The Law of Radioactive Decay
A radioactive nucleus is one that will eventually disintegrate and
release some energy. But when?
One might imagine that a radioactive nucleus would begin to “age” from
the moment of its birth, and that after the passage of a predetermined
time, the disintegration process would take place. This is how
radioactivity _might_ work in a deterministic universe. What actually
happens to a radioactive nucleus, however, is much more interesting.
At any instant of its life, the radioactive nucleus has some probability
of disintegrating in the next second. This probability is unaffected by
its age. No matter how long the nucleus has lived, its chance of
disintegrating in the next second is always the same. It is as if a game
of roulette were being played. The wheel spins, and if its number comes
up, the nucleus disintegrates in the first second. If not, the wheel
spins again. Each time the wheel spins there is some probability of its
number coming up. The precise value of this probability is a
characteristic of each particular radioactive species. The higher the
probability, the more rapidly the nucleus may be expected to
disintegrate. But a given nucleus need not do at any particular time
what is expected of it.
The notion of probability (or chance) has meaning only when applied to a
large number of cases. To say that a given nucleus has one chance in a
hundred of decaying in the next second means that out of some large
number (say 100 million) of such radioactive nuclei, one per cent (one
million) will decay in the next second. But it is absolutely impossible
to say beforehand which nuclei will be the ones to decay. A particular
nucleus may decay immediately or only after some very long time. The
collection as a whole, however, will always do the expected thing. (This
is the principle on which insurance companies operate.)
The situation is best described in terms of a time span which is called
the half-life of the radioactive species. The half-life is defined as
the amount of time which is required for one half of a large number of
identical radioactive nuclei to disintegrate. It makes no difference
what this large number is, provided only that it is large enough.
If the number is not large enough, fluctuations will occur, and instead
of 50 per cent of the nuclei decaying during the period of a half-life,
it may be 40 per cent or 60 per cent. As a matter of fact the 40 per
cent to 60 per cent limits correspond to a sample size of about 100
nuclei. For 10,000 nuclei, the limits will be 49 per cent to 51 per
cent. The number of radioactive nuclei with which we customarily deal,
is about 10²³ (100,000,000,000,000,000,000,000). This is the number, for
example, of radioactive nuclei in about an ounce of radium. For such a
large number of nuclei the deviation from 50-per-cent decay during a
half-life will be utterly negligible. Thus we live in a universe which,
on a macroscopic scale, appears ordered and subject to exact laws; while
underlying these laws, on a microscopic scale, nature plays out a game
of chance, full of randomness and uncertainty in the individual case.
We may draw a graph showing how _N_, the number of the remaining
radioactive nuclei, varies with the time _t_. The graph shows that: in
the first half-life _T_, half of the original number _N_₀ of radioactive
nuclei decay. In the second half-life, half of those remaining decay,
and so on. After the time _T_, one half of the original radioactive
nuclei still remain; after 2_T_, one quarter remain; and so forth.
[Illustration: uncaptioned]
Different radioactive species have different half-lives. Many are only a
small fraction of a second; some are billions of years. N¹⁶ decays to
O¹⁶ (plus an electron and a neutrino) with a half-life of about eight
seconds. A free neutron decays into a proton, an electron, and a
neutrino with a half-life of 13 minutes. Strontium with weight 90 (Sr⁹⁰)
undergoes a beta decay with a half-life of 28 years. (This is an isotope
that is not found anywhere in nature, but is made in fairly large
quantities in the fission process.) Potassium with weight 40 (K⁴⁰),
which is present in the amount of 0.01 per cent in ordinary potassium,
has a half-life of one billion years. It has presumably been left over
from the time when the primordial elements were formed. Half-lives for
gamma decay are extremely short by comparison to those for beta decay.
They usually amount to a small fraction of a second.
Radioactivity is characterized by the kind of particle emitted from the
nucleus (our examples, so far, have been of beta and gamma particles),
by the energy possessed by this particle, and by the half-life in which
the radioactive decay takes place.
The biological hazard from radioactivity depends on all three of these
characteristics. No matter whether the radioactive nuclei are produced
in an atomic explosion or in an atomic reactor, some time will in
general elapse before a human population can become exposed. If this
time is long compared to the half-life of the radioactive species, most
of the nuclei will have disintegrated, and the hazard will thereby be
reduced. If, on the other hand, the half-life is long compared to this
time, as well as to the life-span of a human being, the rate at which
disintegrations occur will be low, and again the hazard will be reduced.
In short the dangerous half-lives are the intermediate ones, not too
long, not too short. Sr⁹⁰ is an example.