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The Interactive Energies
of Protons with Neutrons

This is a continuation of a study concerning the incremental binding energy (IBE) of nucleons in nuclides. The IBE of neutrons their interactions with protons is covered in Neutrons I and Neutrons II.

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 fifth shell with protons in the fifth neutron shell were used to estimate a regression equation of the following form

IBE = c0 + d0p + (c1+d1p)n
which is equivalent to
IBE = c0 + d0p + c1n + d1np

The proposition of interest is that the slope of the relationship between the IBE of a proton in a nuclide and the number of neutrons n in that nuclide is independent of p. 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 protons in the fifth shell with neutrons in the fifth shell.

The results of the regression for the 285 nuclides which involve an interaction between a proton in the fifth shell and a neutron in the fifth neutron shell are:

IBE = 8.11432 − 0.75988p + 0.57805n − 0.000827np
[−4.7] [4.8] [0.2]

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. Thus the coefficient for np is not significantly different from zero.

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

IBE = 6.8602 − 0.72241p + 0.60583n
[−39.5] [37.5]

The coefficient of determination for this equation is 0.87748. The best estimate of the interaction energy between a proton in the fifth shell and a neutron in the fifth shell is 0.60583 million electron volts (MeV).

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

IBE = 8.19905 − 1.18608p + 1.07154n
[−33.1] [29.8]

There are 61 nuclides that represent an interaction between protons in the third shell and neutrons in the third neutron shell. The regression equation for that set is

IBE = 4.38189 − 1.57631p + 2.17253n − 0.02180np
[−2.0] [3.0] [−0.3]

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

IBE = 6.75655 − 1.811168p + 1.95121n
[−12.0] [12.6]

The coefficient of determination for this equation is 0.82668.

There are only 16 nuclides which represent involve an interaction between a proton in the second shell and a neutron in the second neutron shell. The regression equation based upon them is

IBE = 7.8287p − 4.13547p + 1.19745n + 0.267782np
[−2.7] [0.7] [1.8]

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

IBE = − 0.45943 − 1.86560p + 4.02393n
[−2.0] [4.3]

The coefficient of determination for this equation is 0.63747.

The case of the interaction of protons in the sixth shell with neutrons in the sixth neutron shell was left until this point because the results for it are anomalous. There are 355 nuclides which involve such an interaction. The regression equation based upon them is

IBE = −12.64179 − 0.10946p + 0.60967n − 0.0050np
[−0.7] [5.3] [−2.4]

The regression coefficient for the variable np is statistically significantly different from zero at the 95 percent level of confidence whereas that for the variable p is not. However if the variable np is left out the regression equation the reslust is

IBE = 8.35487 − 0.49189p + 0.33446n
[−35.5] [34.4]

The coefficient of determination for this equation is 0.82195.

Interactions Between Nucleons in Shells of Different Numbers

There are 404 nuclides which involve an interaction between a proton in the seventh shell and a neutron in the eighth neutron shell. The regression equation based upon them is

IBE = −23.01013 − 0.08860p + 0.48875n − 0.00251np
[−0.8] [7.0] [−3.4 ]

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

IBE = 11.67890 − 0.46157p + 0.25459n
[−36.4] [28.2]

The coefficient of determination for this equation is 0.77214. This compares with 0.77856 for the regression equation which included the np variable.

There are 732 nuclides involving the interaction of protons of the sixth shell with neutrons in the seventh neutron shell. The regression equation based upon these 732 nuclides is

IBE = 6.97989 − 0.41657p + 0.3616c 0.36165n − 0.00110np
[−8.0] [8.7] [−1.97 ]

In this case the coefficient of the np variable is not statistically significant at the 95 percent level of confidence. A regression with the np variable left out yields

IBE = 14.61445 − 0.5193p + 0.27973n
[−82.4] [57.3]

The coefficient of determination for this equation is 0.90343.

There are 345 nuclides that involve the interaction of protons in the fifth shell with neutrons in the sixth neutron shell. The regression equation for them is

IBE = 25.59067 − 0.90645p + 0.20833n − 0.00416np
[−7.0] [2.0] [−1.9 ]

The regression equation with the np variable left out is

IBE = 14.49351 − 0.66785p + 0.40234n
[−39.1] [39.6]

The coefficient of determination for this equation is 0.85748.

There are 130 nuclides which involve the interaction of protons in the fourth shell with neutrons in the fifth neutron shell. The regression equation for this set is

IBE = 41.96507 − 2.07018p − 0.13566n + 0.03142np
[−4.9] [−0.4] [2.4]

In this equation the coefficient for the np variable is statistically significantly different from zero, but that of the n variable is not. However in order to obtain an estimate of the average interaction energy of protons in the fourth shell with neutrons in the fifth shell a regression was run leaving the np variable out. The result is

IBE = 16.10429 − 1.04757p + 0.66176n
[−21.9] [15.9]

The coefficient of determination for this equation is 0.80415.

There are 56 nuclides which involve an interaction between protons in the third shell and neutrons in the fourth neutron shell. The regression equation for them is

IBE = 23.06981 − 2.05088p 0.62464n + 0.03219np
[−1.5] [0.6] [0.4]

None of the regression coefficients is statistically significantly different from zero at the 95 percent level of confidence. However a regression equation with the np variable left out yields

IBE = 16.13944 − 1.50162p + 1.03291n
[−7.2] [7.7]

The coefficient of determination for this equation is 0.58497.

There are 17 nuclides which involve the interaction of protons in the second shell with neutrons in the neutron third shell. The regression equation for them is

IBE = 10.19971 − 0.72806p + 0.84385n + 0.13081np
[−0.1] [0.2] [0.2]

None of the regression coefficients is statistically significantly different from zero at the 95 percent level of confidence. However a regression equation with the np variable left out yields

IBE = 4.21007 + 0.29743p + 1.60974n
[0.4] [4.3]

The coefficient of determination for this equation is 0.68218.

The above results may be arranged in a table.

The Energies of Interactions Between Neutrons and Protons
Based on the Incremental Binding Energies of Protons
(All figures in MeV)
 Neutron Shell Number
 12345678
Proton
Shell
Number
1
24.023931.03291
31.951211.03291
41.071540.66176
50.531600.4023
60.334460.27973
7

These results can be compared with those from the IBE of neutrons from a previous paper

The Energies of Interactions Between Neutrons and Protons
Based on the Incremental Binding Energies of Neutrons
(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

In principle the figures should be the same in that they are alternate measures of the same quantities. While they are not the same they are generally of the same order of magnitude.


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