Alpha Decay
Many types of heavy atomic nuclei spontaneously decay to produce daughter nuclei
via the emission of
-particles (i.e., helium nuclei) of some characteristic energy.
This process is known as
-decay. Let us investigate the
-decay of a particular type of atomic nucleus of radius
, charge-number
,
and mass-number
. Such a nucleus thus decays to produce a daughter
nucleus of charge-number
and mass-number
,
and an
-particle of charge-number
and mass-number
. Let the characteristic energy of the
-particle
be
. Incidentally, nuclear radii
are found to satisfy the empirical formula
![$\displaystyle R = 1.5\times 10^{-15}\,A^{1/3}\,{\rm m}=2.0\times 10^{-15}\,Z_1^{\,1/3}\,{\rm m}$](img4094.png) |
(11.132) |
for
(Park 1974).
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
![$\displaystyle U(r) = \frac{Z_1\,Z_2\,e^{\,2}}{4\pi\,\epsilon_0\,r}$](img4096.png) |
(11.133) |
for
. Here,
is the magnitude of the electron charge.
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
![$\displaystyle \vert T\vert^{\,2} = \exp\left(-\frac{2\sqrt{2\,m}}{\hbar}\int_{r_1}^{r_2}
\sqrt{U(r)-E}\,dr\right),$](img4098.png) |
(11.134) |
where
and
. Here,
is the
-particle mass, and
is the proton mass. The previous expression
reduces to
![$\displaystyle \vert T\vert^{\,2} = \exp\left[-2\sqrt{2}\,\beta \int_{1}^{E_c/E}\left(\frac{1}{y}-\frac{E}{E_c}\right)^{1/2} dy\right],$](img4103.png) |
(11.135) |
where
![$\displaystyle \beta = \left(\frac{Z_1\,Z_2\,e^{\,2}\,m\,R}{4\pi\,\epsilon_0\,\hbar^{\,2}}\right)^{1/2} = 0.74\,Z_1^{\,2/3}$](img4104.png) |
(11.136) |
is a dimensionless constant, and
![$\displaystyle E_c = \frac{Z_1\,Z_2\,e^{\,2}}{4\pi\,\epsilon_0\,R} = 1.44\,Z_1^{\,2/3}\,\,{\rm MeV}$](img4105.png) |
(11.137) |
is the characteristic energy the
-particle would need in order to escape
from the nucleus without tunneling. Of course,
.
It is easily demonstrated that
![$\displaystyle \int_1^{1/\epsilon}\left(\frac{1}{y} - \epsilon\right)^{1/2} dy \simeq
\frac{\pi}{2\sqrt{\epsilon}}-2$](img4107.png) |
(11.138) |
when
.
Hence.
![$\displaystyle \vert T\vert^{\,2} \simeq \exp\left[-2\sqrt{2}\,\beta\left(\frac{\pi}{2}\sqrt{\frac{E_c}{E}}-2\right)\right].$](img4109.png) |
(11.139) |
Now, the
-particle moves inside the nucleus at the characteristic
velocity
. It follows that the particle bounces backward
and forward within the nucleus at the frequency
, giving
![$\displaystyle \nu\simeq 2\times 10^{28}\,\,{\rm yr}^{-1}$](img4112.png) |
(11.140) |
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
intact nuclei at time
then
there are only
at time
, where
![$\displaystyle dN = - N\,\nu\,\vert T\vert^{\,2}\,dt.$](img4118.png) |
(11.141) |
This expression can be integrated to give
![$\displaystyle N(t) = N(0)\,\exp\left(-\nu\,\vert T\vert^{\,2}\,t\right).$](img4119.png) |
(11.142) |
The half-life,
, is defined as the time which must elapse
in order for half of the nuclei originally present to decay. It follows from
the previous formula that
![$\displaystyle \tau = \frac{\ln 2}{\nu\,\vert T\vert^{\,2}}.$](img4120.png) |
(11.143) |
Note that the half-life is independent of
.
Finally, making use of the previous results, we obtain
![$\displaystyle \log_{10}[\tau ({\rm yr})] = -C_1 - C_2\,Z_1^{\,2/3} + C_3\,\frac{Z_1}{\sqrt{E({\rm MeV})}},$](img4122.png) |
(11.144) |
where
![$\displaystyle C_1$](img4123.png) |
![$\displaystyle = 28.5,$](img4124.png) |
(11.145) |
![$\displaystyle C_2$](img4125.png) |
![$\displaystyle = 1.83,$](img4126.png) |
(11.146) |
![$\displaystyle C_3$](img4127.png) |
![$\displaystyle = 1.73.$](img4128.png) |
(11.147) |
Figure 11.13:
The experimentally determined half-life,
, of various atomic nuclei that decay via
-emission versus the best-fit theoretical half-life
. Both half-lives 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 half-life, the points correspond to the
following nuclei: 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 International Atomic Energy Agency, Nuclear Data Center.)
|
The half-life,
, the daughter charge-number,
, and
the
-particle energy,
, for atomic nuclei that undergo
-decay
are indeed found to satisfy a relationship of the form (11.144). The
best fit to the data (see Figure 11.13) is obtained using
![$\displaystyle C_1$](img4123.png) |
![$\displaystyle = 28.9,$](img4131.png) |
(11.148) |
![$\displaystyle C_2$](img4125.png) |
![$\displaystyle = 1.60,$](img4132.png) |
(11.149) |
![$\displaystyle C_3$](img4127.png) |
![$\displaystyle = 1.61.$](img4133.png) |
(11.150) |
It can be seen that these values are remarkably similar to those calculated previously.