4.5: Alpha Decay

Many types of heavy atomic nucleus spontaneously decay to produce daughter nucleii via the emission of \(\alpha\)-particles (i.e., helium nucleii) of some characteristic energy. This process is know as \(\alpha\)-decay. Let us investigate the \(\alpha\)-decay of a particular type of atomic nucleus of radius \(R\), charge-number \(Z\), and mass-number \(A\). Such a nucleus thus decays to produce a daughter nucleus of charge-number \(Z_1=Z-2\) and mass-number \(A_1=A-4\), and an \(\alpha\)-particle of charge-number \(Z_2=2\) and mass-number \(A_2=4\). Let the characteristic energy of the \(\alpha\)-particle be \(E\). Incidentally, nuclear radii are found to satisfy the empirical formula

\[R = 1.5\times 10^{-15}\,A^{1/3}\,{\rm m}=2.0\times 10^{-15}\,Z_1^{\,1/3}\,{\rm m} \nonumber \]

for \(Z\gg 1\).

In 1928, George Gamow proposed a very successful theory of \(\alpha\)-decay, according to which the \(\alpha\)-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 \(\alpha\)-particle, whose energy is \(E\), is trapped in a potential well of radius \(R\) by the potential barrier

\[V(r) = \frac{Z_1\,Z_2\,e^2}{4\pi\,\epsilon_0\,r} \nonumber \]

for \(r>R\).

Making use of the WKB approximation (and neglecting the fact that \(r\) is a radial, rather than a Cartesian, coordinate), the probability of the \(\alpha\)-particle tunneling through the barrier is

\[|T|^2 = \exp\left(-\frac{2\sqrt{2\,m}}{\hbar}\int_{r_1}^{r_2} \sqrt{V(r)-E}\,dr\right), \nonumber \]

where \(r_1=R\) and \(r_2 = Z_1\,Z_2\,e^2/(4\pi\,\epsilon_0\,E)\). Here, \(m=4\,m_p\) is the \(\alpha\)-particle mass. The previous expression reduces to

\[|T|^2 = \exp\left(-2\sqrt{2}\,\beta \int_{1}^{E_c/E}\left[\frac{1}{y}-\frac{E}{E_c}\right]^{1/2} dy\right), \nonumber \]

where

\[\beta = \left(\frac{Z_1\,Z_2\,e^2\,m\,R}{4\pi\,\epsilon_0\,\hbar^2}\right)^{1/2} = 0.74\,Z_1^{\,2/3} \nonumber \]

is a dimensionless constant, and

\[E_c = \frac{Z_1\,Z_2\,e^2}{4\pi\,\epsilon_0\,R} = 1.44\,Z_1^{\,2/3}\,\,{\rm MeV} \nonumber \]

is the characteristic energy the \(\alpha\)-particle would need in order to escape from the nucleus without tunneling. Of course, \(E\ll E_c\). It is easily demonstrated that

\[\int_1^{1/\epsilon}\left(\frac{1}{y} - \epsilon\right)^{1/2} dy \simeq \frac{\pi}{2\sqrt{\epsilon}}-2 \nonumber \]

when \(\epsilon\ll 1\). Hence.

\[|T|^2 \simeq \exp\left(-2\sqrt{2}\,\beta\left[\frac{\pi}{2}\sqrt{\frac{E_c}{E}}-2\right]\right). \nonumber \]

Now, the \(\alpha\)-particle moves inside the nucleus with the characteristic velocity \(v= \sqrt{2\,E/m}\). It follows that the particle bounces backward and forward within the nucleus at the frequency \(\nu\simeq v/R\), giving

\[\nu\simeq 2\times 10^{28}\,\,{\rm yr}^{-1} \nonumber \]

for a 1 MeV \(\alpha\)-particle trapped inside a typical heavy nucleus of radius \(10^{-14}\) m. Thus, the \(\alpha\)-particle effectively attempts to tunnel through the potential barrier \(\nu\) times a second. If each of these attempts has a probability \(|T|^2\) of succeeding then the probability of decay per unit time is \(\nu\,|T|^2\). Hence, if there are \(N(t)\gg 1\) undecayed nuclii at time \(t\) then there are only \(N+dN\) at time \(t+dt\), where

\[dN = - N\,\nu\,|T|^2\,dt. \nonumber \]

This expression can be integrated to give

\[N(t) = N(0)\,\exp(-\nu\,|T|^2\,t). \nonumber \]

Now, the half-life, \(\tau\), is defined as the time which must elapse in order for half of the nuclii originally present to decay. It follows from the previous formula that

\[\tau = \frac{\ln 2}{\nu\,|T|^2}. \nonumber \]

Note that the half-life is independent of \(N(0)\).

Figure 15: The experimentally determined half-life, \(\tau_{\mathrm{e} x}\) of various atomic nucleii which decay via\(\alpha\)emission versus the best-fit theoretical half-life \(\log _{10}\left(\tau_{t h}\right)=-28.9-1.60 Z_{1}^{2 / 3}+1.61 Z_{1} / \sqrt{E}\). Both half-lives are measured in years. Here, \(Z_{1}=Z-2\). Both half-lives are measured in years. Here, \(Z_{1}=Z-2, \text { where } Z\) is the charge number of the nucleus, and\(E\) the characteristic energy of the emitted |(\alpha\)-particle in MeV. In order of increasing half-life, 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.

Finally, making use of the previous results, we obtain

\[ \log_{10}[\tau ({\rm yr})] = -C_1 - C_2\,Z_1^{\,2/3} + C_3\,\frac{Z_1}{\sqrt{E({\rm MeV})}}, \label{e5.64} \]

where

\[\begin{align} C_1 &= 28.5,\\[4pt] C_2 &= 1.83,\\[4pt] C_3 &= 1.73.\end{align} \nonumber \]

Equation \ref{e5.64} is known as the Geiger-Nuttall formula, because it was discovered empirically by H. Geiger and J.M. Nuttall in 1911 .

The half-life, \(\tau\), the daughter charge-number, \(Z_1=Z-2\), and the \(\alpha\)-particle energy, \(E\), for atomic nucleii which undergo \(\alpha\)-decay are indeed found to satisfy a relationship of the form \ref{e5.64}. The best fit to the data (see Figure [fal]) is obtained using

\[\begin{align} C_1 &= 28.9,\\[4pt] C_2 &= 1.60,\\[4pt] C_3 &= 1.61.\end{align} \nonumber \]

Note that these values are remarkably similar to those calculated previously.