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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 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Capacity | 2 | 4 | 8 | 14 | 22 | 32 | 44 | 58 |
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.
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 n_{i} and p_{i} be the number of neutrons and protons, respectively, in the i-th shell. The equation for explaining binding energies of nuclides is then
A regression based upon this equation gives the following results.
Shell Occupancy | Coefficient | Standard Deviation | t-ratio |
n_{8} | 6.438642308 | 0.086043684 | 74.82992407 |
n_{7} | 7.773687644 | 0.03583642 | 216.9214364 |
n_{6} | 8.268566776 | 0.042067984 | 196.5524837 |
n_{5} | 8.887083671 | 0.074716807 | 118.9435682 |
n_{4} | 9.012779914 | 0.139247815 | 64.72474937 |
n_{3} | 7.5717481 | 0.352915732 | 21.45483302 |
n_{2} | 7.216180455 | 1.052414641 | 6.856784553 |
n_{1} | 0.328731326 | 4.00909448 | 0.081996403 |
p_{7} | 3.985935016 | 0.125705039 | 31.70863361 |
p_{6} | 5.596832273 | 0.051065339 | 109.6013928 |
p_{5} | 8.027003712 | 0.078041739 | 102.8552651 |
p_{4} | 9.58165052 | 0.129579665 | 73.94409072 |
p_{3} | 10.40312863 | 0.30840301 | 33.73225389 |
p_{2} | 9.479514216 | 0.961180626 | 9.862365052 |
p_{1} | 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 |
n_{8} | 6.431821659 | 0.085105711 | 75.57450127 |
n_{7} | 7.775444202 | 0.035445763 | 219.3617414 |
n_{6} | 8.294730226 | 0.041609396 | 199.3475278 |
n_{5} | 7.719855383 | 0.07390231 | 104.4602712 |
n_{4} | 5.44688864 | 0.137729857 | 39.54762417 |
n_{3} | 3.949946917 | 0.349068555 | 11.31567671 |
n_{2} | 1.528062245 | 1.040942141 | 1.467960787 |
n_{1} | 0.53985325 | 3.965390854 | 0.136141246 |
p_{7} | -10.15130348 | 0.124334713 | -81.64496664 |
p_{6} | -8.577467559 | 0.050508669 | -169.8216889 |
p_{5} | -5.906823173 | 0.077190996 | -76.52217838 |
p_{4} | -1.670957413 | 0.128167102 | -13.0373348 |
p_{3} | 1.532755754 | 0.305041071 | 5.024752067 |
p_{2} | 1.149747936 | 0.950702679 | 1.209366462 |
p_{1} | -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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