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The Binding Energetics of Nuclei
with 24 Protons and with 24 Neutrons

Dedicated to Chuck
A friend for sixty years
and an always-willing reader

The Nature of the Nucleonic
Substructures of Nuclei

The mass of a nucleus containing n neutrons and p protons is less than the combined masses of n neutrons and p protons. The difference is called the mass deficit of the nuclide. When the mass deficit is expressed in energy units via the Einstein formula E=mc² the value is called the binding energy of the nuclide.

If BE(n, p) is the binding energy of a nuclide with n neutrons and p protons then the incremental binding energy IBEN of a neutron is

IBEN(n, p) = BE(n, p) − BE(n-1, p)

The incremental binding energy of a proton, IBEP, is

IBEN(n, p) = BE(n, p) − BE(n, p-1)

An incremental binding energy reflects the formation of any of the three types of spin pairs, neutron-neutron, proton-proton and neutron-proton. It also includes the effect of the strong force interactions of the additional nucleon with the nucleon already in the nuclide. The magnitudes involved can be illustrated with the data for chromium isotopes (24 protons). These are shown below in graphic and tabular form.

The Incremental Binding Energies of Neutrons

When the number of neutrons is below 24 the addition of another neutron will result in the formation of a neutron-proton pair. The incremental binding energy for all levels of the number of neutrons shows the net binding energy due to the interaction of the additional neutron with the other nucleons in the nuclide. The binding energy due to the formation of a neutron-neutron spin pair can be estimated by comparing the binding energy for a case in which a neutron-neutron spin pair is formed with the average of the IBE for the adjacent IBE's if the adjacent IBE's do not involve the drop that occurs when the number of the neutrons equals the number of protons. The data includes three ranges in the number of neutrons n: 1. n≤24, 2. 24<n≤28, 3. n>28. Twentyeight neutrons corresponds to the filling of a neutron shell. The estimates of the binding energy due to the formation of a neutron-neutron spin pair are shown in the following table. The values for the cases of twentyfour or less neutrons may also show the effect of the formation of a pair of neutrons with a pair of protons, which may be called an alpha module. There is too little data to tell definitely.

Most of the quantities in the table are in units of millions of electron volts, MeV.

The Binding Energy Quantities
for the Isotopes of Chromium (24 protons)
Neutrons Binding
Energy
(MeV)
IBE
Incremental
Binding
Energy
of Neutron
(MeV)
Average of
Adjacent
IBE
(MeV)
Binding
Energy
Due to
n-n Pair
Formation
(MeV)
18 314.2
19 330.43 16.23
20 349.9 19.47 15.09 4.38
21 363.85 13.95 18.7975 4.8475
22 381.975 18.125 13.5515 4.5735
23 395.128 13.153 17.2295 4.0765
24 411.462 16.334
25 422.044 10.582
26 435.0441 13.0001 9.92185 3.07825
27 444.3058 9.2617 12.5197 3.258
28 456.3451 12.0393 8.60045 3.43885
29 464.2843 7.9392 10.87915 2.93995
30 474.0033 9.719 7.09275 2.62625
31 480.2496 6.2463 8.9877 2.7414
32 488.506 8.2564 5.71015 2.54625
33 493.68 5.174 7.9332 2.7592
34 501.29 7.61 4.582 3.028
35 505.28 3.99 7.315 3.325
36 512.3 7.02 3.995 3.025
37 516.3 4 6.76 2.76
38 522.8 6.5 3.2 3.3
39 525.2 2.4 6.2 3.8
40 531.1 5.9 2.4 3.5
41 533.5 2.4

The plot of the estimates on the binding energy due to the formation of a neutron-neutron spin pair, shown below, indicates that for neutrons less than or equal to 24 neutrons the value is between 4 and 5 MeV. The average for 20 to 24 neutrons is 4.47 MeV. For 25 through 28 neutrons the average is 3.26 MeV. In the higher shell beyond 28 the average is 3.03 MeV. Clearly the binding energy involved in spin pair formation is not the same for all situations. The difference may have to do with the interaction of the spin pair of neutrons with the neutron-proton spin pair formed when the number of neutrons is less than or equal to 24.

The binding energy due to the interaction through the nuclear strong force and the formation of a neutron-proton spin pair may be computed by subtracting the effect of the formation of a neutron-neutron spin pair from the incremental binding energy. Those values are shown in the following graph.

One significant aspect of the display is that the more neutrons there are in the nuclide the smaller is the effect of the strong force interaction. This is an indication that the force between two neutrons is a repulsion. Below shows what the effect of the addition of proton has the binding energy due to the strong force interaction of nucleons.

