< !DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0//EN"> On Determining Whether Nucldes Decay Through Alpha Particle Emmission
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley,
Tornado Alley
& the Gateway
to the Rockies
USA

On Determining Whether Nuclides Decay
Through Alpha Particle Emmission

Nuclei are composed of nucleons (protons and neutrons) in variable proportions. There are almost three thousand nuclides that are stable enough to have their mass measured and their binding energies computed. But most are unstable. The following beautiful display from Wikipedia shows the nature of their instabilities.

As can be seen from the display, beyond proton number 82 the mode of decay is predominantly through the ejection of an alpha (α) particle. There are no stable particles in that range.

The ejection of an alpha particle occurs because its ejection results in an increase in binding energy.

What are sought here are the changes in binding energies involved in these ejections and demonstration that for the cases of nuclides in which alpha particle ejection does not occur it is because it would result in a decrease in binding energy. More basically the purpose of this material is to show the relationship between binding energy and the mode of radioactive decay.

Binding Energy and
Radioactive Decay Products

The total binding energy of a nuclide includes the loss of potential energy that occurs when it is formed from its components, The only binding energy that is available is the binding energy based upon the mass deficits of nuclides. The mass deficit binding energies seem to be closely enough correlated with the binding energies based upon potential energy losses that they are adequate for carrying out empirical analysis.

For the decay by alpha particle ejection the relevant binding energies are:

BE(p, n) => BE(p−2, n−2) + BE(2, 2)

The binding energy of the alpha particle He4, BE(2. 2), is 28.3 MeV.

The theory is that if the true change in binding energy for the transition is positive then alpha decay takes place. If it is not positive (negative or zero) then alpha decay does not take place. But the only binding energies available are those based upon mass deficits. They are correlated with those based upon total binding energies but the correlation is not perfect. The theory then is that if the change in the binding energy based upon mass deficits is strongly positive alpha decay will occur. If it is strongly negative then alpha decay will not occur. If it is small, say less than 1.0 MeV what occurs is uncertain.

What is given in the following table are the binding energies of decay products expressed in millions of electron volts (MeV) for an arbitrarily selected complete sequence of alpha particle decays.

> Yes
Binding Energies for a Sequence of Possible Alpha Transitions
p n BE
(MeV) 		BE+α
(MeV) 		ΔBE
for α tran
(MeV)		 α tran?
104 160 1943.3
102 158 1923.14 1951.435674 8.135674 Yes
100 156 1902.543 1930.838674 7.698674 Yes
98 154 1881.275 1909.570674 7.027674 Yes
96 152 1859.196 1887.491674 6.216674 Yes
94 150 1836.062 1864.357674 5.161674 Yes
92 148 1812.432 1840.727674 4.665674 Yes
90 146 1788.1 1816.395674 3.963674 Yes
88 144 1763 1791.295674 3.195674 Yes
86 142 1737.5 1765.795674 2.795674 Yes

In each case the binding energy of a nuclide in one line is less than the binding energy of the decay products given in the column labeled BE+α in the line below it.

In contrast the nuclide Zr108 (p=40, n=68), which does not decay by alpha particle emission, has a BE of 892.3 MeV but its decay products would have a BE of only 854.1+28.3=882.4 MeV. Thus there is no alpha decay.

Similary the nuclide Nd160 (p=60, n=100), which does not decay by alpha particle emission, has a BE of 1291.6 MeV but its decay products would have a BE of 1259.2+28.3=1287.5 MeV. Thus again there is no alpha decay.

Here is the data for whole sequence of alpha transitions.

Binding Energies for a Sequence of Possible Alpha Transitions
p n BE
(MeV) 		BE+α
(MeV) 		ΔBE
for α tran
(MeV)		 α tran?
88 116 1571.67 1599.965674
86 114 1551.01 1579.305674 7.635674 Yes
84 112 1529.76 1558.055674 7.045674 Yes
82 110 1508.12 1536.415674 6.655674 Yes
80 108 1485.04 1513.335674 5.215674 Yes
78 106 1461.46 1489.755674 4.715674 Yes
76 104 1437.76 1466.055674 4.595674 Yes
74 102 1413.34 1441.635674 3.875674 Yes
72 100 1388.33 1416.625674 3.285674 Yes
70 98 1362.794 1391.089674 2.759674 Yes
68 96 1336.45 1364.745674 1.951674 Yes
66 94 1309.458 1337.753674 1.303674 Yes
64 92 1281.601 1309.896674 0.438674 Uncertain
62 90 1253.108 1281.403674 -0.197326 Uncertain
60 88 1225.032 1253.327674 0.219674 Uncertain
58 86 1197.335 1225.630674 0.598674 Uncertain
56 84 1169.449 1197.744674 0.409674 Uncertain
54 82 1141.877 1170.172674 0.723674 Uncertain
52 80 1109.942 1138.237674 -3.639326 No
50 78 1077.35 1105.645674 -4.296326 No
48 76 1040 1068.295674 -9.054326 No
46 74 1003.3 1031.595674 -8.404326 No
44 72 966.9 995.195674 -8.104326 No
42 70 930 958.295674 -8.604326 No
40 68 892.3 920.595674 -9.404326 No
38 66 854.1 882.395674 -9.904326 No

For nuclides on the borderline between alpha particle ejection and no alpha particle ejection the result of the analysis is ambiguous. The nuclide Hg200 (p=80, n=120) has a BE of 1581.2 MeV whereas that of its supposed products of decay would be 1581.9 MeV. The result indicates there would be essentially no impulse for alpha decay.

Binding Energies for a Sequence of Possible Alpha Transitions
p n BE
(MeV) 		BE+α
(MeV) 		ΔBE
for α tran
(MeV) 		α tran?
80 120 1581.197 Probably
not
78 118 1553.619 1581.914674 0.717674 Probably
not
76 116 1526.117 1554.412674 0.793674 Probably
not
74 114 1498.184 1526.479674 0.362674 Probably
not
72 112 1470.29 1498.585674 0.401674 Probably
not
70 110 1442.5 1470.795674 0.505674

As suggested above, any conflict with theory could arise from using binding energy based only upon the mass deficit rather than total binding energy.

Conclusion

A nuclide decays by alpha particle ejection when the binding energy of its decay products is greater than its binding energy. In other words, nuclides transition to increase binding energy.

(To be continued.)

HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins