San José State University

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Positron emission and
protons not paired
with neutrons
(Version 2)

Background

Electron emission can be explained as being from nuclides in which there are neutrons which are not paired with protons. A similar analysis should explain positron emission but positron emission is much more complicated than electron emission. First of all, some of the nuclei of most unstable structure emit protons instead of positrons. A few such nuclei decay by the capture of an inner orbit electron.

The analysis here is based upon the analysis of the binding energies of almost three thousand different nuclides. That analysis reveals that nuclei are held together by the spin pairing of nucleons. But spin pairing is exclusive in the sense that a neutron can pair with one other neutron and with one proton and no more. The same goes for a proton. But in addition to the spin pairing there is an interactive force between all nucleons which involves like nucleons being repelled from each other and unlike nucleons being attracted. This force between nucleons can be explained by nucleons having a nucleonic charge. If the nucleonic charge of a proton is taken to be +1 then the nucleonic charge of a neutron is −2/3. This results in the minimum energy balance between the numbers of protons and neutrons being where the number of protons is equal to two-thirds of the number of neutrons. For more on this analysis of nuclear structure see What holds a nucleus together?.

In contrast, the conventional analysis of nuclear structure hypothesizes a uniform attractive force between all nucleons, called the nuclear strong force. It is called the strong force because at small separation distances between protons it stronger than the electrostatic repulsion between protons but at greater distances it is weaker. This hypothesis explains the existence of nuclei involving combinations of protons and neutrons but other than that there is no empirical evidence for its validity. The binding energies of nuclides provide evidence for the validity of the alternative explanation of nuclear structure given above.

The Explanations for Electron
and Positron Decay of Nuclei

The mechanisms analogous to those which explain electron emission apply to nuclides in which there are proton spin pairs not involving any spin pairing with neutrons. The conversion of one proton in a proton-proton spin pair into a neutron results in a net release of energy because the energy involved in the electrostatic and nucleonic repulsion between two protons is replaced by the nucleonic attraction between a proton and a neutron. There is no change in the energy associated with the spin pairing; i.e., the energy of a proton-neutron spin pair is the same as that of a proton-proton spin pair. There is also an energy gain from the conversion due to the interaction of a neutron with the rest of the nucleus which has a net protonic nucleonic charge. The energy released in these mechanisms goes to cover the energy required to convert a proton into a more massive neutron.

The Data

Here are the data for the nuclides with an excess of protons. There are only 187 such nuclides. But many of those eject protons rather than positrons. Shown below are life times (half-lives) of nuclides which emit positrons and the half-lives have been measured. They are ordered by their lifetimes and their rates of decay.

