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 $$R = 1.5\times 10^{-15}\,A^{1/3}\,{\rm m}=2.0\times 10^{-15}\,Z_1^{\,1/3}\,{\rm m}$$ 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 $$V(r) = \frac{Z_1\,Z_2\,e^{\,2}}{4\pi\,\epsilon_0\,r}$$ 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 $$|T|^{\,2} = \exp\left(-\frac{2\sqrt{2\,m}}{\hbar}\int_{r_1}^{r_2} \sqrt{V(r)-E}\,dr\right),$$ 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 $$|T|^{\,2} = \exp\left(-2\sqrt{2}\,\beta \int_{1}^{E_c/E}\left[\frac{1}{y}-\frac{E}{E_c}\right]^{1/2} dy\right),$$ where $$\beta = \left(\frac{Z_1\,Z_2\,e^{\,2}\,m\,R}{4\pi\,\epsilon_0\,\hbar^{\,2}}\right)^{1/2} = 0.74\,Z_1^{\,2/3}$$ is a dimensionless constant, and $$E_c = \frac{Z_1\,Z_2\,e^{\,2}}{4\pi\,\epsilon_0\,R} = 1.44\,Z_1^{\,2/3}\,\,{\rm MeV}$$ 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 $$\int_1^{1/\epsilon}\left(\frac{1}{y} - \epsilon\right)^{1/2} dy \simeq \frac{\pi}{2\sqrt{\epsilon}}-2$$ when $\epsilon\ll 1$. Hence. $$|T|^{\,2} \simeq \exp\left(-2\sqrt{2}\,\beta\left[\frac{\pi}{2}\sqrt{\frac{E_c}{E}}-2\right]\right).$$

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 $$\nu\simeq 2\times 10^{28}\,\,{\rm yr}^{-1}$$ 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 $$dN = - N\,\nu\,|T|^{\,2}\,dt.$$ This expression can be integrated to give $$N(t) = N(0)\,\exp(-\nu\,|T|^{\,2}\,t).$$ 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 $$\tau = \frac{\ln 2}{\nu\,|T|^{\,2}}.$$ Note that the half-life is independent of $N(0)$.

Finally, making use of the previous results, we obtain $$\label{e5.64} \log_{10}[\tau ({\rm yr})] = -C_1 - C_2\,Z_1^{\,2/3} + C_3\,\frac{Z_1}{\sqrt{E({\rm MeV})}},$$ where $$\begin{aligned} C_1 &= 28.5,\\[0.5ex] C_2 &= 1.83,\\[0.5ex] C_3 &= 1.73.\end{aligned}$$

Figure 15: The experimentally determined half-life, $\begin{equation}\tau_{\mathrm{e} x}\end{equation}$ of various atomic nucleii which decay via  emission versus the best-fit theoretical half-life $\begin{equation}\log _{10}\left(\tau_{t h}\right)=-28.9-1.60 Z_{1}^{2 / 3}+1.61 Z_{1} / \sqrt{E}\end{equation}$. Both half-lives are measured in years. Here, $\begin{equation}Z_{1}=Z-2\end{equation}$. Both half-lives are measured in years. Here, $\begin{equation}Z_{1}=Z-2, \text { where } Z\end{equation}$ 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 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.

Equation ([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 ([e5.64]). The best fit to the data (see Figure [fal]) is obtained using $$\begin{aligned} C_1 &= 28.9,\\[0.5ex] C_2 &= 1.60,\\[0.5ex] C_3 &= 1.61.\end{aligned}$$ Note that these values are remarkably similar to those calculated previously.

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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