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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 ^{208}Pb) have
an extremely long (such as 10^{150}years) 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 ^{56}Fe has the smallest energy per nucleon and because
conservation laws do not forbid conversion into ^{56}Fe, all substances
but ^{56}Fe 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 ^{63}Cu
→ 9 ^{56}Fe.

**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.

^{209}Bi most likely decays by reaction

^{209}Bi →
^{205}Tl + ^{4}He^{2+} + 2e^{-}

For this reaction, the amount of energy[4]
released, ∆E = ∆m*C^{2}
= (208.98037 - 204.97440 - 4.00260) a.u.*C^{2}
= 0.00337 a.u.*C^{2} = 5.03*10^{-13}J
= 3.14Mev

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

Note: Since the α particle (^{4}He^{2+})
is ejected with a large (1.22*10^{7}m/s) speed, it cannot pick up the
extra electrons from ^{205}Tl.

*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 ^{209}Bi, E^{-0.5}=0.56424, so log R for ^{209}Bi=9.141
and R=8.63*10^{-27}s^{-1}. Thus, average life of atom, T = 1/R
= 1.16*10^{26}s. Thus, the half-life of ^{209}Bi is ≈
2.54 * 10^{18}years.

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

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

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.