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Energy, Binding Energy and Half-life
Statistics for Pure Beta Emitters

Background

Free neutrons are unstable particles and decay with a half-life of a bit over ten minutes into protons and electrons. Within nuclei neutrons are generally stable but there are some nuclides that emit beta rays (electrons). Some of those have additional modes of decay. The ones which emit only beta rays are called pure beta emitters.

Below are compiled the statistics for pure beta emitting nuclides. Those statistics include the average and maximum energies of the emitted electrons as well as the half-lives. The change in binding energy is the difference between the binding energy of the nuclide that results from the conversion of a neutron and the original nuclide. This change in binding energy is expressed in millions of electron volts (MeV). The most notable aspect of those differences is how small they are. The values for all nuclides range from −27 MeV to +27 MeV. See neutron stability for those statistics.

Energy and Half-life Statistics for Pure Beta Emitters
Nuclide Protons Neutrons Change in
Binding
Energy
(MeV)
Max beta
Energy
(MeV)
Average
beta Energy
(MeV)
Half-life
(hrs)
66Cu 29 37 1.8596 2.64 1.111 5.12E+00
209Pb 82 127 -0.138 0.6444 0.1975 2.00E+02
145Pr 59 86 1.023 1.805 0.683 3.60E+02
127Te 52 75 -0.085 0.276 0.0782 5.63E+02
127Te 52 75 -0.085 0.694 0.2247 5.63E+02
121Sn 50 71 -0.3923 0.3889 0.1152 1.58E+03
66Ni 28 38 -0.5566 0.227 0.065 3.31E+03
90Y 39 51 1.4978 2.2814 0.9337 3.68E+03
210Bi 83 127 0.379 1.1615 0.389 7.36E+03
169Er 68 101 -0.431 0.3425 0.0981 1.36E+04
169Er 68 101 -0.431 0.3509 0.1008 1.36E+04
143Pr 59 84 0.152 0.9345 0.3153 1.95E+04
32P 15 17 0.92832 1.7103 0.6949 2.05E+04
33P 15 18 -0.53388 0.2485 0.0764 3.63E+04
89Sr 38 51 0.7127 1.492 0.5833 7.26E+04
91Y 39 52 0.7624 0.3409 0.1008 8.42E+04
91Y 39 52 0.7624 1.5456 0.6049 8.42E+04
188W 74 114 -0.434 0.058 0.0149 9.99E+04
188W 74 114 -0.434 0.285 0.0799 9.99E+04
188W 74 114 -0.434 0.349 0.0997 9.99E+04
35S 16 19 -0.61519 0.1668 0.0486 1.26E+05
123Sn 50 73 0.6202 0.314 0.0905 1.86E+05
123Sn 50 73 0.6202 1.403 0.5255 1.86E+05
45Ca 20 25 -0.5256 0.2568 0.0772 2.34E+05
249Bk 97 152 -0.659 0.1257 0.0324 4.61E+05
106Ru 44 62 -0.743 0.0394 0.01 5.26E+05
171Tm 69 102 -0.686 0.0297 0.0075 9.99E+05
171Tm 69 102 -0.686 0.0964 0.0251 9.99E+05
147Pm 61 86 -0.558 0.2246 0.062 1.37E+06
85Kr 36 49 -0.0949 0.1734 0.04765 5.68E+06
85Kr 36 49 -0.0949 0.6874 0.2516 5.68E+06
3H 1 2 -0.763763 0.0186 0.0057 6.47E+06
113Cd 48 65 -0.465 0.58 0.1854 7.42E+06
241Pu 94 147 -0.7615 0.0208 0.0052 7.57E+06
90Sr 38 52 -0.2369 0.5462 0.1958 1.51E+07
42Ar 18 24 -0.1878 0.6 0.233 1.73E+07
151Sm 62 89 -0.706 0.0548 0.01396 4.74E+07
151Sm 62 89 -0.706 0.0763 0.01968 4.74E+07
63Ni 28 35 -0.7154 0.0669 0.0174 5.27E+07
32Si 14 18 -0.55796 0.225 0.0688 9.05E+07
39Ar 18 21 -0.2173 0.565 0.2188 1.42E+08
14C 6 8 -0.6259 0.1565 0.0495 3.02E+09
99Tc 43 56 -0.4887 0.2935 0.0846 1.10E+11
79Se 34 45 -0.6313 0.1507 0.0558 3.42E+11
10Be 4 6 -0.2265 0.5562 0.2026 7.89E+11
135Cs 55 80 -0.513 0.205 0.0563 1.21E+12
107Pd 46 61 -0.75 0.033 0.0093 3.42E+12
187Re 75 112 -0.78 0.0026 0.0007 2.31E+16
115In 49 66 -0.287 0.497 0.152 2.31E+20
113Cd 48 65 -0.465 0.316 0.0933 4.89E+21

The notation nEm denotes n×10m.

There are duplications because some beta-emitting nuclides have multiple channels for beta decay. Here is the count in terms of the evenness-oddness of the proton and neutron numbers.

Number of Cases
 proton number
EvenOdd
neutron
number
Even1014
Odd224

The effects of the evenness-oddness of the proton and neutron numbers on the change in binding energy due to a neutron conversion are:

Average Change in Binding Energy
Due to Neutron Conversion
(MeV)
 proton number
EvenOdd
neutron
number
Even-0.44366-0.2325
Odd-0.288971.4978

A conversion of a neutron in the even-even case results in the loss of a neutron-neutron spin pair and the gain of a proton-proton spin pair. A neutron conversion in the odd-odd case results in the loss of a neutron-proton spin pair and the gain of a proton-proton spin pair. When the neutron number is odd and the proton number is even then a neutron conversion just results in a loss of a neutron-proton spin pair. When the neutron number is even and the proton number is odd then a neutron conversion results in a loss of a neutron-neutron spin pair and a gain of a proton-proton spin pair.

Geometric Average of Half Lives
(hours)
 proton number
EvenOdd
neutron
number
Even6.3E+69.55E+7
Odd11.3E+3

The predictive power of the change in binding energy is tested by plotting the maximum and average energy of the ejected electron versus the change in binding energy, as shown below.

Except for a few anomalous cases the relationships are nearly perfect linear ones. The regression equations for the relationships are:

Emax = 0.69294 + 0.89402ΔBE
Eav = 0.25647 + 0.36350ΔBE

The coefficients of determination are 0.84665 and 0.83631, respectively. The t-ratios for the coefficients are 16.3 and 15.7, respectively. This means that if a nuclide is known to be a beta emitter the change in binding energy is a good predictor of the amount of energy, average or maximum, carried by the emitted electron. However the change in binding energy is not a good predictor of whether or not a nuclide is a beta emitter.

The relationships appear to intersect the horizontal axis at the same point but that is not quite true. The maximum energy line intersects the horizontal axis at −0.775 MeV whereas the average energy line intersects it at 0.7056 MeV. The emitted electron gets its energy from some other source than the change in binding energy that results from the conversion of a neutron into a proton. That source would be the 1.8 MeV surplus mass of the neutron in excess of the combined mass of the proton and electron.

There should be some inverse relationship between the energy and the half-life of the emitters. There is but it is very weak, as shown below.

Although it is weak there is a definite relationship between the two variables, as is shown by the regression analysis. The regression equation is

log(Eav) = 2.27547 −0.05422log(T)

Although the coefficient of determination (R²) is only 0.119 the t-ratio for the coefficient is 2.5, indicating that the coefficient is significantly different from zero at the 95 percent level of confidence.

(To be continued.)


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