In order to investigate spatial arrangement of nucleons in nuclear shells it is necessary to examine the matter of substructures of nucleons within nuclei. The binding energy of a nuclide is composed of one part due to any substructures such as nucleonic pairs or alpha particles and the other to the binding energy arising from the structural arrangement of nucleons and those substructures.

The matter of alpha particle substructures is examined by creating tables of the binding energy of nuclides with varying number of neutron-proton pairs (deuterons) to see if the binding energy increases significantly more when an additional deuteron enables the creation of an alpha particle compared to when the additional deuteron does not enable the creation of an alpha particle. The incremental binding energy is the increase in binding energy for one more deuteron. What is held constant in these relationships between incremental binding energy and the number of deuterons is the number of excess neutrons. Below is given the display of the cas in which there are no protons or neutrons in excess of those contained in the deuterons.

The graph displays the odd-even fluctuation that would be expected if the addition of another deuteron paired with an unpaired deuteron to form an alpha particle. The magnitude of the increase in binding energy due to the formation of an alpha particle is not always the huge amount of 20⁺ MeV associated with the formation of the first few alpha particles but it is definite and regular.

The above display also shows the phenomenon of significant change in the pattern at specific numbers, usually called magic numbers. The nuclear magic numbers are associated with the completion of shells. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. The case is made elsewhere that 6 and 14 are magic numbers and that 8 and 20 are magic numbers of a different category. In the display there are significant changes in the pattern at {2, 6, 14, 28}. At 8 and 20 something anomalous occurs but not changes in the levels.

When there are two extra neutrons in addition to those contained in the deuterons the display is similar.

Again there is the odd-even fluctuations and again there are shifts in the pattern at the magic numbers. In this case there can be shifts at magic numbers for neutrons and/or magic numbers for protons.

The two displays for very close to being the same as is seen below.

A similar pattern is found for the cases of other even numbers of extra neutrons, say 16, as shown below.

This shows the odd-even fluctuations and the significant shifts at the magic numbers. It is notable that there are significant shifts in the pattern at both 50 neutrons and 50 protons and at both 82 neutrons and 82 protons.

So far the information displayed shows the formation of alpha particles as substructures. The complication comes when the number of extra neutrons is odd. Here is the display for 1 extra neutron.

Here is no clearcut odd-even fluctuation. The shifts take place for some of the magic numbers, but for others it is in the vicinity of the magic number rather than at the magic number. Similar patterns occur for the cases of odd numbers of extra neutrons. This is quite a contrast with the cases of even numbers of extra neutrons.

The superimposition of the graphs, as shown below, shows that the values for the 1 extra neutron case is just about the average of the values for the 0 extra neutrons case.

What appears to be involved is that when there is a spare neutron the addition of a deuteron, a neutron-proton pair, results in the formation of a neutron pair and a resulting enhancement of the incremental binding energy. When another deutreron is added there is the formation of a proton-proton pair for the completion of the formation of the alpha particle. With the magnitudes of the enhancements resulting from the formation of a neutron-neutron pair and a proton-proton pair being approximately equal the incremental binding energies for one and two additional deutrons are approximately equal.

When there is an even number of extra neutrons the neutrons are already combined into pairs so the addition of another deutron does not result in any enhancement from the formation of a neutron pair. It is only when a second deuteron is added that the neutron-neutron and proton-proton pairs are formed and an alpha particle created.

There are also nuclides having extra protons but no extra neutrons. The graph of the data for those with 1 extra proton, as shown below, looks remarkably similar to the graph for 1 extra neutron.

On the other hand, the display for the case of 2 extra protons is very close to that for the cases of an even number of extra neutrons.

How close is demonstrated by their being superimposed.

