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The Interaction of Alpha Particles
and a Proton Pair Within a Nucleus

Suppose that within a nucleus the neutrons and protons form alpha particles whenever possible. The formation of an alpha particle entails a binding energy of 28.295 million electron volts (MeV). The binding energy from the formation of alpha particles accounts for most of the binding energies of nuclides. If BE is the binding energy of a nuclide and a is the number of alpha particles it contains then the excess binding energy XSBE is given by

XSBE = BE − 28.295*a

Below is shown the plot of the XSBE for the nuclides which could contain an integral number of alpha particles, hereafter referred to as the alpha nuclides.

A bent line in which the bends come at the points where the number of protons (and the number of neutrons) is equal to the magic numbers of {2, 6, 14, 28}. The statistical fit is near perfect, as shown by the coefficient of determination (R²) for the regression of 0.999169.

The plot of the excess binding energies of the nuclides which could contain an integral number of alpha particles plus an additional proton pair is shown below.

The bends in this case are not so sharp as in the case of the alpha nuclides. This is because the effects of transitions to a new proton shell and a new neutron shell is spread over two changes in the number of alpha particles.

At this point it is important to note that the crucial variable is the number of protons rather than the number of alpha particles. A comparison of the excess binding energies for the alpha nuclides and the alpha+2proton cases is shown below.

A larger picture of the differences is useful at this point.

Note that the jumps in the level occur at the magic numbers. The number 20 plays a role similar to those of 6, 14 and 28 but to a lesser extent.

The differences can be estimated by a broken-line regression function of the following form

ΔXSBE = c0 + c1p + c2u(p-8) + c3u(p-16) + c4u(n-30) + c5d(n-8) + c6d(n-16) + c7d(n-30)

where a variable of the type u(n-k) is equal to n-k if n>k and zero otherwise. A variable of the type d(n-k) is 1 if n>k and zero otherwise.

The results of the regression are

ΔXSBE = 3.2516 − 1.13286p + 0.83274u(p-8) + 0.07175u(p-16) − 0.00743U(p-30)
+ 5.6203d(p-8) + 3.1274d(p-16) + 3.07004d(p-30)
[-4.4] [3.1] [0.8] [-0.1]
[6.5] [6.2] [7.4]

The figures in the square brackets are the t-ratios for the regression coefficients. The plots of the differences and the regression estimate of the differences nearly coincide.

The coefficient of determination (R²) for the regression equation is 0.93093.

In effect the regression equation is a sequence of linear functions of the number of protons in the nuclide. Those linear functions are:

Proton
Shell
InterceptSlope
First3.2516-1.13286
Second8.87198-0.30011
Third11.99937-0.22837
Fourth15.06941-0.2358

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