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The Statistical Explanation of the
Binding Energies of Nuclides
in Terms of Shell Occupancies
and Numbers of Nucleon Pairs

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. One equation for explaining binding energies of nuclides is then

BE = Σcini + Σdipi

A regression based upon this equation gives the proportion of the variation in the binding energies of 2931 nuclides explained (R²) by the variations in the occupancies of the shells as 99.95 percent. For a regression coefficient to be significantly different from zero at the 95 percent level of confidence its ratio to its standard deviations, its t-ratio, must greater than 2 in magnitude. All of the coefficients except for the ones for the first shells are highly significant. The regression equation above leaves out the effect of nucleon pair formation on binding energy. A regression that takes those effects into account is

BE = enp#np + Σcini + enn#nn + Σdipi + epp#pp

where #np, #nn and #pp stand for the numbers of neutron-proton pairs, neutron-neutron and proton-proton pairs, respectively.

In the previous study the occupancies in the first shells was not statistically significant so the occupancies of the first two shells for both neutrons and protons were combined.

The results of the regression are:

Shell
Occupancy
Coefficient Standard
Deviation
t-ratio
#np 12.60758008 0.323700383 38.94830146
n8 5.384858269 0.350753997 15.35223636
n7 6.726873912 0.34501278 19.49746301
n6 7.240445567 0.345392495 20.96294991
n5 6.807366125 0.350218513 19.43748224
n4 4.749724959 0.370038965 12.83574273
n3 3.51683256 0.449876561 7.81732782
n2+n1 -0.474634975 0.687962444 -0.689914078
#nn 2.103398443 0.687573745 3.059160502
p7 -9.897661568 0.481829177 -20.5418477
p6 -8.322046292 0.473296805 -17.58314488
p5 -5.678440236 0.472020793 -12.03006375
p4 -1.749473764 0.44112536 -3.965933319
p3 1.384229609 0.470924566 2.93938713
p2+p1 -0.305081083 0.727199082 -0.419528972
#pp 2.568145308 0.68766059 3.734611731

The magnitudes of the coefficients and their standard deviations are in units of millions of electron volts (MeV). The coefficient of determination for this equation is 0.99966, thus indicating that 99.966 percent of the variation in the binding energies of the 2931 nuclides is explained by variation in the shell occupancies and the numbers of the nucleon pairs. The magnitudes of the coefficients of all the variables except for the combined occupancies of the first two shells were highly significantly different from zero at the 95 percent level of confidence.

The binding energy due to the formation of a neutron-neutron pair is, according to the regression, equal to 2.10 MeV. That for the formation of a proton-proton pair is 2.57 MeV. These are comparable to values found by other means of estimation. The difference between the values of enn and epp is not statistically significantly different from zero at 95 percent level of confidence.

While the statistical fit is not perfect, it is very good. In a sense the regression equation reduces the information for the binding energies of the 2931 nuclides to 16 parameter values, the 16 regression coefficients. That is notable.

The value for the formation of a neutron-proton pair is 12.61 MeV. This is higher than that found by other means of estimation and is roughly one half of the binding energy involved in the formation of an alpha particle. The inclusion of the number of possible alpha particles in each nuclide does not improve the statistical fit and its regression coefficient is not significantly different from zero at the the 95 percent level of confidence.

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

The shapes of the relationships between the coefficient values and the shell number beyond 2 are quadratic. The regression of the coefficients beyond shell 2 on the shell numbers and their squares gives a coefficient of determination for the neutron case of 0.9515 and 0.9951 for the proton case. Again, while not perfect fits the regressions reduces 11 of the coefficients to the 6 parameters of the two quadratic equations.

Here are the graphs of the coefficients and their regression estimates.

The Interpretation of the Results

The difference in the effect of an additional nucleon in a higher shell compared to a lower shell is the net effect of offsetting influences. A nucleon in a higher shell is subject to the interaction with more nucleons but at a greater distance. The net interaction of a nucleon with other nucleons is the sum of positive terms due to the attraction to nucleons of the opposite type less the negative effect due to repulsion from nucleons of the same type.

Analysis for the Excess Binding Energies of Nuclides

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
#np -1.582341145 0.334349774 -4.732592244
n8 6.380664429 0.362293423 17.61186935
n7 7.722989027 0.356363326 21.67167178
n6 8.235241098 0.356755533 23.08370955
n5 7.804314951 0.361740322 21.5743573
n4 5.745837683 0.382212846 15.03308366
n3 4.557986822 0.464677012 9.808935469
n2+n1 0.422819588 0.710595661 0.595021348
#nn 0.11074406 0.710194174 0.155934903
p7 3.294117542 0.497680833 6.618935914
p6 4.870477768 0.488867756 9.962771549
p5 7.514704851 0.487549765 15.41320577
p4 11.44620778 0.455637905 25.12128083
p3 14.53629126 0.486417473 29.88439368
p2+p1 12.96898112 0.751123142 17.26611843
#pp -23.731127 0.710283877 -33.41076402

The coefficient of determination for this equation is 0.99520. A comparison of the variance of excess binding energy with that of binding energy reveals that the number of potential alpha particles in a nuclide explains 92.42 percent of the variation in binding energies of nuclides. The regression of excess binding energy on the shell occupancies and numbers of nucleon pairs explains 99.52 of the variation in binding energies not explained by the number of alpha particles. Thus together the two procedures explain 99.964 percent of the variation in the binding energies of nuclides. This is essentially the same as the coefficient of determination of the regression equation for binding energies on shell occupancies and numbers of nucleon pairs.


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