symbol p n LT 1/LT p-pn-sp pn sp
11C 6 5 1.22E+03 8.20E-04 0 5 1
13N 7 6 5.98E+02 1.67E-03 0 6 1
15O 8 7 1.22E+02 8.18E-03 0 7 1
14O 8 6 71 1.41E-02 2 6 0
17F 9 8 6.45E+01 1.55E-02 0 8 1
31Cl 17 14 3.10E+01 3.23E-02 2 14 1
21Na 11 10 2.25E+01 4.45E-02 0 10 1
10C 6 4 19.29 5.18E-02 2 4 0
19Ne 10 9 1.73E+01 5.78E-02 0 9 1
23Mg 12 11 1.13E+01 8.84E-02 0 11 1
27Si 14 13 4.16E+00 2.40E-01 0 13 1
29P 15 14 4.14E+00 2.41E-01 0 14 1
22Mg 12 10 3.8755 2.58E-01 2 10 0
31S 16 15 2.57E+00 3.89E-01 0 15 1
33Cl 17 16 2.51E+00 3.98E-01 0 16 1
26Si 14 12 2.234 4.48E-01 2 12 0
24Al 13 11 2.053 4.87E-01 2 11 0
37K 19 18 1.23E+00 8.16E-01 0 18 1
30S 16 14 1.178 8.49E-01 2 14 0
39Ca 20 19 8.60E-01 1.16E+00 0 19 1
35Ar 18 17 8.45E-01 1.18E+00 0 17 1
34Ar 18 16 0.8445 1.18E+00 2 16 0
8B 5 3 0.77 1.30E+00 2 3 0
41Sc 21 20 5.97E-01 1.68E+00 0 20 1
45V 23 22 5.47E-01 1.83E+00 0 22 1
43Ti 22 21 5.09E-01 1.96E+00 0 21 1
47Cr 24 23 5.00E-01 2.00E+00 0 23 1
23Al 13 10 4.70E-01 2.13E+00 2 10 1
20Na 11 9 0.4479 2.23E+00 2 9 0
38Ca 20 18 0.44 2.27E+00 2 18 0
49Mn 25 24 3.82E-01 2.62E+00 0 24 1
36K 19 17 0.342 2.92E+00 2 17 0
51Fe 26 25 3.05E-01 3.28E+00 0 25 1
32Cl 17 15 0.298 3.36E+00 2 15 0
28P 15 13 0.2703 3.70E+00 2 13 0
46Cr 24 22 0.26 3.85E+00 2 22 0
25Si 14 11 2.20E-01 4.55E+00 2 11 1
55Ni 28 27 2.05E-01 4.89E+00 0 27 1
42Ti 22 20 0.1995 5.01E+00 2 20 0
57Cu 29 28 1.96E-01 5.09E+00 0 28 1
29S 16 13 1.87E-01 5.35E+00 2 13 1
59Zn 30 29 1.83E-01 5.48E+00 0 29 1
40Sc 21 19 0.1823 5.49E+00 2 19 0
35K 19 16 1.78E-01 5.62E+00 2 16 1
33Ar 18 15 1.73E-01 5.78E+00 2 15 1
65As 33 32 0.17 5.88E+00 0 32 1
61Ga 31 30 1.68E-01 5.95E+00 0 30 1
48Mn 25 23 0.1581 6.33E+00 2 23 0
50Fe 26 24 0.155 6.45E+00 2 24 0
63Ge 32 31 0.142 7.04E+00 0 31 1
24Si 14 10 0.14 7.14E+00 4 10 0
67Se 34 33 0.133 7.52E+00 0 33 1
25Al 13 12 1.31E-01 7.63E+00 0 12 1
62Ge 32 30 0.129 7.75E+00 2 30 0
9C 6 3 1.27E-01 7.91E+00 2 3 1
21Mg 12 9 1.22E-01 8.20E+00 2 9 1
53Co 27 26 1.15E-01 8.70E+00 0 26 1
52Co 27 25 0.115 8.70E+00 2 25 0
44V 23 21 0.111 9.01E+00 2 21 0
17Ne 10 7 1.09E-01 9.16E+00 2 7 1
54Ni 28 26 0.104 9.62E+00 2 26 0
37Ca 20 17 1.02E-01 9.80E+00 2 17 1
36Ca 20 16 0.102 9.80E+00 4 16 0
71Kr 36 35 0.1005 9.95E+00 0 35 1
47Mn 25 22 1.00E-01 1.00E+01 2 22 1
32Ar 18 14 0.098 1.02E+01 4 14 0
56Cu 29 27 0.093 1.08E+01 2 27 0
20Mg 12 8 0.0908 1.10E+01 4 8 0
75Sr 38 37 0.088 1.14E+01 0 37 1
41Ti 22 19 8.45E-02 1.18E+01 2 19 1
58Zn 30 28 0.084 1.19E+01 2 28 0
43V 23 20 8.00E-02 1.25E+01 2 20 1
49Fe 26 23 7.00E-02 1.43E+01 2 23 1
60Ga 31 29 0.07 1.43E+01 2 29 0
51Co 27 24 6.00E-02 1.67E+01 2 24 1
22Al 13 9 0.059 1.69E+01 4 9 0
79Zr 40 39 0.0565 1.77E+01 0 39 1
44Cr 24 20 0.054 1.85E+01 4 20 0
40Ti 22 18 0.05335 1.87E+01 4 18 0
39Ti 22 17 5.33E-02 1.88E+01 4 17 1
70Kr 36 34 0.052 1.92E+01 2 34 0
45Cr 24 21 5.06E-02 1.98E+01 2 21 1
87Ru 44 43 0.05 2.00E+01 0 43 1
26P 15 11 0.0473 2.11E+01 4 11 0
53Ni 28 25 4.50E-02 2.22E+01 2 25 1
48Fe 26 22 0.044 2.27E+01 4 22 0
50Co 27 23 0.044 2.27E+01 4 23 0
23Si 14 9 4.23E-02 2.36E+01 4 9 1
55Cu 29 26 0.04 2.50E+01 2 26 1
64As 33 31 0.04 2.50E+01 2 31 0
61Ge 32 29 0.039 2.56E+01 2 29 1
57Zn 30 27 3.80E-02 2.63E+01 2 27 1
52Ni 28 24 0.038 2.63E+01 4 24 0
46Mn 25 21 0.037 2.70E+01 4 21 0
66Se 34 32 0.033 3.03E+01 2 32 0
69Kr 36 33 0.0325 3.08E+01 2 33 1
51Ni 28 23 3.00E-02 3.33E+01 4 23 1
60Ge 32 28 0.03 3.33E+01 4 28 0
22Si 14 8 0.029 3.45E+01 6 8 0
27P 15 12 2.60E-02 3.85E+01 2 12 1
35Ca 20 15 2.57E-02 3.89E+01 4 15 1
83Mo 42 41 0.023 4.35E+01 0 41 1
47Fe 26 21 2.18E-02 4.59E+01 4 21 1
43Cr 24 19 2.16E-02 4.63E+01 4 19 1
27S 16 11 1.55E-02 6.45E+01 4 11 1
31Ar 18 13 1.44E-02 6.94E+01 4 13 1
42Cr 24 18 0.014 7.14E+01 6 18 0
28S 16 12 0.0125 8.00E+01 4 12 0
12N 7 5 0.011 9.09E+01 2 5 0
89Rh 45 44 0.01 1.00E+02 0 44 1
91Pd 46 45 0.01 1.00E+02 0 45 1
46Fe 26 20 0.009 1.11E+02 6 20 0
50Ni 28 22 0.009 1.11E+02 6 22 0
13O 8 5 8.58E-03 1.17E+02 2 5 1
45Fe 26 19 1.89E-03 5.29E+02 6 19 1

