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Dmytro Taranovsky
November 27, 1999
Modified:  September 30, 2001


According to quantum mechanics, any reaction can occur if no conservation law forbids it to occur.  Therefore, any storage of free energy can by itself lose free energy and decay.  Half-life is the amount of time that for a given single free energy storage, the probability that the storage will decay is one half.  Many energy storages (such as 208Pb) have an extremely long (such as 10150years) half-life. Such storages are called essentially[1] stable.

Any nuclear reaction will occur if no conservation laws are broken and energy is released[2].  All isotopes with atomic mass larger than 89a.u. can decay so that energy is released; therefore, they are radioactive.  Moreover, there are no two stable isotopes with the same atomic mass number since the heavier [3] isotope can decay in the lighter one.

Because 56Fe has the smallest energy per nucleon and because conservation laws do not forbid conversion into 56Fe, all substances but 56Fe store free energy and thus are essentially stable or unstable.  Many essentially stable substances (such as copper) consist of stable atoms:  Radioactivity is decay of an atom on its own; decay of copper requires more than one atom for reactions such as  8 63Cu → 9 56Fe.

Conclusion/Summary:  A radioactive atom is a storage of free energy.  Moreover, all atoms that are free energy storage can decay and thus are radioactive.  However, the half-life of many such isotopes is so large that they pose no health or other hazard, and their radioactivity had not been detected experimentally.  Therefore, such isotopes are called essentially stable.  In essentially all applications, essentially stable isotopes can be treated as stable.  Scientists should define the exact border between essentially stable and other radioactive isotopes.

Radioactivity of 209Bi

209Bi most likely decays by reaction

209Bi →  205Tl  +  4He2+ + 2e-

For this reaction, the amount of energy[4] released, ∆E = ∆m*C2 = (208.98037 - 204.97440 - 4.00260) a.u.*C2 = 0.00337 a.u.*C2 = 5.03*10-13J = 3.14Mev

Because energy is released, the quantum theory predicts that 209Bi will decay (in this way, or, less likely, in another way).  Therefore, according to the definition of radioactivity, 209Bi is radioactive.

Note:  Since the α particle (4He2+) is ejected with a large (1.22*107m/s) speed, it cannot pick up the extra electrons from 205Tl.


Appendix A:  Estimation of Half-life of 209Bi

Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles by Robert Eisberg and Robert Resnick has, on page 225, a graph of dependence between E-0.5 [in MeV] and log R [in s-1]. The graph is linear.  (E is energy of the alpha particle; the energy is computed from total release of energy and conservation of momentum; R is rate of decay.)

The following are two points on the graph:
(E-0.5 = 0.4, log R = -2.693) and (E-0.5 = 0.5, log R = -16.923).
Therefore, slope of the line is -142.3 log s-1/MeV-0.5.

For 209Bi, E-0.5=0.56424, so log R for 209Bi=9.141 and R=8.63*10-27s-1. Thus, average life of atom, T = 1/R = 1.16*1026s.  Thus, the half-life of 209Bi is ≈ 2.54 * 1018years. 

Using this method, we can find half-life of other isotopes that undergo α decay. 

Note: The half-life of 209Bi is only estimated.  It is possible that the half-life is really several times larger or smaller than stated.

Appendix B:  α Decay is an Example of Barrier Penetration

Legend: x axis is distance from the center of the nucleus, starting at zero
Black line (curved) shows potential energy of α particle.
Blue line indicates total energy of alpha particle.

The sharp increase in potential energy is due to strong nuclear force.  The gradual decrease of potential energy is due to electromagnetic repulsion between nucleus and α particle.  To leave the nucleus, the α particle has to penetrate a region where its total energy is less than potential energy.  Using this model of α decay and quantum theory, the graph between E-0.5 and log R was obtained.

[1] "Virtually", "meta", "apparently", and "practically" are alternative names, but "essentially" best describes the situation.

[2] This is a direct consequence of the first sentence of this paper.

[3] There are no different isotopes (whose half-life is at least an hour) with almost identical mass.

[4] The values for atomic masses were taken from Physics for Scientists and Engineers (Fourth Edition) by Raymond A. Serway on pp. A.4 and A.12.