Positron emission and protons not paired with neutrons

San José State University

applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA

Positron emission and protons not paired with neutrons

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 togerther?.
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 cominations of protons and neutrons but other than that there is no empirical evidence for its validity. The binding energies of nuclides provides 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 nucides 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 replusion 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 and a positron. There is also a neutrino created at the same time as the positron.
Some Data

Here are the data for the nuclides with an excess of proton pairs. There are only 84 such nuclides. There are 103 nuclides which have an unpaired proton along with proton pairs not paired with neutrons. These will be considered later.

Symbol P N p-n eject Life
Time
(sec)

4Li 3 1 2 p 9.10E-23

6Be 4 2 2 2p 5.03E-21
8B 5 3 2 β⁺ 0.77

8C 6 2 4 2p 2.00E-19

10C 6 4 2 β⁺ 19.29

10N 7 3 4 p 2.00E-21

12N 7 5 2 β⁺ 0.011

12O 8 4 4 p 5,8E-22

14O 8 6 2 β⁺ 71

14F 9 5 4 p ?

16F 9 7 2 p 1.10E-20

16Ne 10 6 4 2p 9.00E-21

18Ne 10 8 2 e capture 1.672

18Na 11 7 4 p 1.30E-21

20Na 11 9 2 β⁺ 0.4479

20Mg 12 8 4 β⁺ 0.0908

22Mg 12 10 2 β⁺ 3.8755

22Al 13 9 4 β⁺ 0.059

24Al 13 11 2 β⁺ 2.053

22Si 14 8 6 β⁺ 0.029

24Si 14 10 4 β⁺ 0.14

26Si 14 12 2 β⁺ 2.234

24P 15 9 6 p ?

26P 15 11 4 β⁺ 0.0473

28P 15 13 2 β⁺ 0.2703

26S 16 10 6 2p 0.01

28S 16 12 4 β⁺ 0.125

30S 16 14 2 β⁺ 1.178

28Cl 17 11 6 p ?

30Cl 17 13 4 p <30ns

32Cl 17 15 2 β⁺ 0.298

30Ar 18 12 6 p <20ns

32Ar 18 14 4 β⁺ 0.098

34Ar 18 16 2 β⁺ 0.8445

32K 19 13 6 p ?

34K 19 15 4 p <25ns

36K 19 17 2 β⁺ 0.342

34Ca 20 14 6 p <35ns

36Ca 20 16 4 β⁺ 0.102

38Ca 20 18 2 β⁺ 0.44

36Sc 21 15 6 ? ?

38Sc 21 17 4 p <300ns

40Sc 21 19 2 β⁺ 0.1823

38Ti 22 16 6 2p <120ns

40Ti 22 18 4 β⁺ 0.05335

42Ti 22 20 2 β⁺ 0.1995

40V 23 17 6 p ?

42V 23 19 4 p <55ns

44V 23 21 2 β⁺ 0.111

42Cr 24 18 6 β⁺ 0.014

44Cr 24 20 4 β⁺ 0.054

46Cr 24 22 2 β⁺ 0.26

44Mn 25 19 6 p <105ns

46Mn 25 21 4 β⁺ 0.037

48Mn 25 23 2 β⁺ 0.1581

46Fe 26 20 6 β⁺ 0.009

48Fe 26 22 4 β⁺ 0.044

50Fe 26 24 2 β⁺ 0.155

48Co 27 21 6 p ?

50Co 27 23 4 β⁺ 0.044

52Co 27 25 2 β⁺ 0.115

50Ni 28 22 6 β⁺ 0.009

52Ni 28 24 4 β⁺ 0.038

54Ni 28 26 2 β⁺ 0.104

52Cu 29 23 6 p ?

54Cu 29 25 4 p <75ns

56Cu 29 27 2 β⁺ 0.093

54Zn 30 24 6 2p ?

56Zn 30 26 4 2p 0.036

58Zn 30 28 2 β⁺ 0.084

56Ga 31 25 6 p ?

58Ga 31 27 4 p ?

60Ga 31 29 2 β⁺ 0.07

58Ge 32 26 6 2p ?

60Ge 32 28 4 β⁺ 0.03

62Ge 32 30 2 β⁺ 0.129

60As 33 27 6 p ?

62As 33 29 4 p ?

