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Many types of heavy atomic nucleus spontaneously decay to produce daughter nucleii
via the emission of particles (i.e., helium nucleii) of some characteristic energy.
This process is know as
decay. Let us investigate the decay of a particular type of atomic nucleus of radius , chargenumber ,
and massnumber . Such a nucleus thus decays to produce a daughter
nucleus of chargenumber and massnumber ,
and an particle of chargenumber and massnumber
. Let the characteristic energy of the particle
be . Incidentally, nuclear radii
are found to satisfy the empirical formula

(353) 
for .
In 1928, George Gamov proposed a very successful theory of decay,
according to which the particle moves freely inside the nucleus, and is emitted after tunneling through the
potential barrier between itself and the daughter nucleus. In other words,
the particle, whose energy is , is trapped in a potential well of radius by the
potential barrier

(354) 
for .
Making use of the WKB approximation (and neglecting the fact
that is a radial, rather than a Cartesian, coordinate), the probability
of the particle tunneling through the barrier is

(355) 
where and
. Here,
is the particle mass. The above expression
reduces to

(356) 
where

(357) 
is a dimensionless constant, and

(358) 
is the characteristic energy the particle would need in order to escape
from the nucleus without tunneling. Of course, .
It is easily demonstrated that

(359) 
when .
Hence.

(360) 
Now, the particle moves inside the nucleus with the characteristic
velocity
. It follows that the particle bounces backward
and forward within the nucleus at the frequency
, giving

(361) 
for a 1 MeV particle trapped inside a typical heavy nucleus of radius m.
Thus, the particle effectively attempts to tunnel through the potential
barrier times a second. If each of these attempts has a probability
of succeeding, then the probability of decay per unit time
is . Hence, if there are undecayed nuclii at time then
there are only at time , where

(362) 
This expression can be integrated to give

(363) 
Now, the halflife, , is defined as the time which must elapse
in order for half of the nuclii originally present to decay. It follows from
the above formula that

(364) 
Note that the halflife is independent of .
Finally, making use of the above results, we obtain

(365) 
where
Figure 15:
The experimentally determined halflife, , of various atomic nucleii which decay via emission versus the bestfit theoretical halflife
. Both halflives are measured in years. Here, , where is the charge number of the nucleus, and the characteristic energy of the emitted particle in MeV. In
order of increasing halflife, the points correspond to the
following nucleii: Rn 215, Po 214, Po 216, Po 197, Fm 250, Ac 225, U 230, U 232, U 234, Gd 150, U 236, U 238, Pt 190, Gd 152, Nd 144. Data obtained from IAEA Nuclear Data Centre.

The halflife, , the daughter chargenumber, , and
the particle energy, , for atomic nucleii which undergo decay
are indeed found to satisfy a relationship of the form (365). The
best fit to the data (see Fig. 15) is obtained using
Note that these values are remarkably similar to those calculated above.
Next: Square Potential Well
Up: OneDimensional Potentials
Previous: Cold Emission
Richard Fitzpatrick
20100720