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The Quantification of the Interaction Energy Between
Alpha Particles and Proton Pairs Within Nuclei

This is an examination of the binding energies of nuclides which could contain an integral number of alpha particles plus a fixed number of neutron pairs. It is based upon the presumption 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 α is the number of alpha particles it contains then the excess binding energy XSBE is given by

XSBE = BE − 28.295*α

The difference in the excess binding energies of the alpha+n_pairs of neutrons nuclides and the alpha+(n-1)_pairs of neutrons is the incremental binding energy of the n-th neutron pair.

The Excess Binding Energies of the Alpha+2neutron Nuclides
and the Alpha Nuclides with the Same Number of Neutrons
Number
of Neutrons
Alpha
Nuclides
Alpha+2n
Nuclides
Difference
4 -0.091838 0.973426 1.065264
6 7.274706 8.385852 1.111146
8 14.43664 20.397486 5.960846
10 19.166489 26.624304 7.457815
12 28.482846 36.29154 7.808694
14 38.467172 46.906586 8.439414
16 45.415268 57.549912 12.134644
18 52.054634 65.473768 13.419134
20 59.09526 72.681634 13.586374
22 64.222286 78.93856 14.716274
24 71.913912 86.941986 15.028074
26 79.853238 95.496012 15.642774
28 87.848564 103.914938 16.066374
30 90.55689 110.314364 19.757474
32 93.219216 113.68389 20.464674
34 95.373542 116.559216 21.185674
36 97.777868 119.303542 21.525674
38 100.482194 121.957868 21.475674
40 103.88652 125.390194 21.503674
42 106.690846 128.78652 22.095674
44 108.895172 131.590846 22.695674
46 111.299498 134.895172 23.595674
48 114.303824 138.299498 23.995674
50 117.50815 141.803824 24.295674

The nature of what is involved in the differences must be carefully noted. On the first line of the above table the alpha nuclide that contains 4 neutrons also contains 4 protons. The nuclide whose excess binding energy is given on the first line contains 4 neutrons and 2 protons. Thus the difference reflects the effect of an additional proton pair on excess binding energy. But the difference shown is that of the alpha+2neutron nuclide minus the alpha nuclide so it is the negative of the incremental effect of an additional proton pair. It is how much the excess binding energy would increase if one proton pair were removed from an alpha nuclide.


 

Both cases exhibit a shell-type structure in which the incremental binding energy shifts to a higher level after a shell is filled.

The increments in excess binding energy with respect to increases in the number of alpha particles is the interaction energy between the last alpha particle and the proton pair. This is approximately constant within a shell.

A piecewise linear function can be fitted to the data using two functions u(z) and d(z) where

u(z) = z if z≥0 and u(z)=0 otherwise
d(z) = 1 if z≥0 and u(z)=0 otherwise

The regression equation used is of the form

IXSBE = Σ(ciu(α-(αi+1)) + bid(α-(αi+1)))

The critical levels of alpha particles are {1, 3, 7, 14, 25, 41, 63} and later the values of 4 and 10 come up. These correspond to the magic numbers for protons of {2, 6, 14, 28, 50, 82, 126} and {8, 20}. A comparison of the data values for the effect of the removal of one proton pair and the regression estimates of those data values is shown below.

The fit is so close that the estimates virtually overlay the actual values. The coefficient of determination (R²) for the regression is 0.99861. It is a little easier to see the minor discrepancies between the regression estimates and the actual values if a smooth curve is draw through the points of the two sets of data, as shown below.

The graphs for the effect of the removal of a proton pair from the alpha+4neutron nuclides are shown below.

The levels and slopes for the shells are obtained as the cumulative sums of the corresponding regression coefficients.

Neutron
Pair
Second
Shell
Third
Shell
Fourth
Shell
Fifth
Shell
Sixth
Shell
Coefficient of
Determination
InterceptSlope InterceptSlope InterceptSlope InterceptSlope InterceptSlope
First1.065260.045886.156940.778669.26939 0.6315812.289720.427320.99861
Second-5.27353-3.065315.842771.674027.661920.9591710.08072 0.5609613.82549 0.360000.99925

In the case of the removal of one proton pair from the alpha+6neutron and alpha+8neutron nuclides a new phenomenon develops.

There appears to be subshells for the third shell. For the fourth through fifth alpha particles the addition of an alpha particle substantially increases the excess binding energy. After five alpha particles and through seven the addition of alpha particles do not add to the excess binding energy.

For the removal of one proton pair from the alpha+5neutron nuclides the change in the nature of the third shell continues.

In this case the jump after seven alphas has disappeared. The jump instead comes after ten alphas, which corresponds to the magic number of 20 protons.

The effect of the removal of one proton pair from the alpha+12neutron nuclides is similar to the previous three cases. There are no longer shifts in level and slope after seven alpha particles; instead the shifts come after ten alpha particles.

The data for these graphs can be represented quite well as piecewise linear functions. For example, the case of the third neutron pair, which was shown above, is shown below with the regression estimates.

It is the slopes of the regression lines which are important. These are tabulated below.

The Interaction Binding Energies of Alpha Particles
with Neutron Pairs in the Various Alpha Particle Shells
Alpha Particle Shell
Alpha+Nneutron
Nuclides
Second
Shell
Third
Shell
Fourth
Shell
Fifth
Shell
Sixth
Shell
Coefficient of
Determination
N=20.045880.778660.631580.427320.99861
N=4-3.065311.674020.959170.56096 0.360000.99925
N=65.894700.247201.329650.667070.184000.99845
N=8-0.319501.436150.803540.500000.99783
N=101.660000.547000.513390.875810.626620.99956
N=123.034490.930000.926350.650580.99952
N=143.250000.841001.102860.715510.99849
N=163.450001.550001.115280.748000.99911
N=181.215040.811160.99864
N=201.26950.829860.99904
N=221.356610.867500.99839
N=241.09860.895190.99943
N=260.948600.931280.99935
N=281.63150.954940.99912

The Interaction Binding Energies of Alpha Particles
with Neutron Pairs in the Various Alpha Particle Shells
Alpha Particle Shell
Alpha+Nneutron
Nuclides
Fifth
Shell
Sixth
Shell
Seventhth
Shell
Coefficient of
Determination
N=300.935740.800000.99898
N=326.222070.970450.816000.99736
N=346.758791.010340.849000.99747
N=361.051310.829900.99892
N=381.050810.777500.99783
N=401.094490.662100.99856
N=421.107200.462030.99729
N=441.265500.550930.99418
N=461.287000.648580.99192
N=480.78270.78270.98744
N=500.937670.99774
N=520.949130.99744
N=540.949480.99814
N=560.956090.99759
N=580.964740.99823

The data for the interaction binding energy for a proton pair with any alpha particle in the sixth alpha particle shell are shown below. The initial data for the first few cases are erratic and are left out of the graph.

The more crowded the shell the more intense is the interaction of a proton pair with an alpha particle, until a shell is filled and then the additional pairs go into a higher shell. The number 14 for neutron pairs corresponds to the magic number of 28 neutrons.

(To be continued.)


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