San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
U.S.A.

The Search for the Increment
in Binding Energy Due Strictly to
the Formation of an Alpha Module

Consider the nuclide with 24 protons and 22 neutrons. Its binding energy is 381.975 million electron volts (MeV). That of the nuclide with 24 protons and 24 neutrons is 411.462 MeV, an increase of 29.487 MeV over that of the nuclide with 24 protons and 22 neutrons. That 29.487 MeV increase represents the effect of an additional neutron spin pair, two additional neutron-proton spin pairs and perhaps an increment due strictly to the formation of an alpha module (a combination of two protons and two neutrons equivalent in some way to an alpha particle).


A chain of 4 alpha modules

To determine the effect strictly of the formation of an alpha module consider the binding energies of nuclides with 23 protons and 22 neutrons and with 24 neutrons. The increase in binding energy for these nuclides is 26.273 MeV. This value represents the effect of an additional neutron pair and two additional neutron-proton pairs but not the formation of an alpha module. Thus the increment due strictly to the formation of an alpha module is the difference, 3.214 MeV.

Let BE(n, p) be the binding energy of a nuclide with n neutrons and p protons. The computation given above is given generally by the formula

IAM(n, p) = [BE(n, p) − BE(n-2, p)] − [BE(n, p-1) − BE(n-2, p-1)]
with n=p

Here are the values for a selected number of values of n=p.

Number of
Protons/Neutrons
Increment Due to
Formation of
Alpha Module
(MeV)
4 16.66101
8 7.499476
12 5.81653
16 4.41174
20 3.7798
24 3.214
28 3.3111
32 2.84
36 2.9

These values are independent of any error in the mass of a neutron. When plotted, as shown below, these data show a very systematic inverse relationship.

In logarithmic form this relationship is

The regression equation for this logarithmic relation is

ln(IAM) = 3.7658 − 0.7918ln(n)
[-15.5]
R² = 0.9717

The radius of a nucleus is approximately proportional to the cube root of the number of nucleons in it. The total number of nucleons in a nucleus is strongly correlated with the number of neutrons. If the cube root of the number of neutrons had been in the regression the regression coefficient would be −2.376. So the increment in binding energy due to the formation of an alpha module is roughly inversely proportional to the square of the radius of the nucleus.

Conclusions

There definitely is an increment in binding energy associated with the formation of an alpha module. However its value is inversely related to the scale of the nucleus. For small nuclei it is a dominant factor but for large nuclei it is no more significant than the effect of the formation of spin pairs.


The data for only a selected number of neutrons/protons were displayed above. This was to save time required for the analysis. After seeing the results I decided that it might be worth looking at a more inclusive set of cases. The graph of the result is

There appears to be a slight difference based upon whether the number of alpha modules is even or odd. The logarithmic regression gives virtually the same coefficient as previous case, −0.80.

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