In all known cases there is radioactive decay for these nuclides.

There are numerous cases in which the life time for the nuclide with three proton spin pairs is substantially shorter than the one with two pairs, which in turn is substantially shorter than the one with one proton pair. This is entirely reasonable in that lifetimes are inversely related to the probability of proton conversion and that would be proportional to the number of protons which are vulnerable to decay.

This hypothesis may be tested by using the reciprocals of lifetimes as a measure of the decay rates and hence of probabilities of decay. Here is the graph of the life time reciprocals versus the number of excess protons.

One extreme case was left out of the graph. There is visual confirmation that the decay rate is positively related to the number protons in the nucleus that are not linked with a neutron. This can be tested quantitatively using regression analysis. The equation includes the following variables

Since all of the protons are included in these variables if they are all zero there would be no protons in the nuclide and hence no positron emission. Therefore the constant in the regression equation should be zero.

The regression equation obtained is

(1/LT) = −0.18626pn + 13.76525sp + 11.88868(p-pn-sp)
     [-0.5]      [1.6]      [5.1]

The coefficient of determination (R²) for this equation is 0.31. The t-ratio of 5.1 for the coefficient for protons in pairs and not linked to neutrons strongly indicates that it is such protons which account for positron emission. The low values of the t-ratios for the other variables indicate the protons they represent are not involved in positron emission.

The low value of the R² indicates that some other variables are determining the values of the dependent variable.

Conclusions

Positron emission is associated with the number of proton pairs having no spin pair linkages with neutrons. In such a pair the conversion of a proton releases energy because the repulsions due to the electrostatic and nucleonic force are replaced by attractions. The energy associated with proton-neutron pairing is the same as that due to the proton-proton pairing. The energy released by the conversion of repulsions to attractions goes to supply the energy required to create the more massive neutron from a proton.


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