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The Interactive Energies of Neutrons
with Protons in Shells Below Theirs

This is a continuation of a study concerning the incremental binding energy (IBE) of neutrons and their interactions with protons.

Incremental Binding Energies

The mass of a nuclide, such as helium 4, the alpha particle, is less than the masses of the two neutrons and two protons of which it is composed. The difference is called the mass deficit and that mass deficit expressed in energy units via the Einstein formula E=mc² is called the binding energy. The binding energies have been measured for almost three thousand nuclides. The incremental binding energy of a proton in a nuclide is the difference in the binding energy of that nuclide and the nuclide containing one less proton.

The incremental binding energies for nuclides containing the same number of protons but varying numbers of neutrons can be tabulated.

Protons and neutrons are arranged separately in shells. The numbers corresponding to the shells filled to full capacity are known as the nuclear magic numbers. Conventionally the magic numbers are {2, 8, 20, 28, 50, 82, 126}, but a case can be made for the magic numbers being instead {2, 6, 14, 28, 50, 82, 126} with 8 and 20 being in a different category of magic numbers. For more on this see Magic Numbers.

The structure of the nuclear shells, both for neutrons and protons, is given in the following table.

Shell
Number
1234567
Capacity24814223244
Range1 to 23 to 67 to 1415 to 2829 to 5051 to 8283 to 126

Empirical Studies

The data on the interaction of neutrons in the seventh shell with protons in the sixth proton shell were used to estimate a regression equation of the following form

IBE = c0 + d0n + (c1+d1n)p
which is equivalent to
IBE = c0 + d0n + c1p + d1np

The proposition of interest is that the slope of the relationship between the IBE of a neutron in a nuclide and the number of protons p in that nuclide is independent of n. If the regression coefficient d1 is not significantly different from zero then the proposition is established. The regression with the data for n*p left out would give c1 as the proper value for the interaction energy for neutrons in the seventh shell with protons in the sixth shell.

The results of the regression are:

IBE = −3.36850 − 0.09060n + 0.36759p − 0.00084np
[−2.6] [8.3] [−1.8]

The numbers in the square brackets above are the t-ratios for the corresponding coefficients. The t-ratio for a regression coefficient is its value divided by the standard deviation for that coefficient. For a coefficient to be significantly different from zero at the 95 percent level of confidence its t-ratio must be two or greater in magnitude. Thus the regression coefficient for the variable np is not significantly different from zero at the 95 percent level of confidence.

When the variable np is left out of the regression the results are

IBE = 2.50260 − 0.15315n + 0.28800p
[−37.0] [53.7]

The coefficient of determination for this equation is 0.79227. The best estimate of the interaction energy between a neutron in the seventh shell and a proton in the sixth shell is 0.288 million electron volts (MeV).

There are only a handfull of nuclides which involve an interaction of neutrons in the seventh shell and protons in the fifth proton shell so no regression analysis was attempted.

There are 375 nuclides that represent an interaction between neutrons in the sixth shell and protons in the fifth proton shell. The regression equation for that set is

IBE = 3.52115 − 0.24272n + 0.36715p − 0.00091np
[−3.1] [3.7] [0.5]

Once again the coefficient for the variable np is not statistically significant and with it left out the regression equation is

IBE = 1.12066 − 0.20086n + 0.41951p
[−25.0] [32.6]

There are 151 nuclides that represent an interaction between neutrons in the fifth shell and protons in the fourth proton shell. The regression equation for that set is

IBE = 10.6508 − 0.61984n + 0.37580p − 0.00949np
[−3.1] [3.7] [0.5]

Yet again the variable np is not statistically significant. With it left out the regression equation is

IBE = 2.64615 − 0.37397n + 0.68575p
[−15.6] [22.5]

The coefficient of determination for this equation is 0.78041.

There are only 68 nuclides which represent involve an interaction between a neutron in the fourth shell and a proton in the third proton shell. The regression equation based upon them is

IBE = 10.7942 − 1.07217n + 0.39259p − 0.04103np
[−2.6] [0.7] [1.3]

Again the variable np is not statistically significant. When it is left out the regression equation is

IBE = 1.69296 − 0.55868n + 1.12476p
[−9.2] [11.1]

The coefficient of determination for this equation is 0.68914.

There are only 22 nuclides which represent involve an interaction between a neutron in the third shell and a proton in the second proton shell. The regression equation based upon them is

IBE = −4.2138 − 0.14295n + 2.24100p − 0.06982np
[−0.1] [1.1] [−0.3]

Again the variable np is not statistically significant. With it is left out the regression equation is

IBE = −1.02818 − 0.51226n + 1.63182p
[−3.0] [4.3]

The coefficient of determination for this equation is 0.49993.

There are 412 nuclides which involve an interaction between a neutron in the eighth shell and a proton in the seventh proton shell. The regression equation based upon them is

IBE = −53.14053 − 0.25237n + 0.83106p − 0.00406np
[4.5] [9.3] [−6.8 ]

In this single case the coefficient of the np variable is statistically significant. The best estimate of the interaction energy between a neutron in the eighth shell and a proton in the seventh proton shell is

0.83106 − 0.00406n

In order to obtain a figure on the average intraction energy of neutrons in the eighth shell interacting with protons in the seventh shell a regression was run with the np variable left out. The results were

IBE = 2.80552 − 0.12383n + 0.22505p
[−17.6] [22.7]

The coefficient of determination for this equation is 0.55938. This compares with 0.60445 for the regression equation which included the np variable.

The above results and the results from a previous paper may be arranged in a table.

The Energies of Interactions Between Neutrons and Protons
(All figures in MeV)
 Neutron Shell Number
 12345678
Proton
Shell
Number
1
23.511661.63182
31.638141.12476
40.955380.68575
50.531600.41951
60.296210.28800
70.22505

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