mc² are called their binding energies. The binding energies are known for 2931 nuclides.

The neutrons and protons within a nucleus are organized into substructures. Definitely neutron-neutron, proton-proton and neutron-proton pairs are formed. Definitely there are separate systems of shells for neutrons and protons. There may be alpha particles, Helium 4 nuclei, within nuclei. An alpha particle has a notably high level of binding energy compared to smaller nuclides. Its binding energy is 28.3 million electron volts (MeV) whereas that of a deuteron, a neutron-proton pair, is only 2.2 MeV. The level of binding energies of larger nuclides to a great extent can be explained by the neutrons and protons whenever possible forming alpha particles within a nucleus. In fact, about 98 percent of the variation in binding energy of nuclides is explained by the number of alpha particles which could be formed within a nuclide.

The numbers of nucleons at which shells are filled has come to called magic numbers. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. These were identified by the numbers of stable isotopes and isotones (nuclides with the same number protons or the same number of neutrons). They can also be identified by sharp drops in the incremental binding energies. This method establishes that 6 and 14 are also magic numbers. The sequence {2, 6, 14, 28, 50, 82, 126} is explained by a simple algorithm. This indicates that 8 and 20 are associated with subshells. Using the sequence {2, 6, 14, 28, 50, 82, 126} as the filled shell totals the capacities of the shells are as follows.

Shell
Number
12345678
Capacity2481422324458

The capacities are the same for protons as for neutrons but there are no stable nuclides with proton numbers beyond the seventh shell. The capacity of the eighth shell is computed by the algorithm mentioned above.

A Shell Occupancy Explanation of Binding Energies

Often nuclear phenomena for nucleons are a function of their shell number; i.e., they are relatively the same for nucleons in the same shell. Let ni and pi be the number of neutrons and protons, respectively, in the i-th shell. The equation for explaining binding energies of nuclides is then

BE = Σcini + Σdipi

A regression based upon this equation gives the following results.

Shell
Occupancy
Coefficient Standard
Deviation
t-ratio
n8 6.438642308 0.086043684 74.82992407
n7 7.773687644 0.03583642 216.9214364
n6 8.268566776 0.042067984 196.5524837
n5 8.887083671 0.074716807 118.9435682
n4 9.012779914 0.139247815 64.72474937
n3 7.5717481 0.352915732 21.45483302
n2 7.216180455 1.052414641 6.856784553
n1 0.328731326 4.00909448 0.081996403
p7 3.985935016 0.125705039 31.70863361
p6 5.596832273 0.051065339 109.6013928
p5 8.027003712 0.078041739 102.8552651
p4 9.58165052 0.129579665 73.94409072
p3 10.40312863 0.30840301 33.73225389
p2 9.479514216 0.961180626 9.862365052
p1 1.946388345 4.009740068 0.485415092

The proportion of the variation in the binding energies of 2931 nuclides explained (R²) by the variations in the occupancies of the shells is 99.95 percent. The t-ratios are the ratios of the coefficients to their standard deviations. For a regression coefficient to be significantly different from zero at the 95 percent level of confidence its t-ratio must greater than 2 in magnitude. All of the coefficients except for the ones for the first shells are highly significant.

Here is the graph of the coefficients as a function of shell number.

In light of the fact mentioned above that most of the binding energy of a nuclide can be accounted for as the result of the nucleons within it forming alpha particles wherever possible. The difference between the binding energy of a nuclide and that which could be the result of the formation of alpha particles can be called excess binding energy XSBE. The regression of XSBE on the shell occupancy level gives the following results.

Shell
Occupancy
Coefficient Standard
Deviation
t-ratio
n8 6.431821659 0.085105711 75.57450127
n7 7.775444202 0.035445763 219.3617414
n6 8.294730226 0.041609396 199.3475278
n5 7.719855383 0.07390231 104.4602712
n4 5.44688864 0.137729857 39.54762417
n3 3.949946917 0.349068555 11.31567671
n2 1.528062245 1.040942141 1.467960787
n1 0.53985325 3.965390854 0.136141246
p7 -10.15130348 0.124334713 -81.64496664
p6 -8.577467559 0.050508669 -169.8216889
p5 -5.906823173 0.077190996 -76.52217838
p4 -1.670957413 0.128167102 -13.0373348
p3 1.532755754 0.305041071 5.024752067
p2 1.149747936 0.950702679 1.209366462
p1 -0.677837945 3.966029405 -0.170910973

The coefficient of determination (R²) for this regression is 0.99875. The graph of the coefficients is shown below.


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