In contrast to an additional neutron diminishing the binding energy, an additional proton raises the binding energy.

Considering the data for p=24 (isotopes of Chromium) there appears to be a drop in the value after n=24. The value of that drop is the binding energy involved in the formation of a neutron-proton pair. Its value can be estimated by the use of the following regression equation

IBEN = c0 + c1n + c2m

where m is equal to 1 if n is less than or equal to 24 and 0 otherwise.

The results of the regression are;

IBEN = 28.22 − 0.703n + 0.7676m
               [-12.5]   [2.65]

The numbes in brackets below the regression coefficients are their t-ratios; i.e., the ratio of the coefficient to its standard deviation. The t-ratios indicate that the coefficients are statistically significantly different from 0.

Thus the estimated binding energy associated with the formation of a neutron-proton spin pair is 0.7676 MeV.

The constant in the equation, 28.22 MeV, represents the effect if the number of neutrons were zero. This would be the effect of the 24 protons. That is 1.176 MeV of strong force interaction per proton. The estimated strong force interaction of a neutron with another neutron is −0.703 MeV.

The relative magnitudes of the effects on binding energy are significant. If the strong force charge of a proton is taken to 1 and that of a neutron to be q then the interaction of a proton and neutron would be proportional to q and that of neutron and neutron proportional to q². The ratio of the effects should be equal to q. That ratio is −0.598, a number of the same order of magnitude as the −2/3 found in other analysis.

The Incremental Binding Energies of Protons

Generally the picture of the relationships for protons is the same as for neutrons.

The Binding Energy Quantities for the Nuclides with 24 neutrons
Protons IBE
Incremental
Binding
Energy
of Proton
(MeV)
Average of
Adjacent
IBE
(MeV)
Binding
Energy
Due to
p-p Pair
Formation
(MeV)
Binding Energy
Due to Strong
Force Interaction
and n-p Pair
Formation
(MeV)
12 27.6
13 18.6 24.1 5.5 18.6
14 20.6 17.4 3.2 17.4
15 16.2 19.04 2.84 16.2
16 17.48 13.99 3.49 13.99
17 11.78 15.93 4.15 11.78
18 14.38 10.6175 3.7625 10.6175
19 9.455 13.2726 3.8176 9.455
20 12.1652 8.1721 3.9931 8.1721
21 6.8892 11.2551 4.3659 6.8892
22 10.345 6.0284 4.3166 6.0284
23 5.1676 9.2225 4.0549 5.1676
24 8.1 3.6263 4.4737 3.6263
25 2.085 6.1265 4.0415 2.085
26 4.153 1.0875 3.0655 1.0875
27 0.09 3.4115 3.3215 0.09
28 2.67 -0.885 3.555 -0.885
29 -1.86 1.485 3.345 -1.86
30 0.3

There is a transition to a higher proton shell at the magic number of 28. This shows up as a difference in the level of the estimate above 28 protons compared to the values for 28 and below.

The average binding energy for the formation of a proton-proton spin pair for 25 to 28 protons is 3.50 MeV, roughly the same as 3.26 MeV for the formation of a neutron-neutron spin pair in the range 25 to 28. The value given above for the average binding energy for the formation of a neutron-neutron pair in the 20 to 24 neutron range was 4.47 MeV. The average of the binding energies for the formation of a proton-proton pair for the 20 to 23 proton range is 4.23 MeV. The binding energies for neutron-neutron and proton-proton pair formation appear to be roughly the same.

When the effect of the formation of a proton-proton spin pair is subtracted from the incremental binding energy the result includes the binding energy due to the interaction through the nuclear strong force, but for the cases of the proton number of 24 or below it includes the binding energy due to the formation of a neutron-proton pair. Those values are shown in the following graph.

The above graph does not appear to involve a drop in the level after p=24. The possibility of such a drop can be investigated using the following regression equation.

IBEP = c0 + c1p + c2q

where q is equal to 1 if p is less than or equal to 24 and 0 otherwise.

The results of the regression are;

IBEP = 36.564 − 1.350p − 0.921q
               [-27.4]   [-1.74]

The constant of 36.564 represents the effects of the 24 neutrons at 1.523 MeV per neutron. The effect of q is not statistically significantly different from zero at the 95 percent level of condifence. It happens to be of the wrong sign.

Conclusions

The incremental binding energies of neutrons provides a way to estimate the binding energies due to the formation of two types of spin pairs, neutron-neutron and neutron-proton. The binding energy due to the strong force interactions of the nucleons can be estimated and regression analysis used to separate it into its components.

(To be continued.)

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