64As 33 31 2 β⁺ 0.04

66Se 34 32 2 β⁺ 0.033

68Br 35 33 2 β⁺ <1.2μs

70Kr 36 34 2 β⁺ 0.052

72Rb 37 35 2 p <1.5μs

74Sr 38 36 2 β⁺ <25ms

The β⁺ejection is the ejection of a positron and neutrino. p stands for the ejection of a proton.
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 life times are inversely related to the probability of proton conversion and that would be preportional to the number of protons which are vulnerable to decay.
This hypothesis may be tested by using the reciprocals of life times as a measure of the decay rates and hence of probabilities ofmdecay. Here is the graph of the life time reciprocals versus the number of excess protons.

There is visual confirmation that can be tested quanitatively using regression analysis. The equation obtained is
(1/LT) = 12.81125(p-n) − 19.20248
     [5.6]

The coefficient of determination (R²) for this equation is 0.41. The t-ratio of 5.6 strongly indicates that this relationship is not just due to chance.
The low value of the R² indicates that some other variables are determining the values of the dependent variable. However those other variables do not include the proton number. The regression equation including the proton number is
1/LT = 12.59952(p-n) + 0.57665p − 30.67820
   [5.6]                 [1.6]           [-3.0]

The t-ratio of 1.6 for the coefficient of p indicates that it is not significantly different from zero at the 95 percent level of confidence.
In an effort find what variables besides excess protons (p-n) affect the life time of positron emitting nuclides the number of neutrons was tried. It did the same as than did the number of protons. The net nucleonic charge is (p-(2/3)n). It did no better as an additional explanatory variable than did the number of protons or the number of neutrons.
The numbers of protons and the number of neutrons were used as explanatory variables. The regression equation obtained is:
1/LT = 13.17617p − 12.59952n − 30.67820
   [5.8]                 [-5.6]           [-3.0]

Clearly this result indicate that the relevant variable is (p-n), effectively the excess proton pairs. Further confirmation comes an examination of the cases in which there are only unpaired excess prorons. There are not many such cases. One is the isotope He_3 of helium with two protons and one neutron. It does not decay. The other is Be_5 which is a special case. It does not emit positons. It basically falls apart into two alpha particles and a proton.
Positron Emitters with a Singleton Proton

Symbol P N p-n eject Life
Time
(sec)

1H 1 0 1 p stable

3He 2 1 1 stable

5Li 3 2 1 p 3.70E-21

5Be 4 1 3 p ?

7Be 4 3 1 4.60E+06

7B 5 2 3 p 3.50E-22

9B 5 4 1 p 8.00E-19

9C 6 3 3 β⁺ 1.27E-01

11C 6 5 1 β⁺ 1.22E+03

11N 7 4 3 p 5.90E-21

13N 7 6 1 β⁺ 5.98E+02

13O 8 5 3 β⁺ 8.58E-03

15O 8 7 1 β⁺ 1.22E+02

15F 9 6 3 β⁺ 4.10E-20

17F 9 8 1 β⁺ 6.45E+01

17Ne 10 7 3 β⁺ 1.09E-01

19Ne 10 9 1 β⁺ 1.73E+01

19Na 11 8 3 p <40ms

21Na 11 10 1 β⁺ 2.25E+01

21Mg 12 9 3 β⁺ 1.22E-01

23Mg 12 11 1 β⁺ 1.13E+01

21Al 13 8 5 p <'35ns

23Al 13 10 3 β⁺ 4.70E-01

25Al 13 12 1 β⁺ 1.31E-01

23Si 14 9 5 β⁺ 4.23E-02

25Si 14 11 3 β⁺ 2.20E-01

27Si 14 13 1 β⁺ 4.16E+00

25P 15 10 5 p <30ms

27P 15 12 3 β⁺ 2.60E-02

29P 15 14 1 β⁺ 4.14E+00

27S 16 11 5 β⁺ 1.55E-02

29S 16 13 3 β⁺ 1.87E-01

31S 16 15 1 β⁺ 2.57E+00

29Cl 17 12 5 p <20ns

31Cl 17 14 3 β⁺ 3.10E+01

33Cl 17 16 1 β⁺ 2.51E+00

31Ar 18 13 5 β⁺ 1.44E-02

33Ar 18 15 3 β⁺ 1.73E-01

35Ar 18 17 1 β⁺ 8.45E-01

33K 19 14 5 p <25ns

35K 19 16 3 β⁺ 1.78E-01

37K 19 18 1 β⁺ 1.23E+00

35Ca 20 15 5 β⁺ 2.57E-02

37Ca 20 17 3 β⁺ 1.02E-01

39Ca 20 19 1 β⁺ 8.60E-01

37Sc 21 16 5 p ?

39Sc 21 18 3 p <300ns

41Sc 21 20 1 β⁺ 5.97E-01

39Ti 22 17 5 β⁺ 5.33E-02

41Ti 22 19 3 β⁺ 8.45E-02

43Ti 22 21 1 β⁺ 5.09E-01

41V 23 18 5 p ?

43V 23 20 3 β⁺ 8.00E-02

45V 23 22 1 β⁺ 5.47E-01

43Cr 24 19 5 β⁺ 2.16E-02

45Cr 24 21 3 β⁺ 5.06E-02

47Cr 24 23 1 β⁺ 5.00E-01

45Mn 25 20 5 p ?

47Mn 25 22 3 β⁺ 1.00E-01

49Mn 25 24 1 β⁺ 3.82E-01

45Fe 26 19 7 β⁺ 1.89E-03

47Fe 26 21 5 β⁺ 2.18E-02

49Fe 26 23 3 β⁺ 7.00E-02

51Fe 26 25 1 β⁺ 3.05E-01

49Co 27 22 5 p <35ns

51Co 27 24 3 β⁺ 6.00E-02

53Co 27 26 1 β⁺ 1.15E-01

51Ni 28 23 5 β⁺ 3.00E-02

53Ni 28 25 3 β⁺ 4.50E-02

55Ni 28 27 1 β⁺ 2.05E-01

53Cu 29 24 5 p <300ns

55Cu 29 26 3 β⁺ 0.04

57Cu 29 28 1 β⁺ 1.96E-01

55Zn 30 25 5 2p 2.00E-02

57Zn 30 27 3 β⁺ 3.80E-02

59Zn 30 29 1 β⁺ 1.83E-01

57Ga 31 26 5 p ?

59Ga 31 28 3 β⁺

61Ga 31 30 1 β⁺ 1.68E-01

59Ge 32 27 5 2p ?

61Ge 32 29 3 β⁺ 0.039

63Ge 32 31 1 β⁺ 0.142

61As 33 28 5 p ?

63As 33 30 3 p ?

65As 33 32 1 β⁺ 0.17

65Se 34 31 3 β⁺ <50ms

67Se 34 33 1 β⁺ 0.133

67Br 35 32 3 p ?

69Br 35 34 1 p <24ns

69Kr 36 33 3 β⁺ ,0325

71Kr 36 35 1 β⁺ 0.1005

71Rb 37 34 3 p ?

73Rb 37 36 1 p <30ns

73Sr 38 35 3 β⁺ <25ns

75Sr 38 37 1 β⁺ 0.088

77Y 39 38 1 p 0.0635

79Zr 40 39 1 β⁺ 0.0565

81Nb 41 40 1 β⁺, p <44ns

83Mo 42 41 1 β⁺ 0.023

85Tc 43 42 1 β⁺ <110ns

87Ru 44 43 1 β⁺ 0.05

89Rh 45 44 1 β⁺ 0.01

91Pd 46 45 1 β⁺ 0.05

The β⁺ejection is the ejection of a positron and neutrino. p stands for the ejection of a proton.
The regression of 1/LT on (p-n) yields
1/LT = 21.86518(p-n) − 24.02144        [4.6]       [-1.9]

The coefficient of determination (R²) is only 0.24 but the t-ratio of 4.6 strongly indicates that the rate of decay is proportional to the number of protons not paired with a neutron.
When regression equation includes the number of protons paired with neutrons, which is the same as n, as well as the number of protons paired with protons but not paired with neutrons the results are
1/LT = 24.44016(p-n-1) + 1.46267n − 34.36414
       [5.1]       [2.1]       [-1.9]

The coefficient of determination (R²) is raised to 0.29. The number of neutron is effectively a measure of the size of the nucleus. So the results indicate a weak effect of nucleus size on the decay rate.
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 is 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. There may be a weak influence of the size of the nucleus on its decay rate,

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Symbol	P	N	p-n	eject	Life Time (sec)
4Li	3	1	2	p	9.10E-23
6Be	4	2	2	2p	5.03E-21
8B	5	3	2	β⁺	0.77
8C	6	2	4	2p	2.00E-19
10C	6	4	2	β⁺	19.29
10N	7	3	4	p	2.00E-21
12N	7	5	2	β⁺	0.011
12O	8	4	4	p	5,8E-22
14O	8	6	2	β⁺	71
14F	9	5	4	p	?
16F	9	7	2	p	1.10E-20
16Ne	10	6	4	2p	9.00E-21
18Ne	10	8	2	e capture	1.672
18Na	11	7	4	p	1.30E-21
20Na	11	9	2	β⁺	0.4479
20Mg	12	8	4	β⁺	0.0908
22Mg	12	10	2	β⁺	3.8755
22Al	13	9	4	β⁺	0.059
24Al	13	11	2	β⁺	2.053
22Si	14	8	6	β⁺	0.029
24Si	14	10	4	β⁺	0.14
26Si	14	12	2	β⁺	2.234
24P	15	9	6	p	?
26P	15	11	4	β⁺	0.0473
28P	15	13	2	β⁺	0.2703
26S	16	10	6	2p	0.01
28S	16	12	4	β⁺	0.125
30S	16	14	2	β⁺	1.178
28Cl	17	11	6	p	?
30Cl	17	13	4	p	<30ns
32Cl	17	15	2	β⁺	0.298
30Ar	18	12	6	p	<20ns
32Ar	18	14	4	β⁺	0.098
34Ar	18	16	2	β⁺	0.8445
32K	19	13	6	p	?
34K	19	15	4	p	<25ns
36K	19	17	2	β⁺	0.342
34Ca	20	14	6	p	<35ns
36Ca	20	16	4	β⁺	0.102
38Ca	20	18	2	β⁺	0.44
36Sc	21	15	6	?	?
38Sc	21	17	4	p	<300ns
40Sc	21	19	2	β⁺	0.1823
38Ti	22	16	6	2p	<120ns
40Ti	22	18	4	β⁺	0.05335
42Ti	22	20	2	β⁺	0.1995
40V	23	17	6	p	?
42V	23	19	4	p	<55ns
44V	23	21	2	β⁺	0.111
42Cr	24	18	6	β⁺	0.014
44Cr	24	20	4	β⁺	0.054
46Cr	24	22	2	β⁺	0.26
44Mn	25	19	6	p	<105ns
46Mn	25	21	4	β⁺	0.037
48Mn	25	23	2	β⁺	0.1581
46Fe	26	20	6	β⁺	0.009
48Fe	26	22	4	β⁺	0.044
50Fe	26	24	2	β⁺	0.155
48Co	27	21	6	p	?
50Co	27	23	4	β⁺	0.044
52Co	27	25	2	β⁺	0.115
50Ni	28	22	6	β⁺	0.009
52Ni	28	24	4	β⁺	0.038
54Ni	28	26	2	β⁺	0.104
52Cu	29	23	6	p	?
54Cu	29	25	4	p	<75ns
56Cu	29	27	2	β⁺	0.093
54Zn	30	24	6	2p	?
56Zn	30	26	4	2p	0.036
58Zn	30	28	2	β⁺	0.084
56Ga	31	25	6	p	?
58Ga	31	27	4	p	?
60Ga	31	29	2	β⁺	0.07
58Ge	32	26	6	2p	?
60Ge	32	28	4	β⁺	0.03
62Ge	32	30	2	β⁺	0.129
60As	33	27	6	p	?
62As	33	29	4	p	?
64As	33	31	2	β⁺	0.04
66Se	34	32	2	β⁺	0.033
68Br	35	33	2	β⁺	<1.2μs
70Kr	36	34	2	β⁺	0.052
72Rb	37	35	2	p	<1.5μs
74Sr	38	36	2	β⁺	<25ms

Background

The Explanations for Electron and Positron Decay of Nuclei

Some Data

(1/LT) = 12.81125(p-n) − 19.20248 [5.6]

1/LT = 12.59952(p-n) + 0.57665p − 30.67820 [5.6] [1.6] [-3.0]

1/LT = 13.17617p − 12.59952n − 30.67820 [5.8] [-5.6] [-3.0]

Positron Emitters with a Singleton Proton

1/LT = 21.86518(p-n) − 24.02144 [4.6] [-1.9]

1/LT = 24.44016(p-n-1) + 1.46267n − 34.36414 [5.1] [2.1] [-1.9]

Conclusions

The Explanations for Electron
and Positron Decay of Nuclei

(1/LT) = 12.81125(p-n) − 19.20248
[5.6]

1/LT = 12.59952(p-n) + 0.57665p − 30.67820
[5.6] [1.6] [-3.0]

1/LT = 13.17617p − 12.59952n − 30.67820
[5.8] [-5.6] [-3.0]

1/LT = 24.44016(p-n-1) + 1.46267n − 34.36414
[5.1] [2.1] [-1.9]