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Energy, Binding Energy and Halflife Statistics for Positron Emitters 

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.
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  HalfLife (min)  ΔBE  Modal Energy (MeV)  Maximum Energy (MeV) 
C11  6  5  20.4  2.7649  0.326  0.959 
N13  7  6  10  3.00276  0.432  1.197 
F18  9  9  110  2.4378  0.202  0.633 
O15  8  7  2  3.53633  0.696  1.738 
Fe52  26  26  498  3.1541  
I124  6192  3.942 
The ΔBE statistics are the changes in binding energy resulting from the conversions of a proton into a neutron; i.e.,
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:
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
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 halflives 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:
The coefficient of determination of this regression is only 0.6546 and the tratio 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  Halflife 
14O  8  6  70.606s 
15O  8  7  122.24s 
13N  7  6  9.97m 
11C  6  5  20.4m 
18F  9  9  110m 
22Na  11  11  2.60y 
26Al  13  13  7.4e5y 
82Rb  37  45  1.27m 
38K  18  20  7.636m 
62Cu  29  33  9.74m 
63Zn  30  33  38.47m 
70As  33  37  52.6m 
68Ga  31  37  67.6m 
61Cu  29  32  3.33h 
52Fe  26  26  8.28h 
62Zn  30  32  9.19h 
64Cu  29  35  12.7h 
86Y  39  47  14.74h 
76Br  35  41  16.2h 
55Co  27  28  17.53h 
71As  33  38  65.28h 
74As  33  41  17.77d 
68Ge  32  36  270.8d 
40K  18  22  1.250e9y 
121I  53  68  
120I  53  67  81.0m 
110In  49  61  4.9h 
122Xe  54  68  20.1h 
124I  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.
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.
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