Alpha Decay

Many types of heavy atomic nuclei spontaneously decay to produce daughter nuclei via the emission of $\alpha$-particles (i.e., helium nuclei) of some characteristic energy. This process is known 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

$$R = 1.5\times 10^{-15}\,A^{1/3}\,{\rm m}=2.0\times 10^{-15}\,Z_1^{\,1/3}\,{\rm m}$$ (4.146)

for $Z\gg 1$.

In 1928, George Gamov 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

$$U(r) = \frac{Z_1\,Z_2\,e^{2}}{4\pi\,\epsilon_0\,r}$$ (4.147)

for $r>R$. (See Section 2.1.4.) Here, $e$ is the magnitude of the electron charge.

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

$$\vert T\vert^{2} = \exp\left(-\frac{2\!\sqrt{2\,m}}{\hbar}\int_{r_1}^{r_2} \!\sqrt{U(r)-E}\,dr\right),$$ (4.148)

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, and $m_p$ is the proton mass. The previous expression reduces to

$$\vert T\vert^{2} = \exp\left[-2\!\sqrt{2}\,\beta \int_{1}^{E_c/E}\left(\frac{1}{y}-\frac{E}{E_c}\right)^{1/2} dy\right],$$ (4.149)

where

$$\beta = \left(\frac{Z_1\,Z_2\,e^{2}\,m\,R}{4\pi\,\epsilon_0\,\hbar^{2}}\right)^{1/2} = 0.74\,Z_1^{\,2/3}$$ (4.150)

is a dimensionless constant, and

$$E_c = \frac{Z_1\,Z_2\,e^{2}}{4\pi\,\epsilon_0\,R} = 1.44\,Z_1^{\,2/3}\,\,{\rm MeV}$$ (4.151)

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

$$\int_1^{1/\epsilon}\left(\frac{1}{y} - \epsilon\right)^{1/2} dy \simeq \frac{\pi}{2\!\sqrt{\epsilon}}-2$$ (4.152)

when $\epsilon\ll 1$. Hence.

$$\vert T\vert^{2} \simeq \exp\left[-2\!\sqrt{2}\,\beta\left(\frac{\pi}{2}\!\sqrt{\frac{E_c}{E}}-2\right)\right].$$ (4.153)

Now, the $\alpha$-particle moves inside the nucleus at 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

$$\nu\simeq 2\times 10^{28}\,\,{\rm yr}^{-1}$$ (4.154)

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 $\vert T\vert^{2}$ of succeeding then the probability of decay per unit time is $\nu\,\vert T\vert^{2}$. Hence, if there are $N(t)\gg 1$ intact nuclei at time $t$ then there are only $N+dN$ at time $t+dt$, where

$$dN = - N\,\nu\,\vert T\vert^{2}\,dt.$$ (4.155)

This expression can be integrated to give

$$N(t) = N(0)\,\exp\left(-\nu\,\vert T\vert^{2}\,t\right).$$ (4.156)

The half-life, $\tau$, 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

$$\tau = \frac{\ln 2}{\nu\,\vert T\vert^{2}}.$$ (4.157)

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

Finally, making use of the previous results, we obtain

$$\log_{10}[\tau ({\rm yr})] = -C_1 - C_2\,Z_1^{\,2/3} + C_3\,\frac{Z_1}{\sqrt{E({\rm MeV})}},$$ (4.158)

where

$$C_1 = 28.5,$$ (4.159)

$$C_2 = 1.83,$$ (4.160)

$$C_3 = 1.73.$$ (4.161)

Figure: 4.12 The experimentally determined half-life, $\tau _{{\rm ex}}$, of various atomic nuclei that decay via $\alpha$-emission versus the best-fit theoretical half-life $\log_{10}(\tau_{{\rm th}}) = -28.9 - 1.60\,Z_1^{\,2/3} + 1.61\,Z_1/\sqrt{E}$. Both half-lives are measured in years. Here, $Z_1=Z-2$, 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 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, $\tau$, the daughter charge-number, $Z_1=Z-2$, and the $\alpha$-particle energy, $E$, for atomic nuclei that undergo $\alpha$-decay are indeed found to satisfy a relationship of the form (4.158). See Figure 4.12. The best fit to the data shown in the figure is obtained using

$$C_1 = 28.9,$$ (4.162)

$$C_2 = 1.60,$$ (4.163)

$$C_3 = 1.61.$$ (4.164)

It can be seen that these values are remarkably similar to those calculated previously.