Statistical Investigations

With the information gleaned from the few cases examined above it is possible to specify a regression equation for explaining the incremental binding energies of deuterons in nuclides with #p protons and #n neutrons. First the number of deuterons #d is determined as the minimum of #p and #n. Then the evenness of #d is expressed as a variable e(#d) defined as 1 if #d is even and 0 otherwise. Then the excess number of protons xs_p is determined as #p−#d and likewise xs_n as #n−#d. From the previous work it is known that the incremental binding energy depends upon #p and #n and their relation to the magic numbers. Before carrying out the regression for all the nuclides the analysis is limited to the special case of there being zero extra protons and zero extra neutrons. To capture the shift in the level of the IBE at the magic number m a variable d(#d-m) is created which is 1 if #d is greater than m and zero otherwise. To capture the shift is the slope of the relationship of IBE to #d at magic number m a variable u(#d-m) is created which is zero if #d is less than or equal to m and (#d-m) otherwise. To capture the effect of the evenness of #d a variable e(#d) is created that is 1 if #d is even and 0 if it is odd. Since the magnitude of the odd-even effect may vary depending upon whether #d is above or below a magic number m a variable is created which is the product of e(#d) and d(#d-m). The values of the magic numbers used are {6, 14, 28}. These are designated m2, m3, m4. The number 2 is also a magic number but there are too few cases at #d=2 or below to capture the effect of magic number m1=2.

The regression results are:

The coefficient of determination (R²) for this regression is 0.9829 and its standard error of the estimate is 0.77865 MeV. The coefficient for e(#d) indicates that for #d≤6 the enhancement in IBE produced by a pairing of deuterons is about 20.35 MeV. The t-ratio for this coefficient of 30.6 indicates that it is statistically highly significant. The coefficents for de2 indicates the enhancement for pairing drops by 9.27 MeV when the number of deuterons goes beyond 6. It drops another 4.21 MeV when the number of deuterons goes beyond 14 and another 1.94 MeV for #d beyond 28. The number of deuterons in this case corresponds to the number of protons and the number of neutrons.

The t-ratio of -0.1 for the coefficient of d(#d-m3) indicates that there is no statistically significant shift at the 95 percent level of confidence in the level of IBE at the magic number 14. Likewise the t-ratio of -0.8 for the coefficient of u(#d-m4) indicates that at the 95 percent level of confidence that there is no statistically significant shift in the slope of the relationship between IBE and #d at #d=28.

Now the regression will be applied to the general cases. However for the first such regression the matter of the shifts at the magic numbers is ignored. This first regression equation has the form

IBE = c₄(xs_n) + c₃(xs_p) + c₂e(#d) + c₁#d + c₀

The coefficient c₂ tells how much on average the IBE increases when an additional deuteron completes an alpha particle. Because the relationship is so different when one or the other of xs_p and xs_n is odd the regression equation is estimated with the data grouped into the even-even, even-odd and odd-even sets. (There is no odd-odd set because an odd proton and odd neutron would constitute another deuteron.)

For the set of 1437 even-even cases the regression results are:

IBE = 0.09647(xs_n) − 0.32292(xs_p)
+ 4.66993e(#d) −0.16247#d + 19.389767
(t-ratios)
[24.7] [-7.8}
[65.3] [-68.7] [197.4]
R² = 0.89212
σ = 1.355 MeV

The results indicate that, on average, the pairing of deuterons to form an alpha particle results in an increase in incremenatal binding energy of about 4.67 MeV. The standard error of the estimate, shown as σ, indicates that the regression equation gives estimates which are good to ±1.355 MeV.

The comparison of the one extra proton case with the one extra neutron case indicated that the relationships were virtually identical. Therefore the odd xs_p and the odd xs_n cases are grouped together for the regression. There are altogether 1428 of these cases. The regression results are

IBE = 0.09800(xs_n) − 0.36329(xs_p)
+ 0.51000e(#d) − 0.16308#d + 21.47015
(t-ratios) [26.2] [-7.6]
[7.5] [-72.7] 230.1]
R² = 0.85763
σ = 1.28 MeV

The results indicate that the effect of completing a pair of deuterons is much less for the case in which there is a singleton nucleon, but the t-ratio of 7.5 for the coefficient indicates that it is not zero.

Now a fuller version can be presented in which there may be shifts in the level of IBE at the magic numbers of nucleons.

The coefficient of determination (R²) for this regression is 0.92190. .

The coefficient of determination (R²) for this regression is 0.91869.

These are not the full regressions. The full version would include allowance for the magnitude of the enhancement for a pair of deuteron to vary with the shell. This version would require 26 variables but Excel limits the number of regressors to 16.

(To be continued.)

Conclusions

Substructures of nucleons are formed in nuclei whenever possible. The addition of deuterons will result in the formation of nucleon pairs if there are spare nucleons available; i.e., ones that are not already involved in such pairs. When two deuterons are added an alpha particle is formed.