﻿ Energy, Binding Energy and Half-life Statistics for Positron Emitters
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Energy, Binding Energy and
Half-life Statistics for Positron Emitters

## BACKGROUND

If the general stability of neutrons in nuclei despite the instability of free neutrons is puzzling then the existence of nuclides which display the emission of positrons and the conversion of one proton into a neutron is a hughely puzzling matter. To convert a proton into a neutron and a positron requires an input of energy of slightly over 1.8 million electron volts (MeV) as shown below.

#### Mass energy of a neutron = 939.56563 MeV Mass energy of a positron = 0.510999906 MeV Combined mass energies of a neutron and a positron = 940.0766291 MeV Mass energy of a proton = 938.27231 MeV Difference in mass energies =1.804319906 MeV

Complicating matters further is the fact that when a positron is emitted there is also a neutrino emitted. These emitted neutrinos carry off a significant amount of kinetic energy.

## Some Statistics

There are not a large number of positron emitters but there are some. Here are the statistics for a few.

The Energy and Half Life Statistics for a few Positron Emitters
Isotope proton
number
neutron
number
Half-Life
(min)
ΔBE Modal
Energy
(MeV)
Maximum
Energy
(MeV)
C-11 6 5 20.4 2.7649 0.326 0.959
N-13 7 6 10 3.00276 0.432 1.197
F-18 9 9 110 2.4378 0.202 0.633
O-15 8 7 2 3.53633 0.696 1.738
Fe-52 26 26 498 3.1541
I-124 6192 3.942

The ΔBE statistics are the changes in binding energy resulting from the conversions of a proton into a neutron; i.e.,

#### ΔBE(n, p) = BE(n+1, p-1) − BE(n, p)

The statistic ΔBE for a positron emitter is a good predictor of the maximum and modal energies of the ejected positron, as shown by the following display.

The regression equation for the maximum energy of the ejected positron as function of ΔBE is as follows:

#### Emax = 1.00624ΔBE − 1.82203

The coefficient of determination for this equation is 0.99998. The coefficient of ΔBE is not significantly different from 1.0 at the 95 percent level of confidence. Likewise the constant of −1.82203 MeV is close to the 1.80432 MeV required to create a neutron from a proton and not significantly different from it at the 95 percent level of confidence.

The regression equation for the modal energy of the ejected positron at a function of ΔBE modal = 0.45330ΔBE − 0.91665

The coefficient of determination of this regression is 0.99587.

The statistic ΔBE is not nearly as good a predictor of the half lives of the positron emitters;

The regression equation for the half-lives of the positron emitters as a function of the change in binding energy that results from a conversion of a proton into a neutron is:

#### T = 293.10 − 87.720ΔBE

The coefficient of determination of this regression is only 0.6546 and the t-ratio is −1.95, indicating the coefficient is just barely significantly different from zero at the 95 percent level of confidence.

A more extensive table of positron emitters is given below.

The Half Life Statistics for Positron Emitters
Isotope Number
of Protons
Number
of Neutrons
Half-life
14-O 8 6 70.606s
15-O 8 7 122.24s
13-N 7 6 9.97m
11-C 6 5 20.4m
18-F 9 9 110m
22-Na 11 11 2.60y
26-Al 13 13 7.4e5y
82-Rb 37 45 1.27m
38-K 18 20 7.636m
62-Cu 29 33 9.74m
63-Zn 30 33 38.47m
70-As 33 37 52.6m
68-Ga 31 37 67.6m
61-Cu 29 32 3.33h
52-Fe 26 26 8.28h
62-Zn 30 32 9.19h
64-Cu 29 35 12.7h
86-Y 39 47 14.74h
76-Br 35 41 16.2h
55-Co 27 28 17.53h
71-As 33 38 65.28h
74-As 33 41 17.77d
68-Ge 32 36 270.8d
40-K 18 22 1.250e9y
121-I 53 68
120-I 53 67 81.0m
110-In 49 61 4.9h
122-Xe 54 68 20.1h
124-I 53 71

Positron emmission has something to do with the number of protons compared to the number of neutrons. An excess of neutrons is required for nuclear stability. The graph below the extent of neutrons required.

In the positron emitters there are excess neutrons, just not enough.

## Conclusions

The change in the binding energy due to a conversion of a proton into a neutron is relevant in the statistics for positron emitters but not relevant in explaining why some nuclides emit positrons and